RELATIVE INJECTIVE RESOLUTIONS VIA TRUNCATIONS, PART II

RELATIVE INJECTIVE RESOLUTIONS VIA TRUNCATIONS, PART II ´ ´ OME ˆ WOJCIECH CHACHOLSKI, AMNON NEEMAN, WOLFGANG PITSCH, AND JER SCHERER

Abstract. In the first part of this paper we reinterpreted the by now classical result of Spaltenstein or B¨ okstedt and Neeman of the construction of injective resolutions for unbounded chain complexes: There is a model category of towers of bounded chain complexes which forms a model approximation for unbounded chain complexes. We now wonder in this part whether this point of view can be relativized, i.e. if the same category of towers forms a model approximation for unbounded complexes, but where one changes the weak equivalences (formerly quasi-isomorphisms) to W-equivalences, where W is a class of injective modules. We prove that this approach is valid for any injective class if and only if we work over a Noetherian ring of finite Krull dimension.

Introduction The idea to alter the choice of projective or injective objects to construct resolutions, and hence to do “relative” homological algebra, goes back at least to Adamson [1] for group cohomology and Chevalley-Eilenberg [6] for Lie algebra homology. This was subsumed in a general theory by Hochschild, [10], but the most complete reference for the classical point of view is probably the work of Eilenberg and Moore, [9]. More recently Christensen and Hovey [7] have studied the projective case from the point of view of Quillen’s homotopical algebra. They show that, in many cases, one can equip the category of unbounded chain complexes with a model category structure where the weak equivalences reflect the choice of the projective objects. However this approach is not suitable in the injective case, because of the lack of a dual small object argument. In this article we investigate how far the point of view exposed in the first part [5] of this work can be pushed. Our aim is thus to do relative homological algebra for unbounded chain complexes by constructing a suitable model approximation. Recall that in the absolute case the approximation is given by a category of towers of bounded chain complexes. In the relative setting it is easy to provide the category of cohomological chain complexes concentrated in positive degrees with a model category structure where the weak equivalences are now determined by a class W of modules. This induces in turn a model structure on the category of towers of chain complexes Tow(R, W). 2000 Mathematics Subject Classification. Primary 18G25, 55U15; Secondary 13D09, 13D45, 18G55, 55U35. Key words and phrases. Relative resolutions, model categories, model approximations, Krull dimension, local cohomology, Noetherian rings. The first author was supported by VR grant 2009-6102 and the G. S. Magnuson Foundation, the third and fourth authors were supported by FEDER/MEC grant MTM2007-61545. This research has been supported by the G¨ oran Gustafssons Stiftelse and the Mathematical Science Institute at the ANU. 1

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´ ´ OME ˆ WOJCIECH CHACHOLSKI, AMNON NEEMAN, WOLFGANG PITSCH, AND JER SCHERER

Truncating a chain complex on the left further and further naturally gives rise to such a tower of complexes, and taking the degreewise limit of a tower produces an (a priori) unbounded chain complex. These two constructions provide a pair of adjoint functors. The question is whether the category of towers still approximates the category of unbounded chain complexes, that is whether the pair of adjoint functors between unbounded chain complexes and towers of bounded chain complexes allows us to do homotopical algebra. What kind of classes of modules are suitable to do relative homological algebra? This is the subject of the companion paper [4] where we mostly focus on the nicest classes, i.e. those consisting of injective modules. We showed that these classes form a lattice and proved that they are determined by certain “saturated” sets of ideals. Equipped with this knowledge about what injective classes look like we come back to our objective and prove that our approach does work for any injective class W of injective modules over a Noetherian ring of finite Krull dimension. Theorem 3.1. Let R be a Noetherian ring of finite Krull dimension and W an injective class of injective modules. The category of towers Tow(R, W) forms then a model approximation for Ch(R) equipped with the W-equivalences. However, in the infinite dimensional case, there is always an injective class for which the category of towers does not form a model approximation. Theorem 4.3. Let R be a Noetherian ring of infinite Krull dimension. There exists an injective class W of injective modules such that the category of towers Tow(R, W) does not form a model approximation of Ch(R) equipped with the W-equivalences. More precisely we exhibit a complex X ∈ Ch(R) such that the process of truncating on the left, resolving each of the resulting bounded below complexes in the W-relative setting, and gluing back the resolutions does not give a relative resolution of the complex X. A typical example of a ring for which this fails is Nagata’s well known “bad ring”, [14]. Acknowledgments. We would like to thank Michel van den Bergh for enlightening comments. The last author would like to thank the Mathematics department at the Universitat Aut´onoma de Barcelona for providing terrific conditions for a sabbatical. We would like to also thank the generous support of the G¨ oran Gustafssons Stiftelse. 1. Relative equivalences for chain complexes Let us briefly say how we will do relative homological algebra, where relative means that we change the usual injective modules for some other collection of modules. We start with a reminder from [4] about what an injective class is and then indicate how this determines a new notion of equivalence. From now on we denote by Ch(R) the category of cohomological chain complexes, i.e. the differential raises degrees by one.

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Definition 1.1. Let W be a collection of R-modules. A homomorphism f : M → N is a Wmonomorphism if, for any W ∈ W, f ∗ : HomR (N, W ) → HomR (M, W ) is a surjection of sets. We say that R-Mod has enough W-injectives if, for any object M , there is a W-monomorphism M → W with W ∈ W. Definition 1.2. A collection W of R-modules is an injective class if W is closed under retracts and products, and R-Mod has enough W-injectives. In [4] we proved that injective classes consisting of injective modules are in bijection with certain sets of ideals in R, which we called saturated. The precise definition is given in [4, Definition 2.1], but in this article it will be sufficient for us to know that it is designed in such a way that the class of retracts of products of injective envelopes of quotient modules R/I (for ideals I belonging to a generization closed subset of Spec(R)) forms an injective class. We start now by explaining how a choice of an injective class leads to relative weak equivalences between chain complexes. Throughout the rest of the paper W denotes an injective class of R-modules. Definition 1.3. A morphism f : X → Y in Ch(R) is called a W-weak equivalence if, for any W ∈ W, HomR (f, W ) : HomR (Y, W ) → HomR (X, W ) is a quasi-isomorphism of homological complexes of abelian groups. An object X in Ch(R) is called W-trivial if X → 0 is a W-weak equivalence, i.e. when HomR (X, W ) is an acyclic complex of abelian groups for each W ∈ W. It is clear that W-weak equivalences satisfy the “2 out of 3” property and that isomorphisms are W-weak equivalences. Furthermore the following basic properties hold: Proposition 1.4.

(1) A homotopy equivalence in Ch(R) is a W-weak equivalence. In partic-

ular, a contractible chain complex in Ch(R) is W-trivial. (2) A morphism f : X → Y in Ch(R) is a W-weak equivalence if and only if the cone Cone(f ) is W-trivial. (3) Coproducts of W-weak equivalences are W-weak equivalences. In particular, coproducts of W-trivial complexes are W-trivial. Proof. (1): (2):

This is a consequence of the fact that HomR (−, W ) is an additive functor.

Note that the cone of HomR (f, W ) : HomR (Y, W ) → HomR (X, W ) is isomorphic to the shift

of the complex HomR (Cone(f ), W ), for any W ∈ W. Thus HomR (f, W ) is a quasi-isomorphism if and only if HomR (Cone(f ), W ) is acyclic. Statement (2) follows. (3):

This is a consequence of two facts. First is that HomR (−, W ) takes coproducts in R-Mod

into products of abelian groups and second that products of quasi-isomorphisms of chain complexes of abelian groups are quasi-isomorphisms.

´ ´ OME ˆ WOJCIECH CHACHOLSKI, AMNON NEEMAN, WOLFGANG PITSCH, AND JER SCHERER

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Definition 1.5. Let W be an R-module. The disc in dimension i relative to W is the complex Di (W ): / 0

···

/ W

W

/ 0

/ ···

where the first copy of W is in degree i. Proposition 1.6.

(1) X is W-trivial if and only if, for any i, di : coker(di−1 ) → X i+1 is a

W-monomorphism. (2) Let X be a complex such that, for all i, coker(di−1 ) ∈ W. Then X is W-trivial if and only L if X is isomorphic to a complex of the form i Di (Wi ) with Wi ∈ W. Proof. (1):

The kernel of HomR (di−1 , W ) is given by HomR (coker(di−1 ), W ). Thus the homology

of HomR (X, W ) is trivial if and only if the morphism HomR (X i+1 , W ) → HomR (coker(di−1 ), W ) induced by di is an epimorphism for all i. By definition, this happens if and only if the morphism di : coker(di−1 ) → X i+1 is a W-monomorphism. (2):

If X can be expressed as

L

Di (Wi ), then X is contractible and according to Proposition 1.4(1)

it is W-trivial. Assume now that X is W-trivial. Define Wi := coker(di−1 ). According to (1), the morphism di : coker(di−1 ) → X i+1 is a W-monomorphism. As coker(di−1 ) is assumed to belong to W, it follows that the morphism di : coker(di−1 ) → X i+1 has a retraction. This retraction can be used to define a morphism of chain complexes X → Di (Wi ). By assembling these morphisms L i together we get the desired isomorphism X → D (Wi ). Example 1.7. If W consists of all R-modules, then f : X → Y is a W-weak equivalence if and only if it is a homotopy equivalence. A chain complex is W-trivial if and only if it is isomorphic to L i D(Mi ) for some sequence of modules Mi . The collection W of all injective R-modules is an injective class. A morphism f : X → Y in Ch(R) is a W-weak equivalence if and only if it is a quasi-isomorphism. A chain complex is W-trivial if and only if it has trivial homology. Let l : S-Mod  R-Mod : r be a pair of adjoint functors and W be an injective class of R-modules. According to [4, Proposition 1.4(5)], the collection J of retracts of objects of the form r(W ), for W ∈ W, is an injective class in S-Mod. By applying l and r degree-wise, we get an induced pair of adjoint functors, denoted by the same symbols: l : Ch(S)  Ch(R) : r. A morphism f : X → Y in Ch(S) is a J -weak equivalence if and only if l(f ) : l(X) → l(Y ) is a W-weak equivalence in Ch(R). Example 1.8. Let R be a commutative ring and L be a saturated set of ideals in R, as defined in [4]. Consider the injective class E(L) that consists of retracts of products of injective envelopes E(R/I) for I ∈ L. A morphism f : X → Y in Ch(R) is an E(L)-weak equivalence if and only if Hom(H n (f ), E(R/I)) is a bijection for any n and I ∈ L. This happens if and only if the annihilator

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of any element in ker(H n (f )) or coker(H n (f )) is not included in any ideal that belongs to L, that is, if and only if the associated primes of these modules do not belong to L. Our starting point for studying unbounded chain complexes is the following fundamental result about bounded chain complexes. It is the relative analogue of the classical result of Quillen, [16], which was the starting point for our reinterpretation of the classical case in [5]. One could prove both of these results by constructing explicitly factorizations, but we prefer to simply refer to the general work of Bousfield [3, Section 4.4]. As in [5] we denote by Ch(R)≥n the full subcategory of Ch(R) consisting of chain complexes concentrated in degrees ≥ n. Theorem 1.9. Let W be an injective class of R-modules. The following choice of weak equivalences, cofibrations and fibrations in Ch(R)≥n satisfies the axioms of a model category: • f : X → Y is called a W-weak equivalence if f ∗ : HomR (Y, W ) → HomR (X, W ) is a quasi-isomorphism of complexes of abelian groups for any W ∈ W. • f : X → Y is called a W-cofibration if f i : X i → Y i is a W-monomorphism for all i > n. • f : X → Y is called a W-fibration if f i : X i → Y i has a section and its kernel belongs to W for all i ≥ n. We will often write X → W (X) for a fibrant replacement in this model category, to emphasize the fact that the modules W (X)k all belong to the class W. Here are some basic properties of this model structure on Ch(R)≥n which follow directly from the definition. Remark 1.10. All objects in Ch(R)≥n are W-cofibrant, but a chain complex X is W-fibrant if and only if X i ∈ W for all i ≥ n. More generally, if f : X → Y is a W-fibration, then Ker(f ) is fibrant. Finally a W-fibration f : X → Y is a W-weak equivalence if and only if Ker(f ) is W-trivial. Proposition 1.11.

(1) If f : X → Y is an acyclic W-fibration, then there is an isomorphism

α : Y ⊕ Ker(f ) → X for which the following diagram commutes: α

Y ⊕ Ker(f ) JJ JJ pr JJ JJ JJ $

/ X  f     

Y

(2) A W-fibrant object in Ch(R)≥n is W-trivial if and only if it is isomorphic to a complex of L the form i≥n Di (Wi ) with Wi ∈ W. (3) A W-weak equivalence between W-fibrant chain complexes in Ch(R)≥n is a homotopy equivalence. Proof. (1):

If f : X → Y is an acyclic W-fibration, then because all objects in Ch(R)≥n are

W-cofibrant, there is a morphism s : Y → X for which f s = idY .

´ ´ OME ˆ WOJCIECH CHACHOLSKI, AMNON NEEMAN, WOLFGANG PITSCH, AND JER SCHERER

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(2):

According to Proposition 1.6.(2) we need to show that the assumptions X is W-fibrant and

W-trivial imply that, for all i, Wi := coker(di−1 ) belongs to W. We do it by induction on i. First, coker(dn−1 ) = X n , which belongs to W as X is W-fibrant. Assume now that Wi ∈ W. This combined with the fact that di : Wi → X i+1 is a W-monomorphism (X is W-trivial), implies that di has a retraction. It follows that X i splits as Wi+1 ⊕ Wi . Consequently Wi+1 , as a retract of an element of W, also belongs to W. (3):

This is an immediate consequence of (2).

As a straightforward consequence we see that (inverse) limits behave well with W-fibrancy. Corollary 1.12. Assume that the following is a sequence of W-fibrations and W-weak equivalences in Ch(R)≥n : f2

f1

(· · · X2 −→ X1 −→ X0 ) Then, for any k ≥ 0, the projection morphism limi≥0 Xi → Xk is a W-fibration and a W-weak equivalence. In particular products of complexes which are both W-fibrant and W-trivial are Wtrivial. Just like in the classical case (where weak equivalences are quasi-isomorphisms), we have a model category of towers with relative equivalences. Consider as in [5] the sequence of model categories {Ch(R)≥−n }n≥0 , but this time with the relative model structures given by Theorem 1.9. The inclusion functor in : Ch(R)≥−n ⊂ Ch(R)≥−n−1 takes (acyclic) fibrations to (acyclic) fibrations and hence the following is a sequence of Quillen pairs, where τn is truncation in degree −n: {τn : Ch(R)≥−n−1  Ch(R)≥−n : in}n≥0 We will denote this tower of model categories by T (R, W) and use the symbol Tow(R, W) to denote the category of towers in T (R, W) (see [5, Section 2]). The objects X• are sequences of bounded chain complexes Xn ∈ Ch(R)≥−n together with structure morphisms tn : Xn → in(Xn−1 ) for n ≥ 1. 2. A right Quillen pair for Ch(R) In this section we define a right Quillen pair for Ch(R) that has the potential to form a model approximation. We are going to use the model category Tow(R, W) described in the previous section. Recall also from [5, Lemma 3.7] that there is a “tower” functor tow : Ch(R) → Tow(R, W) which is left adjoint to the limit functor. We are almost ready to prove that this pair of adjoint functors forms a Quillen pair. Lemma 2.1. Let K• ∈ Tow(R, W) be a fibrant object such that, for any n ≥ 0, Kn is W-trivial in Ch(R)≥−n . Then lim(K• ) is W-trivial in Ch(R).

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Proof. Since K• is fibrant in Tow(R, W), K0 is W-fibrant in Ch(R)≥0 and, for n > 0, the structure morphism tn : Kn → Kn−1 is a W-fibration in Ch(R)≥−n . As the bounded chain complexes Kn are W-trivial, the W-fibrations tn are also W-weak equivalences. It then follows from Remark 1.10 that K• is isomorphic to the following tower of chain complexes: pr

pr

pr

· · · → M0 ⊕ M1 ⊕ M2 ⊕ M3 −→ M0 ⊕ M1 ⊕ M2 −→ M0 ⊕ M1 −→ M0 where M0 := K0 and, for n > 0, Mn := Ker(tn : Kn → Kn−1 ). Hence lim(K• ) =

Q

n≥0

Mn .

The fact that Mn is W-trivial and W-fibrant in Ch(R)≥−n implies that Mn is isomorphic L i to i≥−n D (Wn,i ) for some sequence {Wn,i }i≥−n of objects in W (see Proposition 1.11.(2)). Substituting this to the above product describing lim(K• ) we get the following isomorphisms: lim(K• ) ∼ =

Y n≥0

Mn =

Y M

∞ M

Di (Wn,i ) ∼ =

i=−∞

n≥0 i≥−n

Di (

Y

Wn,i )

n≥−i

It is now clear that lim(K• ) is W-trivial. In fact lim(K• ) is even homotopy equivalent to the zero chain complex. Proposition 2.2. The functors tow : Ch(R)  Tow(R, W) : lim form a right Quillen pair for Ch(R) with W-weak equivalences as weak equivalences. Proof. If f : X → Y is a W-weak equivalence in Ch(R), then tow(f ) is a weak equivalence in Tow(R, W) because truncation is a left adjoint. We have thus only to show that, if f• : X• → Y• is a weak equivalence in Tow(R, W) between fibrant objects, then lim(f• ) is a W-weak equivalence in Ch(R). By Ken Brown’s Lemma (see for example [8, Lemma 9.9]), it is enough to show the statement under the additional assumption that f• : X• → Y• is a W-fibration. Let us define K• to be an object in Tow(R, W) given by the sequence {Ker(fn )}n≥0 with the structure morphisms being the restrictions of the structure morphisms of X• . Since all objects in Ch(R)≥−n are W-cofibrant, then so are all objects in Tow(R, W). It follows that there is s• : Y• → X• for which f• s• = id. Applying the functor lim, we then get the following split exact sequence in Ch(R): lim(f• )

0 → lim(K• ) → lim(X• ) −−−−→ lim(Y• ) → 0 Since X• is isomorphic to K• ⊕ Y• , the object K• , as a retract of the fibrant object X• , is also fibrant. Moreover, for any n ≥ 0, Kn is W-trivial in Ch(R)≥−n . This is a consequence of the fact that fn is a W-equivalence in Ch(R)≥−n . We can then apply Lemma 2.1 to conclude that lim(K• ) is a W-trivial chain complex in Ch(R). Thus the morphism lim(f• ) : lim(X• ) → lim(Y• ) must be a W-weak equivalence. It is in fact a homotopy equivalence. There is one question which still remains to be answered: When does this right Quillen pair form a model approximation, or in other words when is it possible to form relative injective resolutions

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´ ´ OME ˆ WOJCIECH CHACHOLSKI, AMNON NEEMAN, WOLFGANG PITSCH, AND JER SCHERER

by the procedure encoded in our pair of adjoint functors? This is the subject of the final part of the article. 3. Noetherian rings with finite Krull dimension In this section we prove our main positive result. If R is a Noetherian ring of finite Krull dimension, then, for all choices of injective classes W consisting of injective modules, the model category of towers always provides a model approximation for the category of unbounded chain complexes with W-equivalences. Theorem 3.1. Let R be a Noetherian ring of finite Krull dimension and W an injective class of injective modules. Then the category of towers forms a model approximation for Ch(R) equipped with W-equivalences. We need some preparation before proving this theorem. The key ingredient is the vanishing of the homology of a W-relative resolution above the Krull dimension of the ring. To make the reading easier we have included in the short Appendix A a few results about local cohomology which will be used here. The p-torsion submodule functor Γp (−) is defined in Definition A.4. Lemma 3.2. Let R be a Noetherian ring, let p be any prime ideal, and let I ∈ Ch(R)≥0 be a complex of injective R–modules. Denote by I(p) the complex Γp (I ⊗ Rp ). Then (1) The complex I(p) can be identified with the subquotient complex of I where, in each module ∼ ⊕E(R/q), we delete all the terms E(R/q) for q 6= p. In = (2) If I is an injective resolution of a module M , then H ∗ (I(p)) = 0 for ∗ > height(p). Proof. By Lemma A.2, I ⊗ Rp is just the quotient complex of I where we delete E(R/q) unless q ⊂ p. Now Lemma A.6 tells us that Γp (I ⊗ Rp ) is the subcomplex of I ⊗ Rp where we only keep the copies of E(R/p); this proves (1). If I is a resolution of M , then by construction I(p) is a complex that computes the local cohomology of Mp as an Rp -module. As the Krull dimension of the local ring Rp is height(p), point (2) follows from Proposition A.8 and Remark A.9. Recall from [4, Corollary 3.9] that one has a good understanding of injective classes of injective modules. They all correspond to a generization closed subset S of Spec(R) and are given by retracts of products of injective envelopes of R/p for prime ideals p ∈ S. Proposition 3.3. Let R be a Noetherian ring of Krull dimension d < ∞ and let W an injective class of injective modules. For any module M and any W-relative resolution W ∈ Ch(R)≥0 of M , we have that H k (W ) = 0 if k > d − 1. Proof. If S be the generization closed subset of Spec(R) corresponding to the injective class W, we define a = a(S) to be the length of the maximal chain of prime ideals in the complement of S. If a = 0 then W consists of all injective modules, so that H k (W ) = 0 for all k > 0.

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We prove the proposition by induction on a; by the preceding paragraph we know the proposition for a = 0, let us therefore assume that a(S) = a ≥ 1, and that the proposition is true for sets S 0 with a(S 0 ) < a. Let T be the set of primes minimal in Spec(R) − S, and let S 0 = S ∪ T . Then S 0 is generization closed, and we let W 0 be the corresponding injective class; since a(S 0 ) < a the proposition is true for W 0 . Let W 0 be a W 0 -relative resolution of M and consider the short exact sequence of complexes

0

/

M

Γp W 0

/ W0

/ W

/ 0

p∈T

That is we define W to be the quotient of W 0 by the direct sum of the torsion subcomplexes Γp W 0 . It is easy to check that W is a W-resolution of M . Induction tells us that H k (W 0 ) vanishes if k > d − 1, and Lemma 3.2 guarantees that H k (Γp W 0 ) vanishes if k > height(p), so it certainly must vanish if k > d ≥ height(p). The inductive step now follows from the long exact sequence in cohomology. Here is a last proposition we will apply in the proof of our main theorem to measure the difference between the resolutions of a bounded complex and a truncation. Recall that W (X) denotes the fibrant replacement of the bounded complex X in the W-relative model structure described in Theorem 1.9, i.e. a W-relative injective resolution of X. Proposition 3.4. Let R be a Noetherian ring with finite Krull dimension d and W an injective class of injective modules. Let X ∈ Ch(R)≥−1 be a bounded complex and τ0 X its zeroth truncation. Then the canonical morphism X → τ0 X induces isomorphisms in cohomology H k (W (X)) → H k (W (τ0 X)) for any k ≥ d − 1. Proof. Let us replace X → τ0 X by a W-fibration W (X) → W (τ0 X) between W-fibrant objects. The kernel K is a chain complex of injective modules in W, and forms therefore a W-fibrant replacement for H −1 (X), the kernel of the canonical morphism. From the previous proposition we know that H k (K) = 0 if k > d − 2. The long exact sequence in cohomology now gives the proposition. Proof of Theorem 3.1. To show that the Quillen pair is in fact a model approximation, we must check that Condition (4) of [5, Definition 1.1] holds, or equivalently that the canonical morphism X → lim W (towX) is a W-equivalence for any unbounded chain complex X. We have learned from Proposition 3.4 that the cohomology of W (τn X) and W (τn−1 X) only differ in low degrees (in degrees < −n + d). This means that the cohomology of the W-fibrant replacement of the tower tow(X) stabilizes. Therefore H k (limW (towX)) ∼ = H k (W (τr X)) whenever r  0.

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4. Noetherian rings of infinite Krull dimension The objective of this section is to show that even under the Noetherian assumption towers do not always approximate unbounded chain complexes. We have seen in the previous section that no problems arise when the Krull dimension is finite. However when the Krull dimension is infinite it is always possible to find a problematic injective class. More precisely, we will give a class W of injective modules and an unbounded complex X which is not approximated by the fibrant replacement of its truncation tower. Let us first briefly recall the typical example of a Noetherian ring with infinite Krull dimension, as constructed by Nagata in the appendix of [14] (see also [2, Exercise 11.4]). Example 4.1. Let k be a field and consider the polynomial ring on countably many variables A = k[x1 , x2 , . . . ]. Choose prime ideals p1 = (x1 , x2 ), p2 = (x3 , x4 , x5 ), p3 = (x6 , x7 , x8 , x9 ), etc., such that the height of pi is precisely i + 1. Take S to be the multiplicative set consisting of elements of A which are not in any of the pi ’s. The localized ring R = S −1 A is Noetherian, but of infinite Krull dimension. In fact its maximal ideals are precisely the ideals mi = S −1 pi , a sequence of ideals of strictly increasing height. As in any (commutative) Noetherian ring a maximal ideal has finite height, in a Noetherian ring with infinite Krull dimension there is a sequence of maximal ideals of strictly increasing height (see [13, Theorem 13.5]). For the rest of this section let us fix a Noetherian ring R of infinite Krull dimension and a sequence (mi )i≥1 of maximal ideals of strictly increasing height. We will denote the height of mi by hi . Without loss of generality we may assume that h1 > 1. Consider the specialization closed subset of Spec(R) consisting of the mi ’s. According to the classification in [15] this determines a smashing localizing subcategory for D(R), namely the one generated by ⊕R/mi . As noticed in our previous paper [4, Remark 3.10], from the point of view of relative resolutions this means that we will do relative homological algebra with respect to the injective class W of injective R-modules generated by the injective envelopes E(R/p) for all prime ideals p 6= mi for all i ≥ 1. More concretely a complex is W-trivial if and only if the only associated primes to its cohomology modules are among our set {mi }i≥1 , compare with Example 1.8. For any integer n and any R-module M , the complex Σn M consists of the module M in degree −n and zero everywhere else. Our bad complex will be M X= Σhi −1 (Rmi ). i≥1

To understand the fibrant replacement of tow(X) it is enough to understand the W-relative resolutions of the modules Rmi . Lemma 4.2. Let j ≥ 1 and I(Rmj ) be a W-injective resolution of the module Rmj . Then k+1 H k (I(Rmj )) = Hm (Rmj ). In particular H hj −1 (I(Rmj )) is a non-zero mj -torsion module. j

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Proof. Let us consider a minimal injective resolution I of the R-module Rmj . Then I is a complex of Rmj -modules, and we have a short exact sequence / Γmj I

0

/ I

/ I0

/ 0

where I 0 = I/Γmj I is a W-injective resolution of Rmj . We note that H k (Γmj I) is by definition k equal to Hm (Rmj ), and H k (I) = 0 if k ≥ 1. The long exact sequence in cohomology gives us j k+1 (Rmj ) for all k ≥ 1. Now mj has height hj ≥ 2, meaning that an isomorphism H k (I 0 ) ∼ = Hm j the dimension of the module Rm is hj ≥ 2, and [11, Theorem 9.3] tells us that H hj −1 (I 0 ) ∼ = j

h Hmjj (Rmj )

6= 0.

Let us now consider the tower approximation of the complex X. By definition it is the limit of the tower given by the W-relative resolution of the truncations of X. From the previous lemma the nth level of this tower is M

Σhi −1 I(Rmi )

i such that hi ≤n+1

and the structure maps are the projections. The complex X has been constructed so that the last non-zero cohomology module of these resolutions accumulate in degree 0. Therefore the limit of Q the tower is the product complex i Σhi −1 I(Rmi ) and its 0-th cohomology module is isomorphic Q hi to the product i Hm (Rmi ). i Theorem 4.3. Let R be a Noetherian ring of infinite Krull dimension. Let (mi )i≥1 be a sequence of maximal ideals of strictly increasing height, and assume that height(m1 ) > 1. Let W be the injective class generated by the injective envelopes E(R/p) for all prime ideals p 6= mi for all i ≥ 1. Then the category of towers Tow(R, W) does not form a model approximation for the category of unbounded chain complexes. Proof. By construction the complex X has no cohomology in degree 0, and we have computed above the cohomology of the limit Y of the fibrant replacement of its truncation tower: The homotopy Q hi fiber of the natural map X → Y is a complex whose cohomology in degree 1 is i Hm (Rmi ). Let i us show that this complex is not W-trivial. As we have seen in Example 1.8 it is enough to prove Q hi that i Hm (Rmi ) has an associated prime which is not one of the primes mi . More concretely, it i Q hi Q` suffices to produce an element a ∈ i Hm (Rmi ) not annihilated by any finite product i=1 mN i . i hi Since Hm (Rmi ) is mi -torsion, the ideal mi is one of its associated primes. By choosing for each i i hi an element in Hm (Rmi ) with annihilator mi we produce an injection i

Y i

R/mi →

Y

hi Hm (Rmi ). i

i

Q Let a = ∆(1) be the image of 1 ∈ R by the diagonal map ∆ : R → i R/mi . Since the ideal Q` N i=1 mi does not lie in m`+1 , it clearly cannot annihilate the ` + 1 component of a = ∆(1).

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Appendix A. Local cohomology In this section R is a commutative Noetherian ring. We recall a few well-known facts about localization, injective envelopes, and local cohomology. For a prime ideal p, we denote by Mp the localization of an R-module M at p. The first lemma will allow us to reduce certain problems to the case of a local ring, namely Rp . Lemma A.1. An R-module M is zero if and only if Mp is zero for all prime ideals p. Proof. Let us assume that M is non-zero, but Mp = 0 for any prime ideal p. We choose a non-zero element x ∈ M and consider its annihilator. This ideal is contained in a maximal ideal m and since Mm = 0, there must exist an element r ∈ R \ m such that rx = 0, a contradiction. A theorem of Matlis, [12], describes the injective modules as direct sums of injective hulls E(R/p) of quotients of the ring by prime ideals. The following two lemmas give some properties of these indecomposable injective modules. Lemma A.2. If q ⊂ p, the module E(R/q) is p-local, and if q * p E(R/q)p = 0. Proof. Assume q ⊂ p and fix r 6∈ p. The multiplication by r on E(R/q) is an isomorphism, so E(R/q) is p-local. Assume now that q 6⊂ p. Then q m 6⊂ p for any m ≥ 1. If x is any element of E(R/q), its annihilator is q m for some positive integer m since E(R/q) is q-torsion. There exists thus an element s ∈ q m which does not belong to p and such that sx = 0. Hence xp = 0. This shows that E(R/q)p = 0. Lemma A.3. The R-module of homomorphisms HomR (E(R/p), E(R/q)) is non-zero if and only if p ⊂ q. Proof. Since E(R/q) is q-local by the previous lemma, any homomorphism factors through the q-localization of E(R/p), which is zero unless p ⊂ q. Let us now introduce local cohomology, a subject which at least one of the authors enjoyed learning in [11]. Definition A.4. Given an ideal p in R, the p-torsion of an R-module M is the submodule Γp (M ) of elements annihilated by pm for some positive integer m. The local cohomology modules Hp∗ (−) with support in p are the right derived functors of Γp . One of the basic properties of local cohomology is that it is indeed “concentrated” at the chosen ideal. Lemma A.5. Let M be an R-module and p an ideal. Then for each k ≥ 0, Hpk (M ) is p-torsion.

RELATIVE INJECTIVE RESOLUTIONS VIA TRUNCATIONS, PART II

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Explicitly, to compute the local cohomology of a module M , we construct an injective resolution I of M and compute Hpj (M ) = H j (Γp (I)). Our last lemma helps us to understand what this p-torsion injective complex look like. Lemma A.6. The p-torsion module Γp (E(R/q)) is equal to E(R/q) if p ⊂ q and is zero otherwise. Proof. Again this follows from the fact that E(R/q) is q-torsion and q-local. Remark A.7. Let R be a local ring with maximal ideal m and let us consider the generization closed subset of Spec(R) given by S = {q | q 6= m}. It yields the injective class W generated by all injective envelopes E(R/q) with q 6= m. Given a module M and an injective resolution I• , we have a triangle in the derived category Γm (I) → I → W , where W is a W-relative injective resolution of M . In particular H k (W ) ∼ = H k+1 (M ) for k ≥ 2. m

Proposition A.8. Let R be a Noetherian ring and p be the radical of the ideal (x1 , . . . , xn ). Then Hpk (M ) = 0 for any k > n and any module M . Proof. Since the torsion functor does not see the difference between an ideal and its radical, we can assume that p = (x1 , . . . , xn ). Then the local cohomology can be computed by means of the ˇ ˇ i , R) ⊗ M , [11, Theorem 7.13]. Here C(x, ˇ R) is the complex 0 → R → Rx → 0 Cech complex ⊗i C(x ˇ concentrated in degrees 0 and 1. The Cech complex is thus concentrated in degrees ≤ n. Remark A.9. If R is a Noetherian local ring of Krull dimension d, then the maximal ideal can always be expressed as the radical of an ideal generated by d elements, see [11, Theorem 1.17].

References 1. I. T. Adamson, Cohomology theory for non-normal subgroups and non-normal fields, Proc. Glasgow Math. Assoc. 2 (1954), 66–76. 2. M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. 3. A. K. Bousfield, Cosimplicial resolutions and homotopy spectral sequences in model categories, Geom. Topol. 7 (2003), 1001–1053 (electronic). 4. W. Chach´ olski, W. Pitsch, and J. Scherer, Injective classes of modules, Preprint (2009), 11 pages. 5.

, Relative injective resolutions via truncations, part I, Preprint (2009), 10 pages.

6. C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124. 7. J. D. Christensen and M. Hovey, Quillen model structures for relative homological algebra, Math. Proc. Cambridge Philos. Soc. 133 (2002), no. 2, 261–293. 8. W. G. Dwyer and J. Spali´ nski, Homotopy theories and model categories, Handbook of algebraic topology, NorthHolland, Amsterdam, 1995, pp. 73–126. 9. S. Eilenberg and J. C. Moore, Foundations of relative homological algebra, Mem. Amer. Math. Soc. No. 55 (1965), 39.

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10. G. Hochschild, Relative homological algebra, Trans. Amer. Math. Soc. 82 (1956), 246–269. 11. S. B. Iyengar, G. J. Leuschke, A. Leykin, C. Miller, E. Miller, A. K. Singh, and U. Walther, Twenty-four hours of local cohomology, Graduate Studies in Mathematics, vol. 87, American Mathematical Society, Providence, RI, 2007. 12. E. Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528. 13. H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986, Translated from the Japanese by M. Reid. 14. M. Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. 15. A. Neeman, The chromatic tower for D(R), Topology 31 (1992), no. 3, 519–532, With an appendix by Marcel B¨ okstedt. 16. D. G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin, 1967.

Wojciech Chach´ olski

Wolfgang Pitsch and J´erˆ ome Scherer

Department of Mathematics, KTH

Departament de Matem` atiques, UAB

10044 Stockholm, Sweden

08193 Bellaterra, Spain

E-mail: [email protected]

E-mail: [email protected]

Amnon Neeman

Last author’s current address

Centre for Mathematics and its

MATHGEOM, Station 8, EPFL

Applications, ANU

1015 Lausanne, Switzerland

Canberra, ACT 0200, Australia

E-mail: [email protected]

E-mail: [email protected]