Homological dimension theory of small categories - Springer Link

Journal of Mathematical Sciences, Vol. 110, No. 1, 2002

HOMOLOGICAL DIMENSION THEORY OF SMALL CATEGORIES A. A. Husainov

UDC 512.664.1

The homological dimension theory of small categories was founded by Mitchell in 1968–1982. It is concerned with the vanishing conditions for the derived functors of the limit. It studies various dimensions of small categories and the global dimension of the functor category. The aim of this paper is to describe problems and results in the homological dimension theory of small categories. Throughout the paper, N will denote the ordered set of nonnegative integers, Z the ring of integers, Ab the category of Abelian groups and homomorphisms, Ens the category of sets, and L : Ens → Ab the functor that assigns to each set E the free Abelian group generated by E and to each map f the homomorphism extending f . For any subset of N, the sup and the inf will be considered in {−1}∪N∪{∞}. In particular, sup ∅ = −1. Let A be an Abelian category. The Yoneda definition of Extn (A, B) in terms of exact sequences of length n from B to A [186] allows one to define the projective dimension p.d. A of an object A in A as the sup of n for which the functor Extn (A, −) : A → Ab is not zero. If A = 0, then there is no n ≥ 0 for which Extn (A, −) = 0. Hence p.d. 0 = −1. The global dimension gl.dim A of an Abelian category A is the sup of the projective dimensions of A ∈ ObA. The global-dimension theory of a functor category arose from ancient achievements of homological algebra such as Hilbert’s syzygy theorem and Maschke’s theorem. Mitchell indicated that most of the homological theory of modules generalizes to functors on preadditive categories. Let A be an arbitrary category and A C the category of functors C → A. For each A ∈ ObA, we consider a functor ∆ C A : C → A with constant values A on objects and 1A : A → A on morphisms. Establishing (∆ C f )c = f for each morphism f : A → B of C, for all c ∈ Ob C, we obtain a diagonal functor ∆ C : A → A C . By the limit functor lim C : A C → A, we mean the right adjoint to the diagonal functor. The left adjoint to ∆ C is called the colimit functor colim C : A C → A. Let R be a ring with 1, C be a small category, and M odR denote the category of left R-modules. The R-cohomological dimension c.d.R C is the projective dimension of ∆ C R ∈ M odRC . We denote c.d.Z C by c.d. C and call it the cohomological dimension. Mitchell obtained complete results about c.d.R C, where C is a category opposite to a directed set [127]. Laudal [110] characterized all small categories of cohomological dimension zero. Let K be a nonzero commutative ring with unit, and let C be a small category. We denote the category opposite to C by Cop . Let K C(a, b) be the free K-module on the set C(a, b) of morphisms a → b in C. The K-module homomorphisms K C(f, g) : K C(a1 , b1 ) → K C(a2 , b2 ) acting on α ∈ C(a1 , b1 ) for f ∈ C(a2 , a1 ), g ∈ C(b1 , b2 ) by α → g ◦ α ◦ f give a functor K C : Cop × C → M odK . Cop × C The Hochschild–Mitchell dimension dim K C is the projective dimension of K C ∈ M odK . When K = Z, we denote the Hochschild–Mitchell dimension by dim C. Mitchell proved that the upper bound of the global dimension of the functor category gl.dim A C ≤ dim C + gl.dim A holds if A is an Abelian category with exact coproducts. In [122], Mitchell conjectured that each cancellative monoid of cohomological dimension c.d. C ≤ 1 is partially free. In [132], Novikov gives counterexamples. It follows from [36] that every cancellative small category of Hochschild–Mitchell dimension dim C ≤ 1 is partially free. An independent proof can Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 78, Algebra-17, 2000. c 2002 Plenum Publishing Corporation 1072–3374/02/1101–2273 $ 27.00 

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be found in [91]. It is a generalization of a theorem of Stallings [177] and Swan [179] about groups of cohomological dimension one, since c.d. C = dim C when C is a groupoid. A category is skeletal if each of its isomorphisms are automorphisms. A delta is a small skeletal category each of whose endomorphisms are identities. Mitchell conjectured and Cheng confirmed that any delta of Hochschild–Mitchell dimension one is a free category [26]. The R-homological dimension of C is defined as h.d. R C = sup{n ∈ N : colimnC = 0}, where colimnC is the nth left derived functor of the colimit functor colim C : M odRC → M odR . We omit R when R = Z. If all components of C are filtered, then colim C is exact, and, therefore, for such categories, we have h.d. C = 0. A conjecture that stood for some time was that the converse is true. This problem was solved by Isbell and Mitchell [74]. Acknowledgment. I want to thank V. I. Kuz’minov for introducing me to the subject. I would like to thank H.-J. Baues, C. C. Cheng, C. U. Jensen, B. V. Novikov, and T. Pirashvili for their help in the preparation of this review. Part of the work was performed in the winter of 1999–2000, while I visited the Ondokuz Mayis University. Thanks are due to the organizer of the seminar, Ali Pancar, for the opportunity to lecture about these materials in several meetings. 1. Cohomological Dimension of a Small Category This section is devoted to the cohomology of small categories with coefficients in functors. We consider interpretations of the cohomology groups of categories. Then the categories with cohomological dimension zero are simply those for which all derivations are inner. A small category C has the cohomological dimension c.d. C ≤ 1 if and only if for every functor F : C → Ab, all fibrations of C by F are split. Categories with c.d. C ≤ n + 1 have corresponding properties. We study properties of the acyclity for projective systems on directed sets and applications to phantom maps. We discuss categories, monoids, and partially ordered sets of cohomological dimension zero and one. 1.1. Cohomology with coefficients in functors. Let C be a small category. Denote by Nn C the α α α set of all sequences of n composable morphisms c0 →1 c1 →2 · · · →n cn in C for n > 0, and the set of all objects c0 ∈ Ob C for n = 0. The nerve of C is a simplicial set N∗ C whose n-simplexes are the elements σ ∈ Nn C and the boundary operators are defined as follows. α α The face operators dni : Nn C → Nn−1 C, 0 < i < n, remove objects ci from the sequences c0 →1 c1 →2 αi+1 αi+1 ◦αi α α · · · →n cn and replace the morphisms ci−1 →i ci → ci+1 by their compositions ci−1 → ci+1 ; for i = 0 and for i = n, we set dn0 (c0 → · · · → cn ) = (c1 → · · · → cn ) and dnn (c0 → · · · → cn ) = (c0 → · · · → cn−1 ). The degenerate operators sni : Nn C → Nn+1 C insert the identity morphism ci → ci in c0 → · · · → cn for 0 ≤ i ≤ n. Cochain complex. Let F : C → A be a functor from a small category C
to an Abelian category A with (infinite) products. For any family {Ai }i∈I of objects, we denote by pri : i∈I Ai → Ai the projections of the product. The cochain complex C ∗ ( C, F ) consists of the products  F (cn ), n ≥ 0, C n ( C, F ) = and the morphisms δn =

n+1 i=0

c0 →···→cn

(−1)i δin : C n ( C, F ) → C n+1( C, F ), where δin are defined so that for each α

α

αn+1

σ = (c0 →1 c1 →2 · · · → cn+1 ) ∈ Nn+1 C they satisfy the relations prσ ◦ δin =



if 0 ≤ i ≤ n

prdn+1 σ i

αn+1

F (cn → cn+1 ) ◦ prdn+1 σ n+1

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if i = n + 1.

Let C n ( C, F ) = 0 for all n < 0. One verifies that δn+1 ◦ δn = 0 for all n ∈ Z. We define the n-th cohomology object of C with coefficients in F as H n ( C, F ) = Ker δn / Im δn−1 Note that if A = M odR , then C n ( C, F ) can be considered as the R-module of all functions  F (c) ϕ : Nn C −→ c∈Ob C α

α

α

with ϕ(c0 →1 c1 →2 · · · →n cn ) ∈ F (cn ). Then δn : C n ( C, F ) → C n+1( C, F ) acts by the formula α

αn+1

(δn ϕ)(c0 →1 · · · → cn+1 ) = αi

αi+1

αn+1

n  α (−1)i ϕ(c0 →1 · · · i=0 αn+1

α

α

→ cˆi → · · · → cn+1 ) + (−1)n+1 F (cn → cn+1 )(ϕ(c0 →1 · · · →n cn )). Right satellites of the limit. The limit of F ∈ M odRC can be defined as follows. For any α ∈ M or C, we denote its domain and codomain by dom α and cod α, respectively. By a thread, we mean any family {xc }c∈ C of elements xc ∈ F (c) such that for
each α ∈ M or C, the relation F (α)(xdom α ) = xcod α holds. Then the set of threads is a submodule of c∈Ob C F (c) that is isomorphic to lim C F . When the product functor in A is exact, the correspondence F → H n ( C, F ) makes an nth right satellite of the limit functor lim C : A C → A in the sense of Grothendieck [59]. In particular, H n ( C, −) are isomorphic to nth right derived functors of the limit functor if A has exact products and enough injectives. If there are right (left) satellites of lim C (colim C ), then we denote them by lim nC (colimnC ). Denote by Hn (N∗ C) the integral homologies of the nerve. It is well known that Hn (N∗ C) ∼ = colimnC ∆ C Z for all n ∈ N. Sometims, we write {F (c)}c∈ C instead of F : C → A. Thus, lim C F and colim C F will be denoted by lim C {F (c)} and colim C {F (c)}, respectively. Let S : C → D be a functor from a small category C into a category D. For d ∈ D, we denote by S/d and d/S the comma-categories S ↓ d and d ↓ S in the sense of [113]. Applying to the functors S op

F op

Cop → Dop → Aop Oberst’s result [140, Theorem 2.3], we obtain the following statement. Theorem 1.1 (Oberst). Let S : C → D be a functor between small categories. Then the following conditions are equivalent: (1) the categories S/d are connected and Hn (N∗ (S/d)) = 0 ∀n > 0; (2) for every Abelian category A with exact products and for every functor F : D → A, the canonical morphisms lim nD F → lim nC (F ◦ S) are isomorphisms for all n ∈ N; (3) the condition of (2) is true for A = Ab. 1.2. Cohomology of directed sets. Any partially ordered set (shortly poset) (I, ≤) will be considered as a small category with ObI = I in which for every a, b ∈ I, the set I(a, b) of morphisms a → b consists of one element if a ≤ b and I(a, b) = ∅ otherwise. A poset is called a directed set if for every a, b ∈ I, there exists c ∈ I such that a ≤ c and b ≤ c. In this subsection, we consider the cohomology of categories opposite to directed sets. Example 1.1 ([43], [76]). Let C be a poset. Denote by Ctop a topological space in which U ⊆ C are declared to be open when y ≥ x ∈ U ⇒ y ∈ U . For every F ∈ Ab C we construct the sheaf F˜ (U ) = lim U F |U . Then lim nC F ∼ = H n ( C, F˜ ) and hence c.d. C is the cohomological dimension of the space Ctop in the sense of sheaf theory [59]. F ∈ Ab C is flasque if the canonical maps lim C F → lim U F |U 2275

are surjective. We see that if F is flasque, then lim nC F = 0 for all n > 0. If I is a directed set and C = I op , ˇ n ( Ctop , F˜ ), where H ˇ n (X, F) is the Chech ˇ then lim nC F ∼ cohomology of a space X with coefficients in =H the sheaf F. Flasque dimension. Let C be a poset, and let F : C → Ab be a functor. We define the flasque dimension of F as the inf of n ∈ N for which there are exact sequences 0 → F → F0 → ··· → Fn → 0 in Ab C with flasque functors F 0 , F 1 , · · · , F n . The flasque dimension f.d. C is the sup of the flasque dimensions of functors F : C → Ab. Theorem 1.2 ([76]). Let C be a totally ordered set. Then  f.d. C = 1 + sup c.d.U : U ⊂ C is open and U = C}. Completions. Some applications [100] arose from the completion construction Aˆ of an Abelian group A by Cauchy sequences. Suppose that a topology on A is defined by a countable base of open neighborhoods of 0 ∈ A consisting of subgroups Ai ⊆ A; then the completion is equal to lim I op {A/Ai }. Hence we obtain the exact sequence 0 −→ ∩Ai −→ A −→ Aˆ −→ lim 1I op {Ai } −→ 0 of Abelian groups, which measures the rejection of A → Aˆ from an isomorphism. Phantom maps. Let X be a connected CW-complex with only a finite number of cells in each dimension, and let Xn denote its n-skeletion. A continuous pointed map f from X to a space Y is called a phantom map if its restriction to each Xn is null-homotopic. Denote by [X, Y ] the set of homotopy classes of continuous pointed maps X → Y . If {E i }i∈Z is an Ω-spectrum that defines a generalized cohomology theory ki (X) = [X, E i ], then the Milnor exact sequence [118] 0 → lim 1Nop {ki−1 (Xn )} → ki (X) → lim Nop {ki (Xn )} → 0 yields that the set of phantom maps X → E i is equivalent to lim 1Nop {ki−1 (Xn )}. Let
{Gn } be a
tower of (noncommutative) groups. Following [57], we define lim 1 {Gn } as follows. Gn → Gn be defined by Let p : n∈N

n∈N

p({gn }) = {gn (pn (gn+1 ))−1 }. Then lim 1Nop {Gn } =



Gn / ∼ =, where {xn } ∼ = ∗ ⇔ (∃{gn }) ({xn } = p{gn }).

n∈N

In [19], the pointed sets lim 1I op {Gi } are defined for every small category I. Let P h(X, Y ) denote the set of homotopy classes of phantom maps from X to Y . As a pointed set, it is isomorphic to the lim 1 term of the tower of groups 0 ← [X, ΩY (1) ] ← [X, ΩY (2) ] ← · · · ← [X, ΩY (n) ] ← · · · , where Y (n) denotes the Postnikov approximation of Y through dimension n. Therefore, we need to study lim 1 for diagrams of noncommutative groups. We refer to [116] and [117] for interesting questions about lim 1 and phantom maps. 2276

Category of projective systems. Let A be a category. A projective (inductive) system of objects and morphisms in A is a functor I op → A (I → A) whose domain I is a directed set. Usually, the limit of a projective system F : I op → A is denoted by lim F , and the colimit of an inductive system F : I → A is ← I

denoted by lim . Let A be an Abelian category with products. To extend the functors lim nIop : AI → I

op

→ A,

where I is a directed set, it is fited the following category pro − A: Objects of pro − A are all projective systems in A. The set of morphisms from F : I op → A to G : J op → A is given by pro − A(F, G) = lim lim {A(F (i), G(j))}. ← J → I

An element of lim {A(F (i), G(j))} consists of an index i plus a morphism fi : F (i) → G(j) modulo the → I

equivalence relation that fi : F (i) → G(j) and fi
: F (i ) → G(j) are equivalent if there is i with i ≥ i and i ≥ i such that fi
◦ F (i ≥ i ) = fi ◦ F (i ≥ i). Hence elements of pro − A(F, G) are represented as pairs (ϕ, {fj }j∈J ), where ϕ : J → I are functions and each fj : F (ϕ(j)) → G(j) is a morphism in A. We state two such (ϕ, {fj }j∈J ) and (ϕ , {fj }j∈J ) to be equivalent if for each j ∈ J, there is i ∈ I with i ≥ ϕ(j) and i ≥ ϕ (j) such that the following diagram commutes: F (i≥ϕ(j))

−−−−−−→ F (ϕ(j))   fj

F (i)   F (i≥ϕ
(j))

fj


F (ϕ (j))

−−−→

G(j)

Let F : → A, G : → A, and H : → A be objects of pro − A. If morphisms f : F → G and g : G → H in pro − A are represented by pairs (ϕ, {fj }j∈J ) and (ψ, {gk }k∈K ), respectively, then g ◦ f : F → H is defined as the morphism that is represented by (ϕ ◦ ψ, {gk ◦ fψ(k) }k∈K ). It is known that if A is an Abelian category, then pro − A is Abelian (see [159]). Let Add(A, Ab) be the category of additive functors A → Ab. Many useful properties of pro − A can be obtained by the isomorphism pro − A ∼ = Lex(A, Ab)op , where Lex(A, Ab) ⊆ Add(A, Ab) is the full subcategory consisting of left exact functors A → Ab. There are set-theoretical problems, but they all have a solution by Grothendieck’s notion of universe (see [171]). A functor T : C → D is cofinal if for every d ∈ Ob D, there is an object c ∈ Ob C such that D(d, T (c)) is nonempty. If I and J are directed sets, then for each cofinal functor T : I → J, there are op natural in F ∈ M odJ isomorphisms lim nJop F → lim nIop F ◦ T op [99,127]. Artin and Mazur gave a natural op . It representation of a morphism F → G in pro − M odR by a natural transformation in some M odK R op follows that pro − M odR is a colimit (over a class of indices) of the categories M odIR . For any directed set I, the diagrams I op

J op

K op

op

M odIR ↓

limn ←

I −→ M odR ↓=

limn ←

pro − M odR −→ M odR n commute, where lim are right derived functors of lim : pro − M odR → M odR . In [47], it is shown that ←

←

limn : pro − Ab → Ab are isomorphic to Extn (∆pt Z, −), where pt is the category with one object 0 and ←

one (identity) morphism 10 . T. Porter in [155, 161] showed that the functors lim are closely related to the torsion theory in ← pro − M odR . Fixed maps. Miminoshvili [120] describes the following two categories. Let K be a category whose objects are pairs (I, X), where I is a directed set and X : I op → Ab is a functor. Any morphism (I, X) → (J, Y ) is given as a functor ϕ : J → I with a natural transformation ω : X ◦ ϕop → Y . 2277

For such morphisms, we obtain the homomorphisms lim n X → lim n Y with a chain homomorphism C ∗ (I, X) → C ∗ (J, Y ). For each poset, I n will denote the set of sequences i0 < i1 < · · · < in in I if n > 0, and I 0 = I for n = 0. Definition 1.1. A fixed map Φ : (I, X) → (J, Y ) is a sequence {ϕn : J n → I 0 }n≥0 such that: (a) ϕ0 : J → I is a function, (b) ϕn (j0 < j1 < · · · < jn ) > ϕn−1 (j0 < · · · < jˆk < · · · jn ), 0 ≤ k ≤ n, (c) for every j ∈ J, there is a homomorphism ωj : X(ϕ0 (j)) → Y (j), (d) for each pair (j0 < j1 ) ∈ J 1 , the composition X(ϕ1 (j0 < j1 )) → X(ϕ0 (j0 )) → Y (j0 ) is equal to the composition X(ϕ1 (j0 < j1 )) → X(ϕ0 (j1 )) → Y (j1 ) → Y (j0 ). Two fixed maps Φ and Φ are called equivalent if for each j ∈ J, there exists i ∈ I such that i > ϕ0 (j), i > ϕ 0 (j) and the diagram X(i) −→ X(ϕ0 (j)) ↓ ↓ Y (j) X(ϕ 0 (j)) −→ is commutative.

whose objects are all the objects of K and whose morphisms are the equivalence Consider a category K ˜ → P ro − Ab of categories class of fixed maps. Miminoshvili [120] proved that there is an isomorphism K n lim ˜ −→ P ro − Ab −→ Ab equals lim n : K ˜ −→ Ab. such that for each n ≥ 0, the composition K Acyclic projective systems. Let R be a ring with identity. A functor F : I op → Ab is acyclic if lim nIop F = 0, ∀n > 0. We saw that flasque projective systems are acyclic. A projective system F : I op → Ab is weakly flasque if for each directed subset J ⊆ I, the natural map lim I op F → lim J op F |J is surjective. Theorem 1.3 ([76]). If F is a weakly flasque projective system, then F is acyclic. Any ring R will be considered with the discrete topology. A topology on a left R-module M is said to be linear if it is invariant with respect to translations and if there exists a fundamental system of neighborhoods of 0 consisting of submodules. Let M be a left R-module with a Hausdorff linear topology. The module M is said  to be linearly (xi + Ni ) is not compact iffor each family {Ni }i∈I of closed submodules and xi ∈ M , the intersection i∈I (xj + Nj ) = ∅ for every finite subset J ⊆ I. empty iff j∈J

Theorem 1.4 ([76]). Let R be a ring with identity, U the forgetful functor from the category of the topological R-modules to M odR , and {Mi }i∈I be a projective system of linearly compact left R-modules such that homomorphisms M (i > j) : Mi → Mj are continuous. Then limnIop {U (Mi )} = 0 ∀n > 0. Corollary 1.5 ([76]). Let R be a ring with identity. Then every projective system of Artinian left Rmodules is acyclic. This corollary is an answer to the question of J.-E. Roos [170], p. 219. Jensen [75] proved it for the first time for either R is Noetherian or Mi are Noetherian (and Artinian) modules. A category C is filtered if: (1) for every pair of objects a, b, there is an object c such that C(a, c) and C(b, c) are nonempty, (2) for every pair of morphisms α, β ∈ C(a, b), there is a morphism γ such that γ ◦ α = γ ◦ β. When C is a poset, (1) simply says that C is directed. The category opposite to being filtered is said to be cofiltered. The directed set I is cofinite if for each i ∈ I, the set {i ∈ I : i < i} is finite. For every filtered category I, there exists a cofinite directed set M (I) and the Mardeˇsi´c cofinal functor init : M (I) → I. 2278

It can be considered a pro-category of functors on cofiltered categories, but the Mardeˇsi´c functor allows us to prove that every object in the pro-category is isomorphic to some projective system over a cofinite directed set. Definition 1.2. A projective system of Abelian groups is said to satisfy the Mittag–Leffler (ML) condition if it is isomorphic in pro − Ab to a projective system whose homomorphisms are surjections. Sometimes, the ML condition is formulated as follows. (ML) For each i ∈ I, there is k ≥ i such that ImF (j ≥ i) = ImF (k ≥ i) for all j ≥ k. A projective system is said to satisfy the strong Mittag–Leffler (SML) condition [48] if it is isomorphic to a projective system F : I op → Ab such that I is a cofinite directed set, and for all j ∈ I, the natural F (k) are surjections. maps F (j) → lim ← {k:k gl.dim M odR . In other words, c.d.(M,R) I op ≤ gl.dim M odR for the category M of finitely generated modules over a commutative Noetherian ring. L. Gruson and C. U. Jensen in [61] proved the following statement. Theorem 1.10. Let {Ai } be a projective system over a directed set I with |Ai | < ℵn for some n ≥ 0, and let T : Ab → Ab be an arbitrary exact functor. Then under the hypothesis 2ℵn = ℵn+1 it is true that limk {T Ai } = 0 for all k ≥ n + 3. In particular, under the assumption 2ℵ0 = ℵ1 , we have limk {Ai } = 0 ← I

for each projective system of countable Abelian groups for all k ≥ 3.

← I

In [61], it is noted that there exists a projective system of countable Abelian groups {Ai } with lim2 {Ai } = 0. ← I

Problem 1.4. Let n ≥ 0 be a natural number, and let M ⊂ Ab be the full subcategory of all Abelian groups with cardinalities ≤ℵn . Prove that c.d.(M,Z) I op ≤ n + 2 for each directed set I. The inequality c.d.(M,Z) I op ≤ n + 2 holds under the assumption that 2ℵn = ℵn+1 by the above Theorem 1.10. It is known [76] that lim 1I op {Ai } = 0 for every projective system of Artinian R-modules. Hence if M ⊂ M odR is the full subcategory of all Artinian modules, then c.d.(M,R) I op = 0 for each directed set I. Let R be a ring with identity. The Krull dimension K − dimM of an R-module M is defined inductively. Let K−dim M = 0 for Artinian M . We define K−dim M = n if the inequality K−dim M < n is not true and for each decreasing sequence M1 ⊇ M2 ⊇ M3 ⊇ · · · of submodules in M all of Mi /Mi+1 , except for a finite set of i, have K − dim(Mi /Mi+1 ) < n. The Krull dimension K − dim R is the sup of K − dim M , where M runs over all finitely generated R-modules. Jensen [76] proved that if R is a local Noetherian ring of Krull dimension one, then limk {Mi } = 0 for all k > 1 ← I

for all projective systems of finitely generated K-modules. Jensen conjectured in [76, p. 82], that a similar result is true for arbitrary Noetherian rings. Gruson [61] proved that if R is a right Noetherian ring and M a projective system of finitely generated R-modules with K − dim Mi < n for all i ∈ I, then limk {Mi } = 0 for k > n. We refer to Porter’s paper [159] for generalizations of these results.

← I

Dependence of lim 1 at the continuum hypothesis. By a ZFC model, we mean a model of set theory for which the Zermelo–Fraenkel axioms and the choice axiom are true. Let (CH) denote the continuum hypothesis 2ℵ0 = ℵ1 . Mardeˇsi´c and Prasolov constructed a projective system that can be acyclic or independent of (CH). Let NN be the set of all sequences n = (n(0), n(1), · · · , n(i), · · · ), n(i) ∈ N. To 2280

order NN , we set n ≤ m, provided that n(i) ≤ m(i) for all i ∈ N. It is clear that NN is a directed set. For every n ∈ N, we set An =

n(i) ∞  

Z

i=0 j=0

and for A(m ≥ n) : Am → An , take the natural projection. 1/n for In the Euclidean space Rk+1 , let S k (n) be the k-sphere centered at (1/n, 0, . . . , 0) of radius (k) be n ≥ 1, and let Y (k) = ∪{S k (n) : n ≥ 1}; Y (k) is called the Hawaiian earring. Let X (k) = ∞ Y i=0 the topological sum of a countable infinite collection of copies of Y (k) . X (k) is a k-dimensional locally compact separable metric space. Let Hp be the strong homology theory of Miminoshvili [119]. Then Hp (Y (p+1) ) is trivial for every p ≥ 0 and Hp (X (p+1) ) = lim 1 A by [114]. Hence the theory Hp is not additive in the sense of Milnor [118] if lim 1 A = 0. Let U and V be arbitrary subsets of N × N and let f : U → Z, g : V → Z be arbitrary functions. We write f ≡ g, whenever the set {(i, j) ∈ U ∩ V : f (i, j) = g(i, j)} is finite. Mardeˇsi´c and Prasolov [114] proved that lim 1 A = 0 if and only if the following question [114, Question 5] has a positive answer. Let (fn , n ∈ NN ) be a set of functions fn : Un → Z, where Un = {(i, j) ∈ N × N : 0 ≤ j ≤ n(i)}. If fn ≡ fm for any pair n, m ∈ NN , does there exist a function f : N × N → Z such that f ≡ fn for every n ∈ NN ? In [114], there is a proof showing that under the assumption of (CH), the answer to their Question 5 is “yes.” Consequently, (CH) implies that lim 1 A = 0. Dow, Simon, and Vaughan [44] showed that (CH) in the statement of that result can be replaced by a weaker hypothesis. Dow, Simon, and Vaughan [44, Theorem 1.2] proved that there exist ZFC models such that lim 1 A = 0. On the other hand, there exists [87] a projective system F such that lim 1 F = 0 if (CH) is true, and lim 1 F = 0 if 2ℵ0 = ℵ2 . Applications to convergence conditions. Let i

D −→ D ↑k ↓j E = E

(1)

be an exact pair of bigraded Abelian groups and homogeneous homomorphisms such that i, j, and k have bidegrees (−1, 1), (1, 0), and (0, 0), respectively. Then we can construct a sequence of exact pairs for r = 1, 2, . . . . i

r Dr Dr −→ ↑ kr ↓ jr Er = Er

(2)

whose first member is (1), where ir , jr , and kr have bidegrees (−1, 1), (r, −r + 1), and (0, 0), respectively. Following [162], we say that (1) is an exact pair of the inverse limit type if for each n there exist r0 and s0 such that Drs,n−s = 0 for all r ≥ r0 and s ≤ s0 . Given such a pair, for every s and t and for some number r0 , we have Drs−r,t+r−1 = 0 for r ≥ r0 and, therefore, Ers,t ⊃ Ers,t ⊃ Ers,t ⊃ ··· 0 0 +1 0 +2 2281

Denote

 r

s,t Ers,t by E∞ . In [162], the spectral sequence Ers,t generated by the exact pair (1) is said to be

n if for each n, there is a sequence of epimorphisms convergent to a graded Abelian group D∞

· · · → Qns → Qns−1 → Qns−2 → · · · n = lim such that D∞

← s∈N

s,n−s {Qns } and Ker (Qns → Qns−1 ) ∼ for all s ∈ Z. Prasolov [162] proved that = E∞

if (1) is an exact pair of the inverse limit type of bigraded Abelian groups such that projective systems n , where D n is {Ers,t }r∈N are acyclic for every (s, t), then the spectral sequence {Ers,t } converges to D∞ ∞ the inverse limit of the sequence i

i

i

· · · → Ds,n−s → Ds−1,n−s+1 → · · · . Using this result, Prasolov proved that for each projective system {F (i)}i∈I of complexes of Abelian groups, there exists a spectral sequence of the right half-plane with E2s,t = lims {H−t F (i)} that converges ← I

in the above sense to the strong homology H −s−t F of Miminoshvili [119]. Recall that for any small category C and a functor F : C → Chain into the category of chain complexes of Abelian groups, the complexes Rp,q = C p ( C, F−q ) are those we have considered in Subsection 1.1 for F = F−q , and
H n F = H −n (T ot∗ (F )) are the cohomologies of the total complex {T otn (F ), d} = { Rp,q ,δ + d } of a p+q=n

cochain bicomplex Rp,q F =



F−q (cp ),

Rp,q = 0 f or p < 0,

c0 →c1 →···→cp

with

d p,−q

=

(−1)p ∂q

where ∂∗ is the differential of F∗ .

Theorem 1.11 ([162]). Let F : C → Chain be a functor. Then there is a spectral sequence with E2s,t = lim sC {H−t (F (c))} that converges in the above sense if lim1 {Ers,t } = 0 for each s and t. ← r

lim sC {H−t (F (c))}

= 0 for all s ≥ s0 or n0 such that Hn (F (c)) = 0 If there is either s0 such that for all n ≥ n0 and c ∈ Ob C, then the spectral sequence of this theorem converges in the usual sense to s,t H −s−t F , i.e., H −s−t F admits a finite filtration whose consecutive quotients are isomorphic to E∞ . For information on the strong homology and applications, we refer to the review of Sklyarenko [174]. 1.3. Elements of the first cohomology group. For any functor F : C → Ab on a small category, nski [56] and Ford Hoff [63] describes the elements of lim 1C F as classes of crossed homomorphisms. Golasi´ [51] gave another interpretation. The record C+ means that the composition in C is denoted by “+.” Crossed homomorphisms. A crossed product of F and C is the set  F (b) × C(a, b) (F ×χ C)+ = (a,b)∈Ob C×Ob C

with the composition law defined by (z , f ) + (z, f ) = (z + F (f )z, f ◦ f ) for f and f such that the composition f ◦ f exists. The groupoid sum of groups c∈Ob C F (c) is a subcategory of the crossed product. A crossed
1 1 homomorphism of C in F is ψ ∈ f ∈M or C F (cod (f )) = C ( C, F ) such that d ψ = 0. A principal crossed homomorphism is any ψ ∈ Im d0 . Crossed homomorphisms are exactly those functions ψ : M or C → c∈Ob C F (c) for which ψ(f ) ∈ F (cod (f )) and ψ(f ◦ f ) = ψ(f ) + F (f )(ψ(f ))

if f ◦ f is defined.

It is principal if there exists τ ∈ C 0 ( C, F ) such that ψ(f ) = τ (cod (f )) − F (f )τ (dom (f )) ∀f ∈ M or C. 2282

Let B 1 ( C, F ) be the subgroup of elements in the group Z 1 ( C, F ) of crossed homomorphisms that are principal. We can consider the elements in lim 1C F as the classes of crossed homomorphisms modulo the principal crossed homomorphism, since lim 1C F ∼ = Z 1 ( C, F )/B 1 ( C, F ). The inner automorphism of (F ×χ C)+ defined by τ ∈ C 0 ( C, F ) is the functor from the crossed product to itself which acts as (z, f ) → (τ (cod (f )) + z − F (f )(τ (dom (f )), f ). Proposition 1.12 ([63]). The group of all those invertible functors from the crossed product to itself that induce the identity functors on both the subcategory c∈Ob C F (c) and the quotient category C is isomorphic to the group Z 1 ( C, F ) of crossed homomorphisms. Under this isomorphism, the inner automorphisms of (F ×χ C)+ defined by τ ∈ C 0 ( C, F ) correspond to the principal homomorphisms. Derivations. By [51] and [56], a derivation is an element of Z 1 ( C, F ). Elements of B 1 ( C, F ) are called of functors L : Ens → Ab and hc . Let J( C) be given inner derivations. Denote by Lhc the composition by the exact sequence 0 → J( C) → c∈Ob C Lhc → ∆ C Z → 0. Then there is a natural isomorphism Ab C (J( C), F ) ∼ = Der( C, F ) n ∼ n for n ≥ 1. This leads to the natural isomorphisms lim n+1 C F = R (Der( C, F )), where R Der( C, −) are the nth right derived functors of Der( C, −). 1.4. Categories of cohomological dimension zero. The limit functor lim C : M odRC → M odR is always left exact. It is natural to ask when it will be exact. This problem amounts to characterizing all small categories C with c.d. C = 0. Oberst’s conjecture. A category has a right zero if there exists an endomorphism ε of an object that maps to all objects such that α ◦ ε = β ◦ ε whenever the equation has a sense. For R = Z, Oberst conjectured that c.d. C = 0 if and only if every connection component has a right zero [140]. This conjecture was proved by Laudal in [110]. Theorem 1.13 ([110]). The limit functor Ab C → Ab is exact if and only if the components of C have right zeros. Categories of R-cohomological dimension zero. For each ring R with identity, the inequality c.d.R C ≤ c.d. C holds. Hence, Laudal characterized small categories such that c.d.R C = 0 for every ring R with identity. Cheng obtain more complete information about categories with c.d.R C = 0. Definition 1.5 ([27]). A C-set is a functor C → Ens. A C-set is said to be decomposable if it is a disjoint union of two C-sets. It is indecomposable if it is not decomposable. A functor RA : C → M odR is defined so that RA(p) for p ∈ Ob C are free R-modules generated by A(p). Denote by h∗ : Cop → Ens C the Yoneda functor. Every C-set A is a disjoint union of indecomposable C-sets Ai . In this case, h∗ /A has components h∗ /Ai , and RA is a direct sum of RAi . Denote α E = {α ◦ ε : ε ∈ E}. Theorem 1.14. Suppose that A is an indecomposable C-set such that A(α) is injective for all α in C. Then RA is projective if and only if there exists an object e of (h∗ /A)op that maps to all objects such that (h∗ /A)op (e, e) contains a finite subset E satisfying: (a) αE = βE for all morphisms α and β whenever the equation has a sense (i.e., α and β have the same domain e and a common codomain). (b) The order of E is invertible in R. 2283

Theorem 1.15. Let C be a connected small category, and let R be a ring with identity. Then c.d.R C = 0 if and only if there exists an object e ∈ Ob C that maps to all objects such that C(e, e) contains a finite subset E satisfying the following conditions: (a) αE = βE whenever the equation has a sense; (b) the order of E is invertible in R. Here αE denotes the set {α ◦ ε : ε ∈ E}. Corollary 1.16 ([30]). Suppose that A is an indecomposable C-set such that A(α) is injective for all α. Then ZA is projective if and only if (h∗ /A)op has a right zero. Monoids of cohomological dimension zero. Corollary 1.17. Suppose M is a monoid. Then c.d.R M = 0 if and only if there exists a finite subset E of M such that: (a) mE = E for all m ∈ M ; (b) the order of E is invertible in R. Hence if G is a group, then c.d.R G = 0 if and only if G is finite with its order invertible in R. ˆ of a monoid M is defined to be the value on M of the left adjoint of the The group reflection M forgetful functor from the category of groups to the category of monoids. By Corollary 1.17 we obtain the following conclusion. Corollary 1.18 ([33]). Suppose M is a finitely generated Abelian monoid. Then c.d.R M = 0 if and only ˆ = 0. if c.d.R M 1.5. Extensions of a category. Let C be a small category, and let F : C → Ab be any functor. Hoff [63] and Mitchell [131] proved that elements of the group lim 2C F can be considered as classes of extensions of C by a functor F : C → Ab. Datuashvili [42] and Golasi´ nski [56] generalized these constructions to elements of lim n F . Extension of functors by a category. An extension of categories is a sequence of functors i

π

H −→ E −→ C such that C is the quotient category of E, i is an injection, and π is a projection; in particular, π is the full functor and π is bijective on objects. Furthermore, we require that for all f ∈ M orE and g ∈ M orE, the relation π(f ) = π(g) is true if and only if there exists z ∈ M orE such that g = i(z) + f . (We denote the composition in E by “+” and identify every object e ∈ ObE with π(e) ∈ Ob C.) Let F : C → Ab be a functor. An extension of F by C is an extension of categories  i π F (c) −→ E −→ C, c∈Ob C

such that i(F (π(b))) ⊆ Aut(b) for each b ∈ ObE. i π i
π
Two extensions c∈Ob C F (c) → E → C and c∈Ob C F (c) → E → C are congruent if there is a functor µ : E → E such that the following diagram is commutative: i π C c∈Ob C F (c) −→ E −→ ↓= ↓µ ↓= i
π
−→ C, c∈Ob C F (c) −→ E Let Opext1 ( C, F ) be the set of all congruence classes of extensions of F by C. Hoff proved that there exists a one-to-one natural correspondence w1 : Opext( C, F ) → lim 2C F . 2284

In [64], the notion of the extension for an interpretation of non-Abelian cohomologies with not necessary functorial coefficients was generalized. Fibrations. Mitchell [131] introducted the notion of fibration. A functor π : E → C is called a fibration by groupoids if the following two conditions hold: (a) For all objects y ∈ ObE and b ∈ Ob C and for every morphism β : b → π(y), there exist an object x ∈ ObE and a morphism α : x → y such that π(α) = β. (b) For all objects x, y and z and morphisms α1 : x → z and α2 : y → z of the category E, and for any morphism β : π(x) → π(y) of C that satisfies the condition π(α2 ) ◦ β = π(α1 ), there exists a unique morphism α : x → y such that π(α) = β. For any c ∈ Ob C, the morphisms α ∈ π −1 (1c ) form a subcategory of E, called the fiber over c. Every fiber is a groupoid. If the fibers Gc = π −1 (1c ) for all c ∈ Ob C are groups, then we say that π is the fibration by groups Gc . Let π : E → C be a fibration by groups Gc , c ∈ Ob C. Then π is a bijection on objects, and so we identify the objects of C and E via π. We see that E has the following properties: (a) If α ∈ E(a, b) and g ∈ Gb , then gα = αg for some g ∈ Ga . (b) If α ∈ E(a, b) and αg = αg for some g, g ∈ Ga , then g = g . We have π(α) = π(α ) if α = αg for some g ∈ Ga . Since π is full, it can be identified with the natural functor from E onto a quotient category. Conversely, let us begin with a category E equipped with a subgroup Ga ⊆ Aut(a) for each a ∈ ObE such that (a) and (b) hold. Define α ∼ α if they are parallel morphisms such that α = αg for some g ∈ Ga . Condition (a) and the property that the Ga are groups imply that ∼ is a congruence relation on E. Let C denote the quotient category E/ ∼. Then condition (b) implies that the natural functor π : E → C is a fibration with groups Ga as fibers. Consider a fibration π : E → C by groups Ga . Let us denote the element g of condition (a), which is unique by condition (b), by G(α)(g). Then G(α) : Gb → Ga is a group homomorphism, and we have G(α α) = G(α)G(α ),

G(1a ) = 1Ga .

In other words, G : E op → Grp is a functor into the category of groups. For each morphism x ∈ M or C, let γx be a morphism of E such that π(γx ) = x. Such a family γ is called a cleavage for the fibration π. If there exists a cleavage γ such that γ : C → E is a functor, then π is called a split fibration. Now even if γ is not a functor, the composition γ

G

Cop → E op → Grp

(3)

can be a functor. Observe that if g ∈ Gb , h ∈ Ga , and α ∈ E(a, b) then gαh = αg h = αhh−1 g h, and, therefore, G(αh)(g) = h−1 G(α)(g)h. If Ga are Abelian, then composition (3) is a functor independent of the cleavage γ. Now consider a small category C and a fixed functor F : Cop → Ab. A fibration π : E → C by Abelian groups such that composition (3) yields the given functor F will be called a fibration of C by F . Two such fibrations π : E → C and π : E → C are equivalent if there is a functor T : E → E for which the diagram π C c∈Ob C F (c) ⊆ E −→ ↓= ↓T ↓= π
−→ C, c∈Ob C F (c) ⊆ E is commutative. Such a functor T should be an isomorphism. Therefore, this relation is an equivalence relation on fibrations of C by F . Theorem 1.19 ([131]). If C is a small category and F : C → Ab is a functor, then there is a one-to-one correspondence between the equivalence classes of fibrations of C by F and the elements of the cohomology group H 2 ( Cop , F ). 2285

Problem 1.6 ([131]). If all fibrations of a category C by Abelian groups split, then do all fibrations of C by not necessarily Abelian groups split? Is this true at least in the case where C is a poset? Operations with extensions. In [56], Golasi´ nski generalizes the Hoff result. Let C be a category, and let F : C → Ab be a functor. Let F + = c∈Ob C F (c) denote a groupoid that is obtained by a disjoint union of the groups F (c). For an extension E : F + → E → C and a natural transformation ϕ : F → F , the pushout square α F + −→ E + ↓γ ↓ϕ α


F + −→ E β


α


yields an extension E : F + → E → C and the commutative diagram α

F + −→ ↓ ϕ+

E ↓

α


β

−→ β


F + −→ E −→

C ↓ 1C C



We set E = E ◦ ϕ and call it the composite of E and ϕ. Oppositely, for an extension E : F + → E → C and a functor δ : C → C, the pullback square β


C ↓δ

β

C

E −→ ↓γ −→

E

yields an extension E : (F γ)+ → E → C and the commutative diagram α


β


C ↓δ

β

C

(F γ)+ −→ E −→ ↓ ↓γ F+

α

−→

E

−→

We set E = γ E and call it the composite of γ and E. The set Opext1 ( C, F ) becomes an Abelian group under the Baer sum, which is defined as follows. Let ∆ C : C → C × C denote the diagonal functor, and let ∇F : ((F × F )∆ C )+ → F + be the + codiagonal morphism for F given as the addition (∇F )c : F (c) × F (c) → F (c) at every c ∈ Ob C. Given α


β

β


two extensions E : F + → E → C and E : F + → E → C , we define their product as the extension α

α×α


β×β


E × E : (F × F )+ → E × E → C . Proposition 1.20 ([56], Proposition 1.1). The set Opext1 ( C, F ) is an Abelian group with the operation that assigns to the congruence classes of the extensions E and E the congruence class of the extension E + E = ∇F (( E × E )∆ C ). The zero element of Opext1 ( C, F ) is the congruence class of the extension F + → F ×χ C → C, where F ×χ C is the crossed product of F and C. ∼ =

Theorem 1.21 ([56]). There is a natural isomorphism w1 : Opext1 ( C, F ) → H 2 ( C, F ) in F . Golasi´ nski [56] defines an n-fold extension of F by C as a sequence of categories and functors E:

α

αn−1

α

α

+ → Fn →n C, F + →0 F1+ →1 · · · → Fn−1

where F, F1 , · · · , Fn−1 are functors C → Ab, Fn is a category, and there exist extensions α

β1

E 0 : F + →0 F1+ → G1 , 2286

βi+1

γi

+ E i : Fi+ → Fi+1 → Gi+1 , for i = 1, 2, · · · , n − 2, γn−1

βn

E n−1 : Fn−1 → Fn → C such that γi ◦ βi = αi for i = 1, 2, · · · , n − 1. We write E as the composite of n extensions E i in the form E = E0 ◦ E1 ◦ · · · ◦ En−1 . Any morphism of such n-fold two extensions E and E is given as a sequence + , γn , id C ) such that the following diagram is commutative: of the functors (idF + , γ1+ , · · · , γn−1 α

0 F + −→ ↓ 1F +

α


α

1 F1+ −→ ··· ↓ γ1+

α


0 1 F + −→ F + −→ ··· 1

−→

αn−1

+ Fn−1 −→ + ↓ γn−1

−→ F + n−1

α
n−1

−→

α

n Fn −→ C ↓ γn+ ↓ 1C

α


n Fn −→

C



and E is congruent to E iff there is a sequence of extensions E 0 , E 1 , · · · , E n such that E 0 = E, E n = E , and for every 0 < i ≤ n, either there is a morphism from E i−1 to E i or a morphism from E i to E i−1 . Let Opextn ( C, F ) be the set of all congruence classes of n-fold extensions of F by C. Define the composite of an n-fold extension of F by C with a “matching” map, namely, if E = E 0 ◦ E 1 ◦ · · · ◦ E n−1 , then we define Eϕ whenever ϕ : F → F is a natural transformation and γ E whenever γ : C → C is a functor by the formulas γ E = (γ E 0 ) ◦ E 1 ◦ · · · ◦ E n−1 , Eϕ = E 0 ◦ E 1 ◦ · · · ◦ ( E n−1 ϕ). nski proved The Baer sum is defined for E and E by the familiar formula E + E = ∇F ( E × E )∆ C . Golasi´ that the Baer sum makes Opextn ( C, F ) an Abelian group and that the family Opextn is a connected sequence of functors. Moreover, he obtained the following result. Theorem 1.22. There is an isomorphism w = {wn : Opextn ( C, −) → lim n+1 C (−), n ≥ 1} of the connected sequences of functors. 1.6. Categories of cohomological dimension one. A small category is called cancellative if each of its morphisms is monomorphic and epimorphic. Hence a category C is cancellative if and only if α ◦ x ◦ β = α ◦ y ◦ β ⇒ x = y for all α, β, x, y ∈ C. Let Γ be a (directed) graph, and let W Γ be the category of paths in Γ. Denote by Γ1 the set of all arrows in Γ. If a category C isomorphic to the fraction category of W Γ obtained by inverting the elements of some subset Σ ⊆ Γ1 , then C is called partially free or a bridge category. Mitchell proved that any partially free category has cohomological dimension ≤1. Monoids of cohomological dimension one. Every monoid will be considered as a small category with one object. Stallings [177] and Swan [179] proved that all groups of cohomological dimension one are free. Mitchell [122] showed that this result is not true for commutative monoids. Conjecture 1.7 ([122]). Let C be a cancellative monoid. Then c.d. C ≤ 1 if and only if C is partially free. This conjecture was refuted by B. V. Novikov, who constructed counterexamples in [132]. Novikov proved that all submonoids in the additive group Z of integers have cohomological dimension ≤1 in [134]. A submonoid in N is free if and only if it is isomorphic to N and, hence there are nonfree commutative cancellative monoids of cohomological dimension 1. For any semigroup S that does not contain the identity we define c.d.S as c.d.(S ∪ {1}). Novikov [134] proved that all commutative cancellative semigroups of cohomological dimension 1 are isomorphic either to Z or to a subsemigroup of N. We mention here [33], since it contains this affirmation as a corollary. Using his results [135], Novikov proved in [136] a weak Mitchell’s conjecture that each cancellative semigroup of cohomological dimension one has an embedding in a free group. We refer to a review of Novikov [138] for more complete information on the cohomology of monoids. 2287

DCC categories and partially ordered sets of cohomological dimension one. Cheng [25] characterized those finite posets C for which c.d.R C ≤ 1. Recall that the height of an element x in a finite poset is the greatest integer n such that there is a chain x0 < x1 < · · · < xn = x. If x, y ∈ C are such that ]x, y[= ∅, then we say that y is a cover for x and x is a cocover for y. Define an element of C to be superfluous if it is of height 0 (minimal) with only one cover or if it has height one and only one cocover. Iterating as many times as possible the process of eliminating a superfluous element, we obtain a finite poset E( C). For each a ∈ C, we define the poset Ca = {x ∈ C : x ≤ a}. Let C = pt be the poset consisting of one element. Theorem 1.23 ([25], Theorem 1.8). Let R be a ring with 1 and C be a poset such that Ca are finite for all a ∈ C. Then c.d.R C ≤ 1 if and only if E( Ca ) = pt for all a ∈ C. An n-crown Cn for n ≥ 2 is the set consistings of 2n + 1 elements {e1 , e2 , . . . , en , f1 , f2 , . . . , fn , g} ordered by ei < fi < g for 1 ≤ i ≤ n, ei < fi−1 for 2 ≤ i ≤ n and e1 < fn . e1 rP r P

en

rrr

PP rrr f1 rP r r rPr ✏ Pr fn P❅✏ Pr ✏

g n-crown Cn A category satisfies the descending chain condition (DCC) if for every sequence of morphisms α1 , α2 , · · · with dom (αi ) = cod (αi+1 ) for all i, there exists k such that αi = 1 for all i > k. It is clear that every DCC category is a delta. All DCC categories of cohomological dimension one were characterized by Cheng in [29]. Let C be a small category. A (singular ) n-crown (x1 , . . . , xn , y1 , . . . , yn , z1 , . . . , zn ) in a category C is a diagram C n : Cn → C with xi = C n (fi < g), yi = C n (ei < fi ), for all i = 1, · · · , n, z1 = C n (e2 < f1 ), z2 = C n (e3 < f2 ), . . . , zn−1 = C n (en < fn−1), zn = C n (e1 < fn ). A 1-crown C 1 in C is a diagram of the form y

x

• −→ −→ • −→ • z

in C, where x y = x z and y = z. A 1-crown C 1 in C is supported in C if there exists an n-crown (x1 , . . . , xn , y1 , . . . , yn , z1 , . . . , zn ) in C with y = x1 y1 and z = xn yn , where all xi = 1. An n-crown (x1 , . . . , xn , y1 , . . . , yn , z1 , . . . , zn ) in C is supported between yk and zk in C if there exists a commutative diagram s1 r

r

rrr rrr

t1

r

sp r tp r

r r rrr P ✏ vP Pr ✏ vp 1 ❅✏

xk

r

in C satisfying yk = v1 s1 and vp tp = zk such that for every i with 1 ≤ i ≤ p, there exist j = k and a commutative diagram • −→ • ↓ xj ↓ vi xk • −→ • If C n is supported between yk and zk in C for all k, then C n is supported in C. 2288

Theorem 1.24 ([29]). Let C be a DCC category, and let R be a ring with identity. Then c.d.R C ≤ 1 if and only if every n-crown contained in C, n ≥ 1, is supported in C. We point out the following corollary of Cheng’s result. Theorem 1.25 ([31]). Let C be a poset with the descending chain condition, and let R be any nonzero ring. Then c.d.R C ≥ 2 if and only if for some n ≥ 2, C contains Cn as a retract in the category of posets and order-preserving maps. The chain condition cannot be removed, since there are totally ordered sets of arbitrarily large cohomological dimension [126], whereas a totally ordered set obviously cannot contain Cn as a retract. To see that Theorem 1.25 is not true in general when C is not a poset, in [29] Cheng considered the category D generated by the graph r r P ❅PPP❅ PPr a b❅ c d✏❅ ✏ ❅ ✏ ✏ ✏ ❅r r✏ ❅ x❅r y

with the relations x a = y b and x c = y d. The category D contains a 2-crown (x, y, a, d, c, b) that is not supported in D, and, therefore, c.d.R D ≥ 2. But D does not contain any 2-crown as a retract. We consider every subset S of a poset as a full subcategory whose objects are elements of S. ˇ be the poset obtained from C by adding two incomparable elements a Let C be a poset, and let C and b with the relations a < c and b < c for each c ∈ C. If C is any poset, we define a tail of C as a subset M such that M op is a maximal directed subset of Cop . If M is a tail of C, let M be the subset of M consisting of those elements that do not belong to any other tail. If M is nonempty, then M op is a cofinal subset of M op . In general, M can be empty, but not under the assumption that each element of C is contained in only a finite number of tails [32, Lemma 12]. In this case, let Mi , i ∈ I, be tails of C, ˆ be the disjoint union and let C ˆ = ( C − ∪i∈I Mi ) ∪ {ei : i ∈ I}, C where the order on the first term is inherited from C, and the order relations involving ei are of the form ˆ there is i ∈ I such that ei ≤ x and ei are minimal. ei < p, where p ∈ Mi − Mi . Therefore, for every x ∈ C, Cheng and Mitchell obtained following result. Theorem 1.26. Let C be a poset in which every element is contained in only a finite number of tails. ˆ contains neither Then c.d.R C ≤ 1 if and only if M op has a countable cofinal subset for each tail M and C op Cn nor γˇ as a retract, where 2 ≤ n < ∞ and γ is a limit ordinal. Dimension of the fraction category. By one result of Cheng, Wu, and Mitchell [37], it is follows that if C is a small category and Σ ⊆ M or C is an arbitrary subset, then for any commutative ring K with identity, the natural map lim 2Σ−1 C F → lim 2C F is a monomorphism for any functor F : Σ−1 C → M odK . This generalizes a result of Barrat [6], where C is a monoid and Σ is the whole C. Corollary 1.27. Let C be a small category and K be a commutative ring with identity. If c.d.K C ≤ 1, then for each Σ ⊆ M or C, the inequality c.d.K Σ−1 C ≤ 1 holds. 1.7. Cohomological dimension of constructions. Colimits of categories. Let I be a poset, and let { Ci }i∈I be a family of small categories such that Ci ⊆ Cj for all i ≤ j. A category C is locally covered by { Ci }i∈I if for each n ∈ N the following conditions hold:

Nn Ci ; (1) Nn C = i∈I

2289

(2) for every σ ∈ Nn Ci ∩ Nn Cj , there exists k ≤ i, j such that σ ∈ Nn Ck . If C is locally covered by { Ci }i∈I , then C = colimI { Ci }. In [82], it is proved that if { Ci }i∈I is a functor I → Cat from any poset I into the category of small categories such that either the poset I is directed or colimI { Ci } is locally covered by { Ci }, then for every Abelian category A with exact products and a functor F : colimI { Ci } → A, there exists a first-quadrant spectral sequence E2p,q = lim pIop {lim qCi (F ◦ ini )} =⇒ lim p+q F, colimI { C } i

(4)

where ini : Ci → colimI { Ci } are the canonical morphisms of the colimit. In [106], Laudal found the spectral sequence (4) for the cover of a poset by open subsets. In each of these cases, we have c.d.R colimI { Ci } ≤ c.d.I op + sup{c.d.R Ci }.

(5)

Mitchell [126] and Datuashvili [42] proved (5) for the directed colimit of categories. Ford [51] proved that if C is the amalgamation of the categories C1 and C2 over D, then there exists an exact sequence · · · → lim nC F → lim nC1 F | C1 ⊕ lim nC2 F | C2 → lim nD F | D → lim n+1 C F → ··· In this case, c.d. C ≤ sup{1 + c.d. D, c.d. C1 , c.d. C2 }. Product of categories. In [82], the spectral sequence lim pC {lim qD {F (c, d)}} =⇒ lim p+q C× D F is found. It follows that the inequality c.d.R C × D ≤ c.d.R C + c.d.R D is true for every nonzero ring R and small categories C and D. ˆ is the Group reflection of a monoid. If M is Abelian and cancellative, then the group reflection M ˆ group of quotients of M and the canonical homomorphism ηM : M → M is injective. Theorem 1.28 ([33]). If M is an Abelian monoid that is either cancellative or finitely generated, then ˆ. c.d.R M = c.d.R M 2. Homological Dimension of a Small Category We introduce the homology of small categories as opposite to the cohomology. This section contains an interpretation of the first homology module, the characterizations of categories of homological dimension zero, the characterization of posets of homological dimension one, and the comparison with the cohomological dimension. 2.1. Homology of small categories. Let A be an Abelian category with exact coproducts, and let C be a small category. For any functor F : C → A, we have the functor F op : Cop → Aop , where Aop is Abelian with an exact product. The colimit colim C F in A can be considered as the limit lim Cop F op in the category Aop . Hence the left satellites colimnC : A C → A can be considered as the right satellites of op lim Cop : (Aop ) C → Aop .  Chain complex. For any family {Ai }i∈I , we denote by ini : Ai → i∈I Ai the canonical morphisms into the coproduct. The chain complex C∗ ( C, F ) consists of the coproducts  Cn ( C, F ) = F (c0 ), n ≥ 0, and the morphisms ∂n = each

n+1 i=0

c0 →···→cn

(−1)i ∂ni : Cn+1 ( C, F ) → Cn ( C, F ), where ∂ni are defined as satisfying for α

α

αn+1

σ = (c0 →1 c1 →2 · · · → cn+1 ) ∈ Nn+1 C 2290



the relations ∂ni

◦ inσ =

if 1 ≤ i ≤ n + 1

indn+1 σ i

α1

indn+1 σ ◦ F (c0 → c1 ) if i = 0. 0

Let Cn ( C, F ) = 0 for all n < 0. The nth homology object of C with coefficients in F is defined as Hn ( C, F ) = Ker ∂n−1 / Im ∂n . We have that if A is an Abelian category with exact coproducts, then there is an isomorphism colim C F ∼ = Hn ( C, F ) n

that is natural in F ∈ A C . Example 2.1 (Cyclic homology). In [40], Connes introduced a cyclic category Λ with ObΛ = {Λn : n ∈ N} whose morphisms f ∈ Λ(Λn , Λm ) are homotopy classes of continuous maps ϕ : S 1 → S 1 of degree 1 for which ϕ(Zn+1 ) ⊆ Zm+1 . Here S 1 is the set of complex numbers λ ∈ C with |λ| = 1 and Zn ⊂ S 1 are the subgroups of numbers satisfying λn = 1. There is an isomorphism Λ ∼ = Λop [40]. The small category Λ is obtained as a quotient of the category EΛ [41]. The latter has one object (Z, n) for each n and the morphisms f : (Z, n) → (Z, m), f : Z → Z,

f (x + n) = f (x) + m ∀x ∈ Z

are nondecreasing maps (n, m ≥ 1). One has Λ = EΛ/Z for an obvious action of Z by translation. A cyclic object in a category A is a functor Λ → A. Let k be a ring with identity. A k(Λ)-module E k(Λ) is a cyclic object in A = M odk . Then the cyclic homologies are defined by HCn (E) = T orn (∆Λ k, E). n n ∼ Thus, HCn (E) ∼ = colimΛ n E. Connes also studied the modules Ext (∆Λ Z, E) = lim Λ E in [40, IV], but n n cyclic cohomologies are defined as HC (E) = Extk(Λ) (E, ∆Λ k). Spectral sequence of a direct image. Let A be a category. Consider a functor S ∗ : A D → A C that acts on F ∈ A D by S ∗ (F ) = F ◦ S. If A is cocomplete (complete), then the functor S ∗ has a left (right) adjoint functor, which is called a left (right) Kan extension LanS : A C → A D (RanS : A C → A D ) along S. If A is an Abelian category with exact coproducts, then LanS has left satellites LanSq : A C → A D and there are natural in F ∈ A C , d ∈ D, isomorphisms LanS F (d) ∼ = colimS/d (F ◦ Qd ), q

q

where Qd : S/d → C are the forgetful functors [52, Application 2]. Theorem 2.1 ([1]). Let S : C → D be a functor between small categories, and let A be an Abelian category with exact coproducts. Then there exists a (third-quadrant) spectral sequence C colimpD {colimS/d q F ◦ Qd }d∈ D ⇒ colimp+q F.

Remark 2.2. If A is an Abelian category with exact products, we obtain the first-quadrant spectral sequence lim pD {lim qd/S F ◦ Q d }d∈ D ⇒ lim p+q C F,

where Q d : d/S → C is the forgetful functor. Here and in Andre’s theorem, the exactness of products (coproducts)  is essential. For example, let E be a set, and let A be an Abelian category in which the coproduct e∈E : AE → A is not exact. Let C consist of two objects a and b and two morphisms f0 : a → b and f1 : a → b (except for identities 1a and 1b ). Then for the projection S : E × C → E the Andre theorem is not true. We refer to [4] for applications of the Andre spectral sequence in the combinatorics of posets. Spectral sequence of a restriction. Let A be an additive cocomplete category and C a small category. The symbolic tensor functor is an additive bifunctor ⊗ C : Ab C × A C −→ A, op

2291

relative to which there is an isomorphism op A(G ⊗ C F, A) ∼ = Ab C (G, A(F (−), A)),

which is natural in G ∈ Ab C , F ∈ A C , and A ∈ A. The existence of such a natural isomorphism characterizes ⊗ C up to natural isomorphism and shows that ⊗ C preserves the colimit in each variable. If A is an Abelian category with coproducts, we define T orkC (G, F ) = Hk (P∗ ⊗ C F ), where P∗ → G is the op projective resolution of G ∈ Ab C . Thus, T orkC (G, F ) are the left derived functors in the first variable. G is said to be pointwise free if G(c) are free for all c ∈ Ob C. Let A have exact coproducts. Then for each op pointwise free G ∈ Ab C , the functor T orkC (G, −) : A C → A is the kth left satellite of G ⊗ C (−). If G is op op a projective object of Ab C , then G ⊗ C (−) is exact. Hence, for any G ∈ Ab C , the functors T orkC (G, −) compose a homological ∂-functor in the sense of Grothendieck [59]. Applying the Grothendieck spectral op op op op sequence to the composition of LanS : Ab C → Ab D and (−) ⊗ D F : Ab D → A, F ∈ A D , we obtain the following result of Oberst [139]. op

Theorem 2.2. Let S : C → D be a functor between small categories, and let A be an Abelian category with exact coproducts. Then there exists a (third-quadrant) spectral sequence C (F ◦ S), T orpD (LanSq ∆ Cop Z, F ) ⇒ colimp+q op

which is natural in F ∈ A D . Remark 2.3. The symbolic hom functor is an additive bifunctor Hom C (−, =) : Ab C

op

× A C −→ A,

relative to which there is an isomorphism A(A, Hom C (G, F )) ∼ = Ab C (G, A(A, F (−))), which is natural in G ∈ Ab C , F ∈ A C , and A ∈ A. This characterizes Hom C (−, =) up to a natural isomorphism. Let A be an Abelian category with an exact product. For F ∈ A C , we denote by ExtkC (−, F ) the right derived functors of Hom C (−, F ). Then for each functor S : C → D from a small category into a small category D, there exists the first-quadrant spectral sequence ExtpD (LanSq ∆ C Z, F ) ⇒ lim p+q C (F ◦ S). 2.2. Flows on categories. For a (directed) graph Γ, we denote by A(Γ) the set of arrows; let V (Γ) be the set of vertices. Definition 2.4. Let Γ be a graph, R be a ring with 1, and F : W Γ → M odR be a functor from a free category generated by Γ into the category of left R-modules. A flow on Γ with coefficients in F is any family {fγ }γ∈A(Γ) such that the following conditions hold: (1) fγ ∈ F (dom γ), ∀γ ∈ A(Γ); (2) the set {γ ∈ A(Γ) : fγ = 0} is finite;   F (γ)(fγ ) = fγ hold. (3) for all c ∈ V (Γ), the equations Let Φ(Γ, F ) ⊆

c=cod (γ)



c=dom (γ)

F (dom (γ)) be the submodule of flows.

γ∈A(Γ) Γ Proposition 2.3. For every functor F : W Γ → M odR , there is an isomorphism Φ(Γ, F ) ∼ = colimW 1 F.

Let C be a small category, and let F : C → M odR be a functor. There exist a graph Γ and a set of relations R in the sense of [126, §19] for which C is the quotient category of W Γ with respect to R. Let π : W Γ → C be the projection. If v = αm · · · α1 and w = βn · · · β1 are two paths in Γ such that (v, w) ∈ R, then for each f ∈ F ◦ π(dom (w)), we have the flow fα1 = f, fα2 = F (α1 )f , · · · , fαm = F (αm−1 · · · α1 )f ; 2292

fβ1 = −f, fβ2 = −F (β1 )f , · · · , fβn = −F (βn−1 · · · β1 )f. We denote this flow by [f, v, w]. A flow ϕ ∈ Φ(Γ, F ◦ π) is called internal with respect to R if there exist k  (vi , wi ) ∈ R and fi ∈ F ◦ π(dom (vi )), 1 ≤ i ≤ k, such that ϕ = [fi , vi , wi ]. Let I(Γ, R, F ) be the i=1

R-module of all internal flows. Theorem 2.4. Let C be a small category, and let F : C → M odR be a functor. If Γ is a graph, R is a set of relations in Γ, and π : W Γ → C is the canonical projection on the quotient category W Γ/R = C, then Φ(Γ, F ◦ π)/I(Γ, R, F ) ∼ = colim1C F. 2.3. Categories of homological dimension zero. It is true for categories all of whose components are filtered that h.d. R C = 0. The conjecture owing to U. Oberst [140] is that the converse is true when R = Z. Affinization of a small category. Several special cases of Oberst’s conjecture were proved [71], [140]. Let C be a small category. We denote by Z C the universal preadditive category generated by C. Thus, Ob(Z C) = Ob C and Z C(a, b) are free Abelian groups generated by C(a, b) for a, b ∈ Ob C. Note that the notation Z C can be used for the functor Z C(−, =) : Cop × C → Ab as well. For every small category C, we denote by aff C the subcategory (nonadditive) of Cadd consisting of those morphisms, the sum of whose integer coefficients is equal to 1. Theorem 2.5 ([73]). h.d. C = 0 if and only if aff C has filtered components. The theorem says that the above conjecture is true if C is a poset, since in this case aff C = C. Isbell [72] proposed a counterexample to the conjecture. He considered the category ∆f ace of all finite ordinals and order-preserving injections and showed that aff∆f ace is filtered. ∆f ace is clearly not filtered, since all of its morphisms are monomorphisms. Fixed-point property. In [74], Isbell and Mitchell gave details of a work sketched in [73]. Recall that a C-set is a functor C → Ens. Let F be a C-set, and let τ : F → F be a natural transformation; then x : C(c, −) = hc → F is a fixed point for τ if τ ◦ x = x. The small category C has the fixed point property if any endomorphism of an indecomposable C-set has a fixed point. The exactness of colim C implies the fixed-point property, and in some cases, both conditions turn out to be equivalent to the property that the components of C are filtered [74]. Problem 2.5. Is the exactness of colim C equivalent to the fixed-point property? Applications to a flatness. An object G ∈ Ab C is flat if G ⊗ C (−) : Ab C → Ab is exact. For op M ∈ Ens C , we denote by ZM the composition of M with L : Ens → Ab. op

Theorem 2.6 ([30]). If M is a C-set such that M (α) is an injection for all α ∈ M or C, then ZM is flat if and only if aff(h∗ /M ) has filtered components. Oberst’s characterization. Oberst [140] characterized the small categories of homological dimension zero as follows. Let Z C be the universal additive category generated by Z C, and let Z be the extension of the op constant functor ∆ Cop Z : Cop → Ab to the category Z C . Theorem 2.7 ([140]). Let C be a small category. Then the following conditions are equivalent: (1) the functor colim C : Ab C → Ab is exact; (2) for every Abelian category A satisfying (AB5 ), the functor colim C : A C → A is exact; op (3) the category Z C/Z, where Z is considered as the object of Add(Z C , Ab) is filtered. 2.4. Homological dimension one. 2293

Groups of homological dimension one. There are no theorems on the homological dimension one. A group is called locally free if all of its finitely generated subgroups are free. It is easy to see that locally free groups have homological dimension one. In [105], it is shown that if G is a group such that h.d. G = 0, then G = {1}. Problem 2.6 ([105]). Prove that if G is a group of homological dimension one, then G is locally free. Partially ordered sets of homological dimension one. Cheng [29] characterized all partially ordered sets of homological dimension one. Recall that Cn is the n-crown, n ≥ 2. Theorem 2.8. Let C be a poset. Then h.d. R C ≤ 1 if and only if C does not contain Cn as a retract. 2.5. Comparison for homological dimensions. Homological dimension of the group reflecˆ is isomorphic to the quotient of the free group of M tion. Let M be a monoid. The group reflection M by the normal subgroup generated by elements of the form xyz −1 , where xy = z and x, y, z ∈ M . ˆ. Theorem 2.9 ([33]). If M is an Abelian monoid, then h.d. R M = h.d. R M 2.5.1. Comparison with the cohomological dimension. D. M. Latch and B. Mitchell stated the following inequalities relating the homological dimension to the cohomological dimension: Theorem 2.10 ([105]). If the cardinal number of the set of morphisms of C is ℵn , then h.d. R C ≤ cdRop Cop ≤ n + 1 + h.d. R C. This theorem was proved by Latch in [102] under a rather restrictive assumption on C. 3. Cohomology with Coefficients in Natural Systems The object of this section is to study the connection between the cohomologies and the extensions of small categories. We first give the definition of the cohomology H n ( C, F ) introduced by Baues and Wirsching [17]. Then we compare this cohomology with the values of the derived functors of the limit and give interpretations of H n ( C, F ) for n ≤ 3. 3.1. Baues–Wirsching cohomologies and derived functors of limits. Let C be a small category. For each a ∈ Ob C, we identify the morphism 1a with the object a; therefore, Ob C ⊆ M or C. The factorization category C is the category whose objects are all morphisms of C, and the set of morphisms C (f, g) for every f, g ∈ M or C consists of pairs (α, β), α, β ∈ M or C, for which the diagram β

b −→ b ↑f ↑g α a ←− a is commutative. The composition is defined as (α , β ) ◦ (α, β) = (α ◦ α , β ◦ β); in particular, (α, β) = (α, 1) ◦ (1, β) = (1, β) ◦ (α, 1). The functors D : C → Ab are called natural systems on C. Denote β∗ = D(1, β) and α∗ = D(α, 1). Let D : C → Ab be a natural system on C, and let N∗ C be a nerve of C. For every integer n ≥ 0, we define the n-cochain group  D(α1 ◦ α2 ◦ · · · ◦ αn ). C n ( C, D) = α

α

α

1 2 n c0 ←c 1 ←··· ←cn

Considering elements of C n ( C, D) as maps ϕ : Nn C → ∪g∈M or C D(g) with ϕ(α1 , · · · , αn ) ∈ D(α1 ◦ α2 ◦ · · · ◦ αn ), where ϕ(1c ) ∈ D(1c ) for n = 0, we define the coboundary dn : C n ( C, D) → C n+1 ( C, D) for 2294

n > 0 by the formula (dn ϕ)(α1 , · · · , αn+1 ) = D(1, α1 )ϕ(α2 , · · · , αn+1 ) +

n  (−1)i ϕ(α1 , · · · , αi−1 , αi ◦ αi+1 , αi+2 , · · · , αn+1 ) i=1

+ (−1)n+1 D(αn+1 , 1)ϕ(α1 , α2 , · · · , αn ). For n = 0, we define (d0 ϕ)(α) = D(1, α)ϕ(dom α)−D(α, 1)ϕ(cod α). The cohomology groups H n (C ∗ ( C, D)) are called the nth cohomology groups of C with coefficients in the natural system D. These groups are isomorphic to the cohomology groups with coefficients in the diagram D on the factorization category. Theorem 3.1 ([17]). For each natural system F on C and for all n ≥ 0, there are isomorphisms H n ( C, F ) ∼ = lim n
F. C



Let (dom , cod ) : C → C × C be a functor that carries every object α ∈ M or C of the factorization category to the pair (dom α, cod α). The functor (dom , cod ) assigns the morphism (f, g) ∈ M or( Cop × C) to each morphism (f, g) : α → β. The comma-categories (dom , cod )/(a, b) have the trivial nth integral homologies for n > 0, and hence, by Oberst’s theorem [139], we have the following statement. op

Theorem 3.2 ([17]). For every functor F : Cop × C → Ab, there are natural in F isomorphisms H n ( C, F ◦ (dom , cod )) ∼ = Extn (Z C, F ) ∀n ≥ 0. For all a ∈ Ob C, we have Hn (cod /a, Z) = 0 if n > 0 and H0 (cod /a, Z) = Z. Applying the Oberst theorem [141], we obtain the following statement. Theorem 3.3 ([17]). For every functor F : C → Ab, there are natural in F isomorphisms H n ( C, F ◦ cod ) ∼ = lim n F, ∀n ≥ 0. Let S : C → D be a functor between small categories. For any morphism γ : a → b of D, we denote by S < γ > the following category. α

β

The objects of S < γ > are pairs of morphisms a → S(x) → b such that β ◦ α = γ. The morphisms α

β1

α

β2

between a →1 S(x1 ) → b and a →2 S(x2 ) → b are given by f ∈ C(x1 , x2 ) satisfying S(f ) ◦ α1 = α2 and β2 ◦ S(f ) = β1 . It is easy to verify by Oberst’s theorem that the canonical morphisms H n ( D, −) → H n ( C, (−) ◦ S) of ∂-functors are isomorphisms if and only if Hn (N∗ (S < α >)) are zero for all n > 0 and H0 (N∗ (S < α >)) ∼ = Z. Baues and Wirsching [17] proved that H n ( D, −) → H n ( C, (−) ◦ S) are isomorphisms if S is an equivalence of categories. 3.2. Derivations and linear extensions. Derivations. Let C be a small category, and let D be a natural system on C. A derivation [17] ∆ : C → D is a function that carries every morphism f : a → b C to an element ∆(f ) ∈ D(f ) such that ∆(g ◦ f ) = D(1, g)(∆(f )) + D(f, 1)(∆(g)). A derivation is inner [17] if there exists a function ∇ that carries each object a ∈ C to an element ∇(a) ∈ D(1a ) such that ∆(f ) = D(1, f )∇(dom f ) − D(f, 1)∇(cod f ) ∀f ∈ M or C. Let Der( C, D) and Ider( C, D) be the sets of derivations and inner derivations, respectively. These sets are Abelian groups with respect to pointwise addition, and the quotient group is isomorphic to the first cohomology groups ([12, 17]): H 1 ( C, D) ∼ = Der( C, D)/Ider( C, D). 2295

We refer to the Pirashvili work [151] for Unso¨ld’s interpretation of elements in H 1 ( C, D) by equivalence classes of coverings of C with fiber D. Linear extensions. Let D be a natural system on C. We say that p

+

D −→ E −→ C is a linear extension of the category C by D if the following conditions hold. (1) The categories E and C have the same objects, and p is a full functor that is the identity on objects. (2) For each morphism f : a → b in C, the Abelian group D(f ) acts transitively and effectively on the subset p−1 (f ) ⊆ E(a, b). We write (f0 , α) → f0 + α for the action of α ∈ D(f ) on f0 ∈ p−1 (f ). (3) The action satisfies the linear distributivity law (f0 + α) ◦ (g0 + β) = f0 ◦ g0 + f∗ β + g∗ α. +

p

+

p


Two linear extensions D −→ E −→ C and D −→ E −→ C are equivalent if there is an isomorphism of the categories ε : E → E with p ◦ ε = p and ε(f0 + α) = ε(f0 ) + α for f0 ∈ M orE, and α ∈ Dpf0 . The +

p

extension D −→ E −→ C is split if there is a functor σ : C → E with p ◦ σ = 1. Theorem 3.4 ([17]). Let M ( C, D) be the set of equivalence classes of linear extensions of C by D. Then there is a canonical bijection ψ : M ( C, D) ∼ = H 2 ( C, D), which maps the split extension into the zero element. We obtain a representing cocycle ∆t of the cohomology class {E} = ψ(E) ∈ H 2 ( C, D) of an extension p D → E → C as follows. Let t : M or C → M orE be a function that associates with each morphism f ∈ M or C a morphism f0 = t(f ) in E such that p(f0 ) = f . Then t yields a cocycle ∆t by the formula +

t(g ◦ f ) = t(g) ◦ t(f ) + ∆t (g, f ) +

p

with ∆t (g, f ) ∈ D(g ◦ f ). The cohomology class of ∆t is trivial if and only if D → E → C is a split extension. For examples and applications of linear extensions, see [8–10, 14, 152]. Leech’s extensions of monoids. Let M be a monoid considered as a small category with one object. By a diagram of Abelian groups over M , we mean a natural system on M . For every diagram F of Abelian groups over M , the groups H n (M, F ) are the cohomology groups of M in the sense of Leech [111]. The Abelian Leech extension of M by a diagram F over M is a monoid T , together with an epimorphism of monoids p : T → M , such that the following conditions hold: (1) For every u ∈ M , the Abelian group F (u) acts transitively and effectively on the subset σ −1 (u) ⊆ T. We denote this action as (α ∈ F (u), x ∈ σ −1 (u)) → a ∗ x. (2) For all u, v ∈ M , x ∈ σ(u), y ∈ σ −1 (v), a ∈ F (u), and b ∈ F (v), the relations (a ∗ x)u = F (1, v)(a) ∗ (xy) and x(b ∗ y) = F (u, 1)(b) ∗ (xu) hold. Two Abelian extensions σ1 : T1 → M and σ2 : T2 → M are equivalent if there is an isomorphism α : T1 → T2 such that σ1 = σ2 ◦ α and α(a ∗ x) = a ∗ α(x). The equivalence classes of Abelian extensions are isomorphic to H 2 (M, F ) [138]. For any epimorphism σ : T → M of monoids and a natural system F : M → Ab, the Oberst spectral sequence
Extp (Lanσq ∆T
Z, F ) =⇒ lim p+q T
(F ◦ σ ) 2296

yields the exact sequence of Leech [111]


0 → H 1 (M, F ) → H 1 (T, F ◦ σ ) → AbM ({H1 (σ < α >)}α∈M
, F ) → H 2 (M, F ) → H 2 (T, F ◦ σ ). 3.3. 2-categories. Track categories. Let C be a small category, and let D : C → Ab be a natural system on C. Let (E, ×, 1) be a category with finite products. The category enriched by E is the class ObK with the family K(A, B) ∈ ObE for the pairs (A, B) ∈ ObK × ObK and with morphisms of the category E: µA,B,C : K(A, B) × K(B, C) → K(A, C), iA : 1 → K(A, A) such that the following diagrams in E are commutative: µ×1

K(A, B) × K(B, C) × K(C, D) −→ K(A, C) × K(C, D) ↓1×µ ↓µ µ −→ K(A, B) K(A, B) × K(B, D) (iA ,1)

K(A, B) −→ ↓ 1K(A,B)

1K(A,B)

K(A, B) −→ ↑ 1K(A,B) K(A, B)

(1,iB )

−→

K(A, A) × K(A, B) ↓µ K(A, D) ↑µ K(A, B) × K(B, B).

Definition 3.1. A track category denoted by T K or by → p T K −→ C → is a category K enriched by the category of groupoids, together with a full functor p : K → C that is the identity on objects and satisfies the condition p(f ) = p(g) for morphisms f, g of K(A, B) ⇔ T (f, g) = ∅, where T (f, g) denotes the set K(A, B)(f, g) of 2-morphisms f → g. We write H : f , g if H ∈ T (f, g). We call H a track from f to g. The (vertical) composition of 2morphisms is denoted by + : T (f, g)×T (g, h) → T (f, h); let 0 be an identity, and let − : T (f, g) → T (g, f ) be an inverse morphism. The category C is isomorphic to the quotient category of K by the relation f , g ⇔ T (f, g) = ∅. For any G ∈ M orK(A, B) and H ∈ M orK(B, C), we denote by H ∗ G the horizontal composition µA,B,C (G, H). We set f∗ G = 0f ∗ G, g∗ H = H ∗ 0g . For any H : f , f 1 and G : g , g1 , the following distributivity law holds: H ∗ G = f∗ G + (g1 )∗ H = g∗ H + (f 1 )∗ G. A functor between two track categories t : T K → T K is defined as a functor t : K → K with functions tf,g : T (f, g) −→ T (tf, tg) such that t(0) = 0, t(H + G) = (tH) + (tG), t(−H) = −(tH) (these three relations show that t is a groupoid morphism) and t(f∗ H) = (tf )∗ (tH), t(g∗ H) = (tg)∗ (tH). A functor between two track categories induces a functor t : K/ ,→ K / , between the quotient categories. 2297

Let C be a category, and let D be a natural system on C. The linear track extension T K of C by p p + → D, which is denoted by D → T → → K −→ C, is defined by a track category T → K −→ C, functor p, and action of D on T as follows. The action of D on T is given by isomorphisms of groups ∼ =

σf : D(pf ) −→ T (f, f ), f ∈ M orK such that the following diagrams are commutative: σf

D(pf ) −→ T (f, f ) a → σf (a) ↓ g∗ ↓ (pg)∗ ↓ ↓ σf g (a) ◦ g. D(pg, 1)(a) → σ f D(pf pg) −→ T (f g, f g) The morphism g∗ acts by T (f, f ) → T (f g, f g) and is equal to (−) ◦ g. The morphism (pg)∗ : D(pf ) → D(pf pg) is equal to D(pg, 1): σg

D(pg) −→ T (g, g) a → σg (a) ↓ (pf )∗ ↓ f∗ ↓ ↓ σf g D(1, pf )(a) → f ◦ σ g (a). D(pf pg) −→ T (f g, f g) The morphism f∗ acts by T (g, g) → T (f g, f g) and is equal to f ◦ (−). The morphism (pf )∗ : D(pg) → D(pf pg) equals D(1, pf ). For all 1-morphisms f and h and 2-morphisms H : f , h, we consider the functors T (f, H) : T (f, f ) → T (f, h), x → H + x, T (H, h) : T (h, h) → T (f, h), x → x + H. The linear track extension for all f, h, and H ∈ T (f, h) should satisfy the commutativity condition of the diagram σf

−→

D(ph) = D(pf ) ↓ σh

T (f, f ) ↓ (−) + H = T (f, H)

T (H,f )=H+(−)

−→ T (f, h) T (h, h) Consider maps between linear track extensions. Let C be a category, D be a natural system on C and T K and T K be linear track extensions of C by D. A D-equivariant map t : T K → T K over C between linear track extensions is a functor t : T K → T K between the track extension which satisfies p t = p, and tf,f σf = σtf for all f ∈ M orK. Hence all linear track extensions of C by D and D-equivariant maps over C form a category, which is denoted by T rack( C, D). Two objects in this category are equivalent, T K ∼ T K , if there exist two maps T K ←− T K −→ T K in T rack( C, D). The set of equivalence classes π0 T rack( C, D) = Ob(T rack( C, D))/ ∼ is the set of connected components of the category T rack( C, D). Theorem 3.5 ([13, 149]). There is a canonical bijection ∼ H 3 ( C, D). ψ : π0 T rack( C, D) = Example 3.2. The category T op∗ of pointed topological spaces is a track category. Let I be the unit interval, and let IX = I × X/I × {∗} be the reduced cylinder of X ∈ T op∗ . We have the maps (i0 ,i1 )

p

X ∨ X −→ IX −→ X, 2298

where X ∨ X is the one-point union. Here we set it (x) = (t, x) and p(t, x) = x, t ∈ I, x ∈ X. For the maps f, g : X → Y ∈ T op∗ , let T (f, g) = [IX, Y ](f,g) be the set of homotopy classes relative to X ∨ X of the map H : IX → Y with H(i0 , i1 ) = (f, g). An element H ∈ T (f, g) is termed a track H : f , g. This defines the track category → T T op∗ → T op∗ / ,, → which yields the following linear track extensions. Theorem 3.6 ([8]). (A) Let K be a full subcategory of T op∗ such that the objects of K are suspensions. Then there is a natural system DΣ on K/ ,, together with a linear track extension → + DΣ → T K → K/ , . → (B) Let K be a full subcategory of T op∗ such that the objects of K are loop spaces. Then there is a natural system DΩ on K / ,, together with a linear track extension → + DΩ → T K → K / , . → It follows from Theorem 3.5 that these linear track extensions correspond to some cohomological classes < K >Σ ∈ H 3 (K/ ,, DΣ ) and < K >Ω ∈ H 3 (K / ,, DΩ ). These classes are called the universal Toda brackets for K and K . All classical triple Toda brackets < f, g, h > can be obtained by [13] or [8] from the universal Toda bracket < K >Σ . We note that Baues and Dreckmann in [13] considered a more general cohomology theory of categories. Bicategorical interpretation. M. Jibladze [79] constructed another interpretation of H 3 ( C, D). A bicategory B consists of (a) a class Ob B of objects; (b) a family of categories B(X, Y ) where X, Y ∈ Ob B; (c) a family of objects IX ∈ B(X, X) where X ∈ Ob B; (d) a family of functors MX,Y,Z : B(X, Y ) × B(Y, Z)→ B(X, Z), where X, Y, Z ∈ Ob B; (e) families of natural isomorphisms λX,Y : 1 B(X,Y ) → MX,X,Y ◦ (IX × 1 B(X,Y ) ), ρX,Y : 1 B(X,Y ) → MX,Y,Y ◦ (1 B(X,Y ) × IY ), µX,Y,Z,T : MX,Y,T ◦ (1 × MY,Z,T ) → MX,Z,T ◦ (MX,Y,T × 1), which satisfy certain coherence conditions. A homomorphism P : B → C between bicategories assigns to every object X in B an object P X in C and an isomorphism PX : IP X → P (IX ), to every pair X, Y of objects a functor PX,Y : B(X, Y ) → C(P X, P Y ), and to each pair (G : X → Y, F : Y → Z) an isomorphism P (F, G) : (P F )(P G) → P (F G), satisfying coherence conditions. For each set S, we denote by Ad(S) the antidiscrete category with Ob(Ad(S)) = S. 2299

Let C be a small category, and let D be a natural system on C. Consider C as a bicategory for which all C(X, Y ) are discrete. A homomorphism P : B → C of bicategories is called a linear biextension of C by D if (1) P is identical on objects; (2) for a morphism f : X → Y of C, let P −1 (f ) be a full subcategory of B(X, Y ) that is carried by P to f ; then the obvious functor P −1 (f ) −→ Ad(S) has a linear-extension structure by the constant natural system D(f ); (3) moreover, the actions of D(f ) on p−1 (f ) are coherent with the composition law, that is, for every diagram, g


f
−→ Z −→ f



X −→ −→ Y g



g

g

and ϕ : in B and 2-morphisms ψ : → P (ψ) = 1g , and P (ϕ) = 1f the relations

f

→ f for which P (f ) = P (f ) = f , P (g ) = P (g ) = g,

f (b + ψ) = f b + f ψ,

(a + ϕ)g = a g + ϕ g

hold for all a ∈ D(f ), b ∈ D(g). This defines a morphism between biextensions. If there is a morphism between P1 : B 1 → C and P2 : B 2 → C, then P1 and P2 are said to be equivalent. M. Jibladze proved that there is a bijection between H 3 ( C, D) and the equivalence classes of biextensions by D. Iterated linear extensions. Let C be a small category, and let F : C → Ab be a natural system. Pirashvili [150] introduced an n-fold linear extension for n ≥ 1 ∂n−1

γ

p

∂

0 → F → Fn−1 → · · · →2 F1 → E → C of C by F as an exact sequence γ

∂n−1

∂

0 → F → Fn−1 → · · · →2 F1 of natural systems on C, together with a linear extension +

p

Coker∂2 → E → C of C by the natural system Coker∂2 . All of the n-fold linear extensions of C by F form a category whose set of connected components is denoted by M n ( C, F ). Pirashvili proved that M n ( C, F ) ∼ = H n+1 ( C, F ) for all n ≥ 1. 3.4. Baues–Wirsching dimension. The Baues–Wirsching dimension of a small category C is defined by Dim C = c.d. C , where C is the factorization category. Let Σ ⊆ M or( C) be any set of morphisms. In [17], Baues and Wirsching proved that if Dim C ≤ 1, then Dim (Σ−1 C) ≤ 1. Note that Laudal’s and Cheng’s theorems on categories of cohomological dimension 0 characterize the categories with Dim C = 0. If a small category is cancellative, then Dim C = dim C; therefore, the properties of the Baues– Wirsching dimension for such categories will be considered in the next section. 4. Hochschild–Mitchell Dimension of a Small Category The Hochschild–Mitchell dimension has its origin in ring theory. We begin with a generalization of the Hochschild cohomology and its interpretations to algebroids. Then we consider cancellative categories of Hochschild–Mitchell dimension one. Then we describe the dimension of finite posets and totally ordered sets. The end of this section deals with the Jibladze–Pirashvili applications to cohomology of algebraic theories. 2300

4.1. Hochschild dimension of algebroids. Let A be a preadditive category, and let C(A) denote the class of endomorphisms of the identity functor 1A . Then C(A) is a commutative ring with unit (neglecting the fact that it cannot be a set), which is called the center of A. If Λ is a ring considered as the preadditive category, then C(Λ) is isomorphic to the subring of all elements c ∈ Λ such that cλ = λc for all λ ∈ Λ. Let K be a commutative ring. A K-category is a preadditive category A, together with a ring homomorphism K → C(A). Equivalently, a K-category is a category A equipped with a K-module structure on each hom set in such a way that the composition A(A, B) × A(B, C) −→ A(A, C) is K-bilinear. A one-object K-category is simply an associative K-algebra, and, therefore, Mitchell calls a small K-category a K-algebroid. If A and B are K-categories, then a functor F : A → B is a K-functor if A(A, A ) → B(F A, F A ) is a K-module homomorphism for all A and A . Let B A denote the category of all K-functors A → B for a K-category B and a K-algebroid A. The tensor product A ⊗K B of K-algebroids can be formed in an obvious way so as to ensure the isomorphism Add(A ⊗K B, C) ∼ = Add( B, Add(A, C)). An A-module is an additive functor M : A → Ab or, equivalently, a K-functor M : A → M odK . Hochschild dimension. Let A be a K-algebroid. The enveloping algebroid of A is the category Ae = Aop ⊗K A. If M is an Ae -module, then the Hochschild cohomology groups of A with coefficients in M are defined by H n (A, M ) = ExtnAe (A, M ), where A denotes the functor A(−, =) : Aop ⊗K A → Ab (equivalently, the K-functor from Ae into M odK ). The Hochschild dimension dim K A is the projective dimension of A(−, =) in the category Add(Ae , Ab). Let C be a small category, and let K C denote the K-category generated by C. Then dim K K C Cop × C equals the projective dimension of K C(−, =) ∈ M odK . Hence the Hochschild–Mitchell dimension op dim K C is equal to the projective dimension of K C(−, =) ∈ Ab C × C . The Hochschild–Mitchell dimension dim C can be defined by [17] dim K C = sup{n ≥ 0 : lim nC
(−) ◦ (dom , cod ) = 0},


C into M od . where lim nC
are the functors from M odK K

Derivations of K-algebroids. If M is an Ae -module, then a derivation [37] into M is a family of K-module homomorphisms d = da,b : A(a, b) → M (a, b) satisfying d(λµ) = (dλ)µ + λ(dµ). The principal derivations are those of the form dλ = λma − mb λ, λ ∈ A(a, b), for some family ma ∈ M (a, a). The quotient K-module of derivations modulo principal derivations is isomorphic to H 1 (A, M ). Extensions over K-algebroids. Let A be a K-algebroid. Recall that a (2-sided) ideal of A is a subfunctor I ⊆ A. One can then form the quotient algebroid A/I. An extension over A is a Kalgebroid morphism E → A that is identical on objects and such that fa,b : E(a, b) → A(a, b) are injective homomorphisms of K-modules for all a, b ∈ ObA. If M (a, b) are kernels of fa,b , then M is an ideal of E, and we have A ∼ = E/M . The two-sided action of E on M induces a two-sided action of A on M if and 2 only if M = 0. Here M 2 is an ideal generated by m1 m2 with m1 , m2 ∈ M . Denote the inclusion M → E g

f

by g; we then say that the family of short exact sequences of K-modules M → E → A is a singular extension over A with kernel M . Two such extensions (g1 , f1 ) and (g2 , f2 ) with the same kernel M are equivalent if there is a morphism K : E 1 → E 2 of K-algebroids such that kg1 = g2 and f2 k = f1 . It then follows that k is an isomorphism. An A-module is K-projective if M (a) are projective K-modules for all a ∈ ObA. 2301

If A is K-projective as an Ae -module, then the set of equivalence classes of all singular extensions over A with kernel M is equivalent to H 2 (A, M ) [37]. 4.2. Hochschild–Mitchell dimension of weak cancellative categories. It was proved in [17] that c.d. C ≤ dim C ≤ Dim C. For the idempotent monoid E = {1, e} consisting of 2 elements, it was shown in [92] that dim C = 0 but Dim C = 1. Categories of dimension zero. If C is a category without idempotents, except identities, then dim C = 0 if and only if C is equivalent to a discrete category. Two small additive categories C and D are Morita equivalent if categories Add( C, Ab) and Add(D, Ab) are equivalent. Mitchell [128] supposed that dim C = 0 if and only if Z C is Morita equivalent to Z D for some discrete category D. He characterized all small categories of dimension zero in [131]. Cheng has a different characterization for monoids of dimension zero in [28]. We recall that a delta is a small skeletal category all endomorphisms, are of which identities. If C is a delta with dim K C = 0 for a commutative ring K, then C is discrete [126, Sec. 33]. Dimension of deltas. Let C be a delta. If a, b ∈ Ob C and C(a, b) = ∅, we write a ≤ b. A full subcategory of a delta with the object set {x ∈ Ob C : a ≤ x ≤ b} is called a muscle. A delta is weak if each of its muscles has only a finite number of morphisms. Weak deltas with dim K C ≤ 2 are characterized by Mitchell [126]. Mitchell proved in [126] that if a small category C is free, then dim K C ≤ 1. Conjecture 4.1 ([126]). Let C be any delta. If dim K C ≤ 1, then C is a free category. Mitchell raised the conjecture in the case where C is either a weak delta or a poset [126, p. 151]. The conjecture was proved by Cheng in [26]: Theorem 4.1. Let K be a nonzero commutative ring with identity, and let C be a delta. Then dim K C ≤ 1 if and only if C is a free category. Dimension of cancellative categories. Theorem 4.2 ([92]). Let C be a cancellative small category. Then Dim C = dim C. Corollary 4.3. Let C be a cancellative small category, and let G be a subgroupoid of C. Then dim G ≤ dim C. For groups C this statement is known as Shapiro’s lemma. Cheng, Wu, and Mitchell in [37] proved the following statement. Theorem 4.4. If dim K C ≤ 1, then for any Σ ⊆ M or C, the inequality dim K Σ−1 C ≤ 1 holds. It follows that any bridge category C has the dimension dim K C ≤ 1. Theorem 4.4 holds for Kalgebroids A and any subset Σ ⊆ M orA, where A is the K-projective Ae -module. For the cohomological dimension, the Stallings–Swan theorem is not true. But if in Conjecture 1.7, the cohomological dimension is replaced by the Hochschild–Mitchell dimension, then it becomes true. Theorem 4.5 ([91]). Let C be a cancellative category such that dim C ≤ 1. Then C is a bridge category. A category C is weakly cancellative if for every α, β ∈ M or C, the following two conditions hold: (i) αβ = α or βα = α ⇒ β = 1; (ii) αvβ = αβ with an invertible morphism v ⇒ v = 1. It is torsion-free if αn = 1 for α ∈ M or( C) and n > 0 implies α = 1. 2302

Theorem 4.6 ([35]). Suppose C is weakly cancellative and torsion-free. Then dim K C ≤ 1 if and only if C is a bridge category. Problem 4.2 ([128]). Let C be a small category of Hochschild–Mitchell dimension one without idempotents except identities. Is C a partial category of a free category? 4.3. Hochschild–Mitchell dimension of posets. Let C be a poset. Each poset is a delta, and hence dim K C = 0 if and only if C is discrete; dim K C ≤ 1 if and only if C is isomorphic to a free category. Locally finite posets of dimension 2. Using [89, Corollary 4.3], together with the relations dim K C = sup{dim K [a, b] : a, b ∈ C} and gl.dim Ab C = sup{gl.dim Ab[a,b] : a, b ∈ C} [129], we have gl.dim Ab C = 1 + dim C for every locally finite set C. ˆ n for n ≥ 2 consists of 2n + 2 elements {d, e1 , e2 , . . . , en , f1 , f2 , . . . , The (suspended) n-crown [124] C fn , g} ordered by relations d < ei < fi < g for 1 ≤ i ≤ n, ei < fi−1 for 2 ≤ i ≤ n, and e1 < fn . Figure 1 ˆ 2 by inserting x such that e1 < x < f1 and e2 < x < f2 . is obtained from the 2-crown C dr

❅r e2 ❅r x f1 r ❅r f2 ❅r

e1

r

g

d

✏PP r r r Pr en PP r r r PP f1 rP r r r r ✏ Pr fn P❅✏ Pr ✏ ✏ e1 r✏ P r

g

ˆn Fig. 1 n-crown C ˆ It is easy to show that dim C ≥ 3 for any poset C containing Cn as a full subset, n ≥ 3. If C ˆ 2 is a retract in C and dim C ≥ 3 again. ˆ 2 as a full subset but does not contain Fig. 1, then C contains C ˆ n as a full subset for some n ≥ 2 with the above additional We say that C contains a crown if it contains C condition in the case n = 2. Let C be a locally finite poset. Then the inequality dim C ≤ 2 is equivalent to gl.dim Ab C − gl.dim Ab ≤ 2. It follows from [25] that the inequality dim C ≤ 2 holds if and only if C does not contain ˆ n for every n ≥ 2. a suspended crown C Dimension of the product. Theorem 4.7. Let C and D be locally finite sets. Then inequality dim C × D < dim C + dim D is strong if and only if the following two conditions hold: (1) 3 < dim C < ∞, 3 < dim D < ∞; (2) for all a < b in C and x < y in D and for p = dim C − 2 and q = dim D − 2, the groups Hp (]a, b[) and Hq (]x, y[) are zeros and the groups Hp−1(]a, b[) and Hq−1 (]x, y[) do not contain elements with equal order. Mitchell first gave examples when this inequality is strong. Locally finite posets of dimension three. Let C be a locally finite poset, A be a category, and A ∈ A be an object. Define the functor A C : Cop × C → A by A C(a, b) = A if a ≤ b and A C(a, b) = 0 otherwise. ˆ n is the suspended n-crown with n ≥ 2, then pdA C = 3 + pdA. In [124], Mitchell proved that if C = C In [126], he conjectured that this relation is true for each locally finite poset with dim C = 3. It was confirmed in [93]: 2303

Theorem 4.8. Let C be a locally finite poset of Hochschild–Mitchell dimension 3, A be an Abelian category, and A be an object of A. Then pdA C = 3 + pdA. Counterexamples. There exist examples of surjective nondecreasing maps of lattices C1 → C2 such that dim C1 < dim C2 [89]. 4.4. Hochschild–Mitchell dimension of a totally ordered set. Let C be a poset. Then the factorization category C is isomorphic to the set of closed intervals [a, b] = {x ∈ C : a ≤ x ≤ b} ordered by “ ⊆.” Mitchell proved in [126] that if C is a totally ordered set whose closed intervals all have cardinal number at most ℵn , then dim K C ≤ n + 2. In particular, the ordered set of rationals have dimension ≤2. Well-ordered sets. Let ωn be the smallest ordinal of cardinality ℵn . Mitchell [126] computed dim C for all ordinals C excluding ω1 , ..., ωn , . . . . Herman Brune computed dim ωn for all n ∈ N. He proved that dim K ωn = n + 1 [20]. Theorem 4.9 ([129]). Let K be a commutative ring with identity, and let α be an ordinal of cardinality ℵn . Then dim K α = n + 2 if α > ωn and dim K α = n + 1 if α = ωn . The set of rationals includes ω0 + 1 as a retract and, hence, it is of dimension 2. Density. Let R denote the ordered set of real numbers. Mitchell raised the problem of computing dim R. This problem was solved by Stanislaw Balcerzyk in [5]. He has constructed a projective resolution d

d

d

0 → T3 →3 T2 →2 T1 →1 T0 −→ KR(−, =) → 0 ×R and shown that the monomorphism d3 does not split. This gives the following in the category M odR K answer to Mitchell’s question. op

Theorem 4.10. The Hochschild–Mitchell dimension dim K R of the ordered set of real numbers is equal to 3 for any commutative ring K with identity. Mitchell [126] asked if countability characterizes the dimension ≤2? For subsets of reals this was proved in [86]. Theorem 4.11. Let C ⊆ R be a subset of reals such that 2ℵ0 < 2| C| . Then dim C = 3 . In particular, if C ⊆ R and | C| = 2ℵ0 , then dim C = 3. Under the assumption 2ℵ0 < 2ℵ1 , the dimension of every uncountable C ⊆ R equals 3. Problem 4.3. Let C be a totally ordered set with dim C ≤ 2. Prove that |[a, b]| ≤ ℵ0 for all closed intervals [a, b] in C. In [86], it was proved that dim is nondecreasing on the class of totally ordered sets ordered by inclusion. Theorem 4.12. Let C be a totally ordered set. Then dim X ≤ dim C for any subset X ⊆ C. Definition 4.4. Let C be a totally ordered set. A subset I ⊆ C is called half-dense if it has nonempty intersection with any closed interval that contains at least two elements. In [87], the dimension of any totally ordered set C that contains the half-dense subset I ⊆ C such that |I| < | C| was computed. 2304

Theorem 4.13. Let I ⊂ C be a half-dense subset of the totally ordered set, and let | C| > |I| = ℵn for some 1 ≤ n < ∞. If ℵn = 2ℵn−1 and 2ℵn < 2| C| , then dim C = n + 3 and dim I = n + 2. Mitchell tried to compute the dimensions of all totally ordered sets. Conjecture 4.5 (general Mitchell conjecture). Let C be a totally ordered set. Then dim C = n + 2, where ℵn = sup{|[a, b]| : a, b ∈ C}. A lexicographical ordered set Cn = {0, 1}ωn contains the half-dense subset Hn = {ϕ : ωn → {0, 1} : (∃β < ωn ) (ϕ(β) = 1 and ϕ(α) = 0 for α > β)}. Hence, in the case 2ℵn−1 > ℵn the relations dim Cn = n + 3 and dim Hn = n + 2 hold. Choosing the least n for which 2ℵn > ℵn+1 , we obtain dim Cn = n + 3 and |Cn | > ℵn+1 . Therefore, the following corollary, which was first proved in [85], is true. Corollary 4.14. If there exists n such that 2ℵn > ℵn+1 , then the general Mitchell conjecture is not true. In [87], a property of the Hochschild–Mitchell dimension that is equivalent to a set theoretical axiom was found. Theorem 4.15. The following suppositions are equivalent: (CHω ) for all natural n ≥ 0, the relations 2ℵn = ℵn+1 hold; (D) if a totally ordered set C contains a half-dense subset I such that | C| > |I| = ℵn , then dim C = n + 3. Problem 4.6. Is the general Mitchell conjecture equivalent to (CHω )? (By Corollary 4.14 the implication “⇒” is true.) 4.5. Cohomology of algebraic theories. Jibladze and Pirashvili [81] generalized the MacLane cohomology of rings to algebraic theories. Recall that an algebraic theory A is a category with Ob A = {An : n ∈ N}, where An = A1 × · · · × A1 for n ∈ N means the n-fold product of A1 with itself. In particular, A0 is the terminal object in A. An A-algebra in a category A is a functor A → A that preserves finite products. The category of A-algebras and natural transformations in Ens is denoted by Ab . The Yoneda embedding induces a full embedding I A : Aop → Ab whose values I A (An ) are just free algebras generated by n elements. Hence an arbitrary algebraic theory A is equivalent to the opposite of the full subcategory in Ab consisting of finitely generated free A-algebras. For example, if A is the algebraic theory of Abelian groups, then Aop is the category whose objects are 0, Z, Z2 , Z3 , . . . , and morphisms Zm → Zn are given by m × n integral matrices. Let A be an algebraic theory. Denote by Ab( Ab ) the category of A-algebras in Ab. Let U : Ab → Ens be the forgetful functor. Consider the functor Ab( Ab ) → Ab acting as X → U ◦ X. It has a left adjoint (−)ab : Ab → Ab( Ab ). Definition 4.7 ([81]). Let A be an algebraic theory, and let T : Aop → Ab( Ab ) be a functor. The cohomology of A with coefficients in T is defined by H ∗ ( A, T ) = Ext∗ ((I A )ab , T ), where (I A )ab is the composition I

(−)ab

A Ab −→ Ab( Ab ). Aop −→

For a functor T : Aop → Ab( Ab ), denote by T˜ : A × Aop → Ab the functor for which T˜(An , Am ) = T (An )(Am ). 2305

Proposition 4.16 ([81]). Let A be an algebraic theory, and let T : Aop → Ab( Ab ) be a functor. Then there are isomorphisms H ∗ ( A, T ) ∼ = H ∗ ( Aop , T˜), where H n ( Aop , T˜) are the Hochschild–Mitchell cohomologies of Aop with coefficients in T˜. Let T h be a category
of algebraic theories and product-preserving functors that are bijective on objects. Denote Ensω = Ensn , where Ensn = Ens. Consider the functor Ω : T h → Ensω defined n∈N

by Ω( A) = { A(An , A1 )}n∈N . It is known [171] that Ω has some left adjoint L. The theory A is called free if there is an object P in Ensω such that A ∼ = L(P ). In [81], it is proved that if A is a free algebraic theory, then H n ( A, −) = 0 for n ≥ 2. When Ab is the category of R-modules over a ring with identity, the cohomology of algebraic theory is isomorphic to the MacLane cohomology [81] H ∗ (R, M ) ∼ = Ext∗ (I, M ⊗R (−)), where I is the embedding of the category F of free finitely generated left R-modules into M odR and Extn are considered in the category of all functors F → M odR . In [153], the corresponding expression for the MacLane homology was obtained and applied. We refer to [18] for any applications of the homology of functors T : A × Aop → Ab, where A is an algebraic theory of R-modules over a ring R with identity. 5. Global Dimension of the Functor Category This section contains a list of results on extension groups in the functor categories. Starting from a spectral sequence that converges to extension groups of functors, we pass to generalizations of Hilbert’s and Mashke’s theorems. Then we consider generalizations to ringoids of the Quillen–Suslin theorem. We conclude with a description of the global dimension of the functor category on finite posets and on totally ordered sets. 5.1. Global dimension of the module category. Recall that an object A ∈ A is small if hA = A(A, −) preserves coproducts. A is
faithful (or a generator) if hA is faithful. More generally, a family Ai is faithful. Then {Ai }i∈I is called a set of generators. Freyd {Ai }i∈I of Ai ∈ ObA is faithful if i∈I

characterized additive functor categories Add( C, Ab) as Abelian categories with coproducts and faithful set of small projectives. If there exists a ringoid C such that A ∼ = Add( C, Ab), then a category A is called a generalized module category, and its objects are called C-modules. Denote M od C = Add( C, Ab). Let C be a ringoid. The exact inclusion I : M od C → Ab C induces some morphisms In : Extn (T, F ) → G

U

L

Extn (I(T ), I(F )) for all T, F ∈ M od C . For every functor G : C → Ab the composition C → Ab → Ens → Ab is an additive functor. Hence an augmentation LU G → G belong to M od C and induces homomorphisms Extn (I(T ), I(F )) → Extn (LU I(T ), I(F )). In [152], Pirashvili proved that the composition I

n Extn (T, F ) −→ Extn (I(T ), I(F )) → Extn (LU I(T ), I(F ))

is an isomorphism for all n ∈ N, and T, F ∈ M od C . Thus, the extensions in the module category can be reduced to the extensions in the category of all functors. 5.1.1. Extension groups of functors. In [80] and [81], a spectral sequence that converges to extension groups Extn (F, G) for every functor F, G : C → M odR , where R is a ring with identity and C a small category, was constructed. In [84], it was generalized to any Abelian category with enough  projectives. E For a set E, an E-coproduct is a functor A → A that carries any family {Ae }e∈E to e∈E Ae . 2306

Theorem 5.1 ([84]). Let C be a small category, let A be an Abelian category with the Ob C-coproduct and Ob( C/c)-coproducts for all c ∈ Ob C, and let P be a proper class of exact sequences in A in the sense of [112] such that A has enough P -projectives, and for each c ∈ Ob C, the Ob( C/c)-coproduct carries every family of P -epimorphisms to a P -epimorphism. Then for every functor F, G : C → A, there exists a first-quadrant spectral sequence E2p,q = lim pC
{ExtqP (F (dom α), G(cod (α)))} ⇒ Extp+q C P (F, G), where C P is a proper class in A C that consists of all exact sequences 0 → F → F → F → 0 for which sequences 0 → F (c) → F (c) → F (c) → 0 belong to P for all c ∈ Ob C. Using Mitchell’s condition [123] of Ext-preserving, it is proved in [84] that if all categories C/c are finite, then this spectral sequence exists for each Abelian category A and for each proper class P . Global dimension. Let modf p R be the category of finitely presented right R-modules over a ring R with identity. In representation theory as well as other topics of ring theory, it is convenient to consider the categories D(R) = Add(modf p R, Ab) and L(R) = Add((modf p R)op , Ab). Jensen shows in [77] that commutative Noetherian rings R such that gl.dim D(R) ≤ 2 are precisely the Artinian principal ideal rings (or K¨ othe rings). In [23], Brune showed that commutative Noetherian rings R with gl.dim L(R) ≤ 2 are just finite products R = R1 × · · · × Rn of rings, where each Ri is a K¨othe ring or a Dedekind domain. Together with [77, Corollary 1], this result yields, for example, gl.dim D(Z) = 3 and gl.dim L(Z) = 2. In contrast to this example, if R is an Artin algebra, then gl.dim D(R) = 2 if and only if gl.dim L(R) = 2. Recall that a module is pure projective if it is a direct summand of a direct sum of finitely presented modules. In [23], it is proved that if R is a right Noetherian ring with gl.dim L(R) ≤ 2, then every submodule of a pure projective right R-module is pure projective. Pure global dimension. Let C be a ringoid. A tensor product ⊗ : M od C op × M od C → Ab is a functor additive in each argument and defined by the isomorphisms Ab(G ⊗ F, A) ∼ = M od C op (G, hA ◦ F ), which are natural in G ∈ M od C op , F ∈ M od C , and A ∈ Ab, where hA = C(−, A). Recall that a right If there are a finite set I, a family {Ai }i∈I of C-module is a left C op -module. Let F be a left  C-module. A i h → F , then F is said to be finitely generated. If there objects Ai ∈ Ob C, and an epimorphism λ : i∈I

exists an exact sequence F1 → F0 → F → 0 om with finitely generated C-modules F0 and F1 , then F is said to be finitely presented. Following Stenstr¨ [178], an exact sequence 0 → F → F → F → 0 in the category M od C is said to be pure if for each G ∈ M od C op , the induced sequence 0 → G ⊗ F → G ⊗ F → G ⊗ F → 0 is exact. Then an object M ∈ M od C is pure projective if there is a finitely generated C-module P , together with a retraction P → M . The pure projective dimension p.proj.M of a left C-module M is the inf of lengths of pure exact resolutions of M by pure projective left C-modules. The left pure global dimension of a ringoid C is defined by l.p.gl.dim C = sup{p.proj. dim M : M ∈ M od C }. Let r.p.gl.dim C = l.p.gl.dim C op . Let K be a field, and let I be a totally ordered set. We denote by K[I] the ringoid that yields a natural isomorphism M odIK ∼ = M odK[I] of categories. In [22], Brune proved that for a totally ordered set I, the equation l.p.gl.dim K[I] = 0 is true if and only if I is well ordered. Mitchell [126] proved that the notions of left and right semisimplicity generalize to ringoids and coincide. In contrast to this result, Brune proved the following statement. 2307

Theorem 5.2 ([22]). Let K be a field of cardinality ≤ℵ0 , and let n be any ordinal. Then (1) l.p.gl.dim K[ℵn ] = 0; (2) r.p.gl.dim K[ℵn ] = n + 1 if n < ℵ0 ; (3) r.p.gl.dim K[ℵn ] = ∞ if n ≥ ℵ0 . In connection with the pure dimension, C. U. Jensen mentioned the famous (still open) problem whether the left pure global dimension of a ring is zero if and only if its right global dimension is zero. In his book with H. Lenzing [78], similar questions are discussed. 5.2. Categories of dimension one. Note that if C is a monoid, then the category M odRC for any ring R with identity is equivalent to M odR C , where R C is a monoid algebra. Consider the following classical result. Hilbert’s chain-of-syzygies theorem. Let N be the additive monoid of natural numbers. Hilbert’s theorem [62] says that for each field K and an integer m ≥ 1, m

gl.dim M odN K = m, where Nm is the m-fold Cartesian product of N with itself. This can be obtained by induction from the relation gl.dim M odN R = 1 + M odR , which is true for any ring R with identity. Mitchell proved the following generalization. Theorem 5.3 ([128] (generalized syzygy theorem)). Let A be an Abelian category with exact coproducts, and let C be a bridge category that is not equivalent to a discrete category. Then gl.dim A C = 1 + gl.dim A. A converse to the Generalized syzygy theorem is true in the following sense [34].
Let R be a ring with identity, and let M be a monoid. It is clear that if M is of the form ( ni=1 Mi )×H, where Mi are isomorphic to the additive group Z or to the additive monoid N and H is a finite group with |H|R = R, then gl.dim M odM R = n + gl.dim M odR . Theorem 5.4 ([34]). Let K be a commutative Noetherian ring with identity such that gl.dim M odK < ∞. If M is a finitely generated Abelian cancellative monoid with gl.dim M odM K = n + gl.dim M odK , then M  
n is isomorphic to Mi × H, where Mi are Z or N and H is a finite group with |H|K = K. i=1

This computes the global dimension of the functor categories M odM K , where M is a finitely generated Abelian cancellative monoid and K is a Noetherian commutative ring with finite global dimension. Also, this characterizes all such monoids M with dimK M = n. Hereditary deltas. Let R be a ring with identity. A small category C is R-hereditary if gl.dim M odRC ≤ 1. In [35], the problem of characterizing all small categories C such that both C and Cop are R-hereditary, where C is weakly cancellative and torsion free, was solved. This problem is solved by H¨oppner [65] in the case where C is weakly cancellative with all hom-sets finite. Recall that a small category is rigid if xy = x y for some x, y, x , and y implies that either x = x z or x = xz for some morphism z. Theorem 5.5 ([36]). Let C be a nondiscrete delta and let R be a ring with identity. Then C is Rhereditary if and only if the muscles of C are cancellative and rigid with DCC and R is completely reducible (i.e., gl.dim M odR = 0). If, for every sequence of morphisms α1 , α2 , ..., with dom αi = cod αi+1 for all i, there exists k for which all αi with i > k are isomorphisms, then we say that C has a weak DCC. A category C is called locally Artinian if for every c ∈ C, the category c/ C has a weak DCC. 2308

Theorem 5.6 ([65]). For a weakly cancellative category C with all sets C(a, b) finite, the following are equivalent: (1) gl.dim M odRC ≤ 1 + gl.dim M odR for every ring R such that C has no R-torsion; (2) gl.dim M odRC ≤ 1 for some semisimple ring R; (3) C is rigid and locally Artinian. Weak global dimension of the module category. A left module M is flat if the functor (−) ⊗ C M is exact. One defines T ornC (E, M ) in the usual way, using the projective resolution for E or M . The weak (or flat) dimension of E is then given by w.d. E = sup{k : T orkC (E, −) = 0}, or, equivalently, as the length of the shortest flat resolution for E. Then the weak global dimension w.gl.dim M od C is sup of w.d. M , where M ∈ M od C . Brune [21] characterized posets C for which w.gl.dim A C − w.gl.dim A ≤ 1 for all generalized module categories A. A tree is a poset in which all closed intervals are totally ordered. Theorem 5.7 ([21]). The relations w.gl.dim A C = 1 + w.gl.dim A hold for all generalized module categories A if and only if C is a nondiscrete tree. Recall that a ring R with identity is von Newmann if every left R-module is flat. H¨ oppner studied the weak global dimension of the functor category on a weak cancellative delta. Theorem 5.8 ([65]). For any weakly cancellative category C with all monoids C(c, c) finite, the following conditions are equivalent: (1) w.gl.dim M odRC ≤ 1 + w.gl.dim M odR for each ring R such that C has no R-torsion; (2) w.gl.dim M odRC ≤ 1 for some von Newmann regular ring R; (3) C is rigid. 5.3. Global dimension of the group ring. Maschke’s theorem. We now consider one more classical result. Recall that C(A) denotes the center of an Abelian category A. Theorem 5.9. Let A be an Abelian category, and let G be a finite group. If the order of G is invertible in C(A), then gl.dim AG = gl.dim A. Otherwise, gl.dim AG = ∞. Corollary 5.10 (Maschke). If G is a finite group and K is a field whose characteristic does not divide the order of G, then the K-algebra K(G) is semisimple. Global dimension of the G-object category for an Abelian group G. Mitchell proved that if G is a finitely generated Abelian group whose torsion subgroup has order n and A is an Abelian category with exact countable products or coproducts, then gl.dim AG = r(G) + gl.dim A, where n is a unit in C(A) and gl.dim AG = ∞ otherwise. Moreover, the following statement holds. Theorem 5.11 ([122]). Let G be an Abelian group of rank r, and let A have exact products or coproducts indexed by G. Then r + gl.dim A ≤ gl.dim AG . 2309

Furthermore, let P be the set of primes p for which G has a p-torsion. If p is not a unit in C(A) for some prime p ∈ P , then gl.dim AG = ∞. On the other hand, if p is a unit in C(A) for all primes p ∈ P and if G is countable, then gl.dim AG ≤ 1 + r + gl.dim A. In [122], Mitchell asked the following. Is the countability condition necessary in Theorem 5.11? More specifically, does there exist a ring R and an Abelian torsion group G (necessarily noncountable) such that 1 + gl.dim M odR < gl.dim M odG R < ∞? The answer was given by N. H. Chen in [24], and B. Osofsky [147] gave an alternate proof. Theorem 5.12 ([147]). Let G be an Abelian torsion group and |G| = ℵk ≥ ℵ0 . If K is a field such that the order of any element of G is invertible in K, then gl.dim M odG K = k + 1. For example, let G = (Z/pZ)|I| , and let p be a prime number. Let K be a field with charK = p. In the case of 2|I| = ℵk , |I| ≥ ℵ0 , we have gl.dim M odG K = k + 1. Hence, if 2ℵ0 = ℵk and k < ∞, then 1 + gl.dim M odK < gl.dim M odG K = k + 1 < ∞. Consequently, the answers to the Mitchell questions are both positive. 5.4. Projective free ringoids. A ringoid C is projective free if every projective C-module is free. The theorem of Quillen–Suslin says that if C is a finitely generated free commutative monoid and R is a pure integral domain (pid), then R C is projective free. Local ringoids. A ringoid C is local if every representable C(a, −) : C → Ab has a unique maximal ideal. As in the case of rings, C is local if and only if C op is local. Mitchell generalized the Kaplansky theorem and obtained the following statement. Theorem 5.13. A local ringoid is projective free. Corollary 5.14. If K is a division ring and C is a poset, then K C is projective free. Ringoids over posets. Cheng and Mitchell [25, Corollary 3] proved that if a poset C has DCC and if a ring R with identity is projective free, then R C is a projective-free ringoid. They asked: if C is any poset, then is Z C projective free? This problem was solved by H¨ oppner and Lenzing [66]. Theorem 5.15. Let A be a category  of left modules over a nonzero ring R, and let C be a poset. Then Ap hp for some family of objects p ∈ Ob C with each Ap projective every projective in A C is of the form p

in A.

Note that R. Wong in [183] generalized the result of Cheng and Mitchell for DCC categories. Theorem 5.16.  Let pC be a DCC category. Then a functor P : C → M odR is projective if and only if it Ap h , where p runs over a family of objects in C with each Ap projective. is of the form p

Problem 5.1 ([29]). It is not known whether the DCC assumption is necessary in Theorem 5.16. If R is a field, then the DCC assumption can be omitted by [126, p. 89]. 2310

Firoids. A ringoid C has an invariant basis number (ibn) if any two bases for a C-module have the same number of elements. If C has an ibn, then so does C op . If there exists an additive functor from a ringoid C into a ringoid D with ibn, then C has an ibn. In particular, if R is a ring with an ibn and C is any small category, then R C has an ibn. If C has an ibn and if every left ideal is free, then C is called a free left ideal ringoid, or more briefly, a left firoid. C is a right firoid if C op is a left firoid, and it is a two-sided firoid if it is both left and right. It easy to see that a left firoid is a domain (that is, βα = 0 ⇒ α = 0 or β = 0). In particular, if R C is a left firoid, then C is cancellative. Each left firoid is projective free. A complete description of two-sided firoids of the form R C was given by Wong [184]. Theorem 5.17. R C is two-sided firoid if and only if C is a bridge category and R is a division ring, or C is equivalent to a discrete category and R is a fir. Theorem 5.18. If C is a nontrivial monoid, then R C is a two-sided fir (in the sense of Cohn [39]) if and only if R is a division ring and C is partially free. 5.5. Partially ordered sets. Posets of small dimension. Let L be a full subcategory of a small category C, and let L be a complete lattice. Then we can define F : C → L by setting F (c) to be the inf of all members of L greater than or equal to c. The map F is order-preserving, and by fullness is restricted to the identity on L. Consequently gl.dim A C ≥ gl.dim A D for any Abelian category A. This yields the following statement. Theorem 5.19. If C is a discrete poset and A is any Abelian category, then gl.dim A C = gl.dim A. If C is not discrete, then gl.dim A C ≥ 1 + gl.dim A for any Abelian category A. Let 2 be a poset whose elements are 0 and 1 ordered by the relation 0 < 1. Clearly, gl.dim A2×2 = 2 + gl.dim A. Hence if poset C contains 2 × 2, then gl.dim A C ≥ 2 + gl.dim A. Following Mitchell, in this case, we say that C contains contains a square. The Krull dimension of a poset is the sup of the lengths of its chains. A poset C is free as a category if and only if all of its muscles are finite chains and C does not contain a square. Theorem 5.20. If C is a poset whose muscles have finite Krull dimension and if C does not contain a square, then gl.dim A C ≤ 1 + gl.dim A for all Abelian categories A with exact coproducts. A Grothendieck category is an Abelian category with exact direct colimits and generator. Theorem 5.21. [129] Let C be a poset. Then gl.dim A C ≤ 1 + gl.dim A for all Grothendieck categories with enough projectives if and only if all closed intervals of C are well ordered. Furthermore, equality holds if and only if C is not discrete. Note that if gl.dim Ab C ≤ 1 + gl.dim Ab, then C is a tree. Posets containing a suspended crown. It is easy to show that ˆ

gl.dim ACn = 3 + gl.dim A. ˆ n is a finite lattice for n ≥ 3, we have Since C ˆ

gl.dim ACn ≥ 3 + gl.dim A 2311

ˆ n as a full subset, n ≥ 3. If C contains C ˆ 2 as a full subset but does not for any poset C containing C ˆ ˆ contain Fig. 1, then C2 is a retract in C. We say that C contains Cn as a full subset for some n ≥ 2 with the above additional condition in the case n = 2. Theorem 5.22. If C is a poset whose closed intervals have a finite Krull dimension and if C does not contain a crown, then gl.dim A C ≤ 2 + gl.dim A for all Abelian categories A with exact coproducts. On the other hand, if C is any poset containing a crown, then gl.dim A C ≥ 3 + gl.dim A for all Abelian categories A. Totally ordered sets. Brune estimated an upper bound of gl.dim A C , where C is a totally ordered set and A the module category over a commutative Noetherian ring with identity. Let J be a totally ordered set. A subset I ⊆ J is coinitial if for every x ∈ J there exists y ∈ I such that y ≤ x. The coinitiality of J is the inf of the cardinalities of coinitial subsets. Theorem 5.23 ([95]). Let J be a totally ordered set containing at least two elements, and let K be a commutative Noetherian ring with identity. If K is either a Dedekind domain or a local ring, then gl.dim M odJK = n + 2 + gl.dim M odK , where ℵn is the sup of the coinitialities of open subsets U ⊂ J such that U = J. Mitchell’s generalization [126, Corollary 36.12] of the Osofsky statement [145, Corollary 7.5] to the valuation categories yields gl.dim M odJK = n + 2 for each division ring K. Using the Bass theorem [7], Brune [21] proved the inequality gl.dim M odJK ≤ n + 2 + gl.dim M odK for every commutative Noetherian ring K. For the ordered set R of reals we obtain gl.dim M odR Z = 3. This is an answer to the question of Brune [21]. Problem 5.2. Let J be a totally ordered set containing at least two elements and K be a commutative Noetherian ring with identity. Prove that gl.dim M odJK = n + 2 + gl.dim M odK , where ℵn is the sup of the coinitialities of open subsets U ⊂ J with U = J. 5.6. Global dimension of incidence algebras. When a category C has only a finite number of objects, a category Ab C is equivalent to the category M odZ[ C] , where Z[ C] is the incidence ring of C. Hence the relative global dimension of Z[ C] can reduce to the relative global dimension of Ab C . Henceforth, P denotes a proper class of short exact sequences in A in the sense of [112]. Let C be any poset, and let A be an object of A. For each c ∈ C, denote by A[c] : C → A a functor defined as  A if x = c, A[c](x) = 0 otherwise. ˜ n ( C, A) the reduced cohomology of the nerve of C with For any Abelian group A, we denote by H coefficients in A. Let ]a, b[= {x ∈ C : a < x < b}. Let A C be the category of functors C → A. Recall that C P is the proper class of sequences 0 → F → F → F → 0 such that 0 → F (c) → F (c) → F (c) → 0 belong to P for all c ∈ C. 2312

Theorem 5.24 ([96]). Let C be a finite poset. Then for all A, B ∈ A and a < b in C, there exists a first-quadrant spectral sequence of the type ˜ p−2 (]a, b[, Extq (A, B)) ⇒ Extp+q (A[a], B[b]). E2p,q = H P CP If A is the category of left modules over an arbitrary ring K with identity, then we obtain the isomorphisms ˜ n−2 (]a, b[, K) ∼ H = Extn (K[a], K[b]) ∀n ≥ 2. For the fields K these isomorphisms were obtained in [148]. Cibils generalized it to the case where K is any ring with identity. Igusa and Zacharia [70] and Polo [154] applied it to study the extension groups and the global dimension in the category of modules over an incidence ring. We define the relative global dimension gl.dim P A of A with respect to P as the upper bound of n ≥ 0 for which ExtnP (−, =) = 0. Corollary 5.25. Let C be a finite poset and A an Abelian category such that gl.dim P A < ∞. Then gl.dim

CPA

C

= gl.dim P A

if and only if C is discrete. Corollary 5.26. Let C be a finite poset and A an Abelian category such that gl.dim P A < ∞. Then gl.dim

CPA

C

= 1 + gl.dim P A

if and only if C is not discrete and for every pair a < b from C the subset ]a, b[⊆ C is totally ordered. Corollary 5.27. Let C be a finite poset and let gl.dim P A < ∞. Then the following properties of C are equivalent: (1) gl.dim C P A C ≤ 2 + gl.dim P A; ˆ n for n ≥ 3, and if a subset S ⊆ C is isomorphic to C ˆ 2 , then S is contained (2) C does not contain C in Fig. 1. These corollaries generalize the results of Mitchell [123]. Corollary 5.28. Let C be a finite poset, and let q = gl.dim P A < ∞. The strong inequality gl.dim

CPA

C

< dim C + gl.dim P A

is equivalent to the following two conditions: (1) dim C > 3; (2) (∀a, b ∈ C) Hp−2(]a, b[) = 0 for p = dim C, and for all A, B ∈ A, mExtqP (A, B) = ExtqP (A, B), where m is the cardinality of the torsion subgroup of Hp−3 (]a, b[). W. T. Spears [176] constructed the following example of a finite ordered set for which gl.dim A C − gl.dim A, A = M odK , depends on the field K: r

✑◗ ✑ ✁❆ ◗ ✑ ✁ ❆ ◗ ✑ ✁r ◗◗r r ✑ ✟PP✟❆r ✟ ❅ ✟ ❅ P ✟✟ ❅ ✟✟P❅ ✟P ❅✟ ❅ ✟✟ ✟ ✟✟ ❅PP❅ PPr r✟ r✟ ❅ r✟ ❅ ✏r ✦❅ ❛❛ P ✦ ✟r ✏❅ ◗◗ PP❍✟ ✑ ❅ ❛❛ ❍ ✏ ❅ ✑ ✦ P ✏ ✑✦✦ ✟✏ ❍ ◗ P ❛ ❅ ✟ ✏ P✑❍ ✦❅ ✏◗ P ✟❛ ❍ ❅r✏ ✟✏ ❛◗ ❛r✑ ✦✦ P❅ Pr ❍ ◗◗ ✑✑ ◗ ✑ ◗ ✑ ◗✑ r

The Spears counterexample 2313

2 be a cell complex of the division of the subspace Mitchell applied topological methods. Let Dm 2 2 2 {(x, y) : x, y ∈ R, x + y ≤ 1} in R by 2m sectors   kπ (k − 1)π ≤ϕ≤ Sk = (ρ cos ϕ, ρ sin ϕ) : ρ ≤ 1, m m 2lπ where k = 1, 2, · · · , 2m. Identifying every point (cos ϕ, sin ϕ) with the points (cos(ϕ + 2lπ m ), sin(ϕ + m )), where l = 1, 2, · · · , l − 1, we obtain the cell complex whose realization is the pseudoprojective plane. If we then order the cells by inclusion and add an initial element 0 and a terminal element t, we obtain 1 ) be the ring of all rational numbers whose denominators are the m-gem gm of Mitchell [126]. Let Z( m 1 the products of prime factors of m. In [126], Mitchell proved that if A is an Abelian Z( m )-category with exact coproducts, then gl.dim Agm = 3 + gl.dim A,

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