TILTING MODULES OF FINITE PROJECTIVE DIMENSION: SEQUENTIALLY STATIC AND COSTATIC MODULES ALBERTO TONOLO Abstract. In [5], Miyashita introduced tilting modules of finite projective dimension. A tilting module A V of projective dimension less or equal than r furnishes r+1 equivalences between subcategories of A-Mod and End V -Mod: we call static and costatic the modules in A-Mod and End V -Mod, respectively, involved in these equivalences. In this paper we characterize the modules in A-Mod and End V -Mod which have a filtration with static and costatic factors, respectively.
Introduction Tilting modules arose from representation theory of algebras and are known to furnish equivalences between categories of modules. Their definition has its origin in the works of Gel’fand Ponomarev, Brenner and Butler, Happel and Ringel (see [1] for a good reference); since then there have been generalizations in several directions. One of these is the study of tilting modules for arbitrary rings. Let A be an arbitrary associative ring. A left A-module V is said tilting provided it satisfies the following properties: (1) There is an exact sequence 0 → A P1 → A P0 → A V → 0, with P1 and P0 finitely generated projective modules (thus, pd A V ≤ 1). (2) Ext1A (V, V ) = 0. (3) There is an exact sequence 0 → A A → A V1 → A V0 → 0, with V1 and V0 summands of a finite direct sum of copies of V . The main result on tilting modules is essentially due to Brenner and Butler [2]. Stated for a tilting module over a finite dimensional algebra, it has been later generalized by Colby and Fuller [3] to tilting modules over arbitrary associative rings. Theorem of Brenner-Butler Let A V be a tilting module, and B = End A V . Then VB also is a tilting module, and A ∼ = End VB . Moreover there are two category equivalences HomA (V,−)
B Gen(A V ) = Ker(Ext1A (V, −)) − ←− −− −− −→ − Ker(Tor1 (V, −)) VB ⊗−
Ext1A (V,−)
Ker(HomA (V, −)) − ←− −− −− −→ − Ker(VB ⊗ −). TorB 1 (V,−) 1
and
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A. TONOLO
The pairs (Ker(Ext1A (V, −)), Ker(HomA (V, −)))
and
(Ker(VB ⊗−), Ker(TorB 1 (V, −)))
are torsion theories in A-Mod and B-Mod, respectively. Denote by H, H (1) , T and T(1) the functors HomA (V, −), Ext1A (V, −), VB ⊗− and TorB 1 (V, −). We call static the 0-static and the 1-static modules, i.e. the left A-modules in Gen(A V ) = Ker(Ext1A (V, −)) and Ker(HomA (V, −)), respectively. Analogously, we call costatic the 0-costatic and the 1-costatic modules, i.e. the left B-modules in Ker(TorB 1 (V, −)) and Ker(VB ⊗ −), respectively. For each left A-module M (resp. each left B-module N ) we have the following short exact sequence (see [3, Theorem 1.4]) 0 → T H(M ) → M → T(1) H (1) (M ) → 0 (resp. 0 → H (1) T(1) (N ) → N → HT (N ) → 0), where T H(M ) is 0-static and T(1) H (1) (M ) is 1-static (resp. HT (N ) is 0costatic and H (1) T(1) (N ) is 1-costatic). In other words we have the filtrations M ≥ T HM ≥ 0
(resp. N ≥ H (1) T(1) N ≥ 0)
with 1- and 0- static (resp. 0- and 1- costatic) filtration factors. Thus all modules in A-Mod and B-Mod consist sequentially of pieces of static and costatic modules, respectively; in particular each simple left A-module is static, each simple left B-module is costatic. In [5] Yoichi Miyashita introduced tilting modules of finite projective dimension over an arbitrary associative ring A. He proved various fundamental results, in particular a generalization of the Brenner and Butler Theorem. Following his definition, a left A-module A V is said to be a tilting module of projective dimension ≤ r if it satisfies the following three conditions: (1) A V has a projective resolution 0 → A Pr → . . . → A P0 → A V → 0 with each Pi finitely generated. (2) ExtiA (V, V ) = 0, if 1 ≤ i ≤ r. (3) There exists an exact sequence 0 → A A → A V0 → A V 1 → . . . → A V r → 0 with each A Vi summand of a finite direct sum of copies of A V . If r = 0 then A V is a progenerator, if r = 1 then A V is a tilting module. Theorem of Miyashita Let A V be a tilting module of projective dimension ≤ r, and B = End A V . Then VB also is a tilting module of projective dimension ≤ r, and A ∼ = End VB . There are r + 1 category equivalences ExteA (V,−)
KEe (A V ) − ←− −− −− −→ − KTe (VB ), TorB e (V,−)
0≤e≤r
SEQUENTIALLY STATIC AND COSTATIC MODULES
3
where KEe (A V ) = {M ∈ A-Mod : ExtiA (V, M ) = 0, if 0 ≤ i ≤ r and i 6= e} KTe (VB ) = {M ∈ B-Mod : TorB i (V, M ) = 0, if 0 ≤ i ≤ r and i 6= e}. We say e-static the modules in KEe (A V ) and e-costatic the modules in KTe (VB ); the left A-modules in the classes KEe (A V ), e ≥ 0, are called static, the left B-modules in the classes KTe (VB ), e ≥ 0, are called costatic. Increasing from 2 to r + 1 the number of category equivalences, we loose the possibility to interpretate them in terms of torsion theories. Not only, but modules are not more made sequentially of pieces of static or costatic modules: in particular, we have examples of simple modules which are neither static nor costatic (see Example 1.5). Roughly speaking, the main result of this paper characterize the left Amodules (resp. the left B-modules) which are sequentially made of pieces r-, r − 1-,. . . , 1-, 0- static modules (resp. 0-, 1-, . . . , r − 1-, r- costatic modules). Let us be more precise. We say that a left A-module M (a left B-module N ) is sequentially static (resp. sequentially costatic) if for each i 6= j ≥ 0, T(i) H (j) M = 0
(resp. H (i) T(j) M = 0).
This definition is justified by our main result: Theorem (1.10,1.11) Let A V be a tilting module of projective dimension ≤ r, and B = End A V . Then a left A-module M (resp. a left B-module N ) is sequentially static (resp. costatic) if and only if there exists a filtration M = Mr ≥ Mr−1 ≥ Mr−2 ≥ . . . ≥ M0 ≥ M−1 = 0 (resp. there exists a filtration N = N−1 ≥ N0 ≥ N1 ≥ . . . ≥ Nr−1 ≥ Nr = 0) such that Me /Me−1 is e-static (resp. Ne−1 /Ne is e-costatic) for each e ≥ 0. In such a case Me /Me−1 ∼ = T(e) H (e) M (resp. Ne−1 /Ne ∼ = H (e) T(e) N ) for each e ≥ 0. We are pleased to finish the introduction, underlining the striking analogy between our main theorem and the characterization of sequentially CohenMacaulay modules over a commutative Gorenstein ring due to Christian Peskine (see [6, Ch. III, Theorem 2.11], [4, Theorem 1.4]): In fact it is its covariant version, in the more general setting of modules over an arbitrary associative ring, with the tilting module playing the role of the Gorenstein ring and the Cohen-Macaulay condition substituted by its homological characterization. 1. The Main Theorem Let A be an associative ring and A V be a left tilting A-module of projective dimension ≤ r (see Introduction). Denote by B the endomorphism ring End A V . By [5, Theorem 1.5], A VB is a faithfully balanced bimodule and VB is a right tilting module of projective dimension ≤ r.
4
A. TONOLO
Denote by H = H (0) , H (i) , T = T(0) and T(i) the functors HomA (V, −), ExtiA (V, −), VB ⊗ − and TorB i (V, −), respectively. Lemma 1.1. [5, Lemma 1.7] (i) For any injective left A-module I, there is a canonical isomorphism T HI → I, t ⊗ f 7→ f (t). (ii) For any projective left B-module P , there is a canonical isomorphism P → HT P, p 7→ (t 7→ t ⊗ p). (iii) If i ≥ 1, then for all injective left A-module I and all projective left B-module P we have T(i) HI = 0
and
H (i) T P = 0.
We concentrate our attention on left A-modules. Analogous results can be obtained, with the obvious modifications, on the category of left B-modules. Let us fix a left A-module M and an injective resolution 0 → M → I • with differential operator d. We denote by Ji , i ≥ 0, the ith -syzygy of I • , i.e. the kernel of di+1 : Ii+1 → Ii+2 ; J−1 will indicate the module M itself. As usual, we denote by Bi (M ) and Zi (M ), i ≥ 0, the i-boundaries and the i-cycles of the complex 0 → H(I • ) := 0 → HI0 → HI1 → HI2 → . . . . We have the following exact sequences (1) 0 → HM → HI0 → B1 (M ) → 0. (2) 0 → Bi (M ) → Zi (M ) → H (i) M → 0, i ≥ 1. (3) 0 → Zi (M ) → HIi → Bi+1 (M ) → 0, i ≥ 1. Applying the functor T , by Lemma 1.1 we get T1. . . . → 0 → T(i+1) B1 (M ) → T(i) HM → 0 → . . . → 0 → T(2) B1 (M ) → T(1) HM → 0 → (0 →)T(1) B1 (M ) → T HM → T HI0 ∼ = I0 → T B1 (M ) → 0. T2. . . . → T(j) Bi (M ) → T(j) Zi (M ) → T(j) H (i) M → . . . . . . → T(1) H (i) M → T Bi (M ) → T Zi (M ) → T H (i) M → 0. T3. . . . → 0 → T(j+1) Bi+1 (M ) → T(j) Zi (M ) → 0 → . . . → 0 → T(1) Bi+1 (M ) → T Zi (M ) → T HIi ∼ = Ii → T Bi+1 (M ) → 0. From these exact sequences we obtain easily the solid part of the following commutative diagram with exact rows and columns. The dotted arrows are usual factorizations of morphisms, or compositions, or easy consequences.
SEQUENTIALLY STATIC AND COSTATIC MODULES
5
(Diagram (A)) 0
T(1) B1 (M )
0O
T H(M )
wow C1 Pu
T B2 (M ) O
o
PP ' _ _ _ _ _ / _ _ _ _ _ _ / T HI0 ∼ = OI0 0 M O p
p8 8
C2 Nt
T(1) Z1 (M ) / T(1) H (1) M
NN
&
O Oξ O
/ T B1 (M )
/ T HI1 ∼ O = I1 O O '
/ T Z1 (M ) O
/ T H (1) M
/0
/ T(1) B2 (M ) O
/ T(1) Z2 (M )
/ T H (2) M (1)
0
T(2) Z2 (M ) / T(2) H (2) M
0 ... T(i) Zi (M )
...
...
... (i) / T(i−1) Bi (M ) / T(i−1) Zi (M ) / T (i−1) H M
/ T H (i) M (i)
Lemma 1.2. Given the solid part of the commutative diagram 0
0
C
/A
/M
α
0
C
β
0
/B
ϕ
/L
ψ
/L
ϑ
/N
with exact rows and columns, there are unique maps α and β such that the diagram commutes. With these maps the first column is exact; moreover, if ϑ is epic, then so is β. Proof. This follows by diagram chasing. Applying Lemma 1.2 to the commutative triangles from Diagram A T HI0 ∼ = NI0
NNN NNξN NNN N'
T B1 (M )
/ T Z1 (M )
and
T HI0 ∼ = IP0
/ T H(I1 ) ∼ = I1 , O
PPP PPPξ PPP PP'
T Z1 (M )
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A. TONOLO
we get the following exact sequences 0 → C1 → Ker ξ → C2 → 0
η
and 0 → Ker ξ → M → T(1) B2 (M );
if T H (1) M = 0, the morphism η is epic. Since T(i) Zi+1 (M ) ∼ = T(i+1) Bi+2 (M ), i ≥ 1 (see T 3), componing with morphisms from Diagram (A), we obtain the following diagram with exact rows and columns and natural morphisms. It describes how the functors 1Mod , T(i) H (i) , i ≥ 0, are canonically related. (Diagram (B)) T(1) H (2) M O
0o 0
∼ T(2) B3 (M ) / T(1) Z2 (M ) = O
...
0O CO 2 o / Ker ξ O
T HM O
T(1) H (1) M o
T(1) Z1 (M )
/
n6 M ρM nnnn n n nnn
η
/ T(1) B2 (M ) O
T(2) H (2) M O
T(1) B1 (M )
T(2) Z2 (M )
O
0 The map ρM is the counity of the adjunction between the functors T and H. Definition 1.3. We say that a left A-module (resp. a left B-module) is e-static (resp. e-costatic) if it belongs to the class KEe (A V ) = {M ∈ A-Mod : ExtiA (V, M ) = 0, i 6= e} (resp. KTe (VB ) = {N ∈ B-Mod : TorB i (V, N ) = 0, i 6= e}). The e-static and the e-costatic modules, for some e ≥ 0, are called static and costatic modules, respectively. In [5, Theorem 1.16], Miyashita proved that the categories of e-static and that of e-costatic modules “are equivalent under the functors ExteA (V, −) and TorB e (V, −), which are mutually inverse to each other”. Definition 1.4. We say that a left A-module M (a left B-module N ) is sequentially static (costatic) if for each i 6= j ≥ 0, T(i) H (j) M = 0
(H (i) T(j) N = 0).
Why these modules are called sequentially static and costatic will be clear after our Theorems 1.10, 1.11.
/ ...
SEQUENTIALLY STATIC AND COSTATIC MODULES
7
If the projective dimension of the tilting module A T is less or equal than 1, then any left A-module is sequentially static and any left B-module is sequentially costatic ([3, Theorem 1.4]). This is not more true for tilting modules of higher projective dimension: Example 1.5. In this example, k denotes an algebraically closed field, and all rings are finite-dimensional k-algebras given by quivers. If i is a vertex of a quiver, we denote by P (i) the indecomposable projective associated to i, by E(i) the indecomposable injective associated to i, and we denote by S(i) the simple top of P (i), or, equivalently, the simple socle of E(i). a
b
(1) Let A denote the k-algebra given by the quiver 1 → 2 → 3 with relation b ◦ a = 0. It is easy to verify that the minimal injective cogenerator 2 1 ⊕ ⊕ 1 is a tilting left A-module of projective dimension 2. AV = 3 2 c
d
The ring B = End A V is the k-algebra given by the quiver 4 → 5 → 6 with relation d ◦ c = 0. With some calculation, it is possible to see that T(2) H (1) S(2) ∼ = T(2) S(4) ∼ = T(1) S(5) ∼ = S(3) 6= 0. Observe, moreover, that HS(2) ∼ = S(6) 6= 0 and hence the simple module S(2) is not static, since it does not belong to any class KEe (A V ), e = 0, 1, 2. (2) Let R denote the k-algebra given by the quiver a
4
c
2
1=b ==
3
with the relation c ◦ a = 0. It is easy to verify that the minimal injective 2 1 1 cogenerator R V = ⊕ ⊕ ⊕ 1 is a tilting left R-module of projective 4 2 3 dimension 2. The ring S = End R V is the k-algebra given by the quiver d
8
f
7
5=e ==
6
with the relation f ◦ d = 0. With some calculation, it is possible to see that HT(1) S(5) ∼ = HE(3) ∼ = S(6) 6= 0. Observe, moreover, that T(2) S(5) ∼ = T(1) S(7) = S(4) 6= 0; hence the simple left B-module S(5) is not costatic, since it does not belong to any class KTe (VB ), e = 0, 1, 2. Lemma 1.6. Let Q be an injective cogenerator of the category of left Amodules. We denote by V ∗ the left B-module HQ. For each left A-module M and each left B-module N we have:
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A. TONOLO
(1) Im H ⊆ Cogen V ∗ ⊆ Ker T(r) and Im T ⊆ Gen V ⊆ Ker H (r) . (2) If r ≥ 2, T(r−1) HM = 0 and H (r−1) T N = 0. (3) T H (r) M = 0 and HT(r) N = 0. Proof. 1. Let us consider the exact sequences 0 → A M → QX
and B (X) → B N → 0.
Applying the functors H and T we get 0 → HM → V ∗ X
and V (X) → T N → 0.
Therefore HM and T N are respectively cogenerated by V ∗ and generated by V . Let us assume M 0 and N 0 respectively cogenerated by V ∗ and generated by V . Now T(r+1) and H (r+1) are the zero functors; then applying T(r) and H (r) we get 0 → T(r) M 0 → T(r) V ∗ X
and H (r) V (X) → H (r) N 0 → 0.
We conclude since by Lemma 1.1, (iii), T(r) V ∗ X = Tr H(QX ) = 0
and H (r) V (X) = H r T (B (X) ) = 0.
2. Let 0 → M → I • a P • → N → 0 be an injective and a projective resolutions. Applying Tr−1 and H r−1 to the exact sequences 0 → HM → HI0 → B1 (M ) → 0
and 0 → B0 (N ) → T P0 → T N → 0
we have T(r) B1 (M ) → Tr−1 HM → Tr−1 HI0 = 0
and
0 = H (r−1) T P0 → H (r−1) T N → H (r) B0 (N ). Since B1 (M ) is cogenerated by V ∗ and B0 (N ) is generated by V we conclude. 3. Let us prove T H (r) M = 0. Consider an injective resolution 0 → M → • I ; denote by Ji the ith syzygy. It is easy to verify that H (r) M ∼ = H (1) Jr−2 . By [5, Lemma 1.1], Jr−1 belongs to KE0 (V ) and hence HJr−1 belongs to KT0 (V ) (see [5, Lemma 1.8]). From the commutative diagram with exact rows T HIr−1
∼ =
Ir−1
/ T HJr−1 ∼ =
/ Jr−1
(r) M / T H (1) J ∼ r−2 = T H
/0
/0
we get T H (r) M = 0. In the same way we can prove HTr N = 0 for each left B-module N . Lemma 1.7. If T(j+1) H (j) M = 0, for each j ≥ 1, then Ti Zi (M ) = 0 for each i ≥ 1. Moreover T1 B1 (M ) ∼ = T1 Z1 (M ) = 0.
SEQUENTIALLY STATIC AND COSTATIC MODULES
9
Proof. Since Zr (M ) ≤ HIr ∈ Cogen V ∗ , by Lemma 1.6, (i), we have T(r) Zr (M ) = 0. Assume T(j) Zj (M ) = 0, j ≥ 2; from the exact sequence (T2) 0 = T(j+1) H (j) M → T(j) Bj (M ) → T(j) Zj (M ) = 0 we get T(j) Bj (M ) = 0 and hence, by T 3, T(j−1) Zj−1 (M ) = 0. Theorem 1.8. Let M be a left A-module such that T(j) H (j+1) M = 0 = T(i+1) H (i) M for each j ≥ 0 and i ≥ 0. Then there exists a filtration M = Mr ≥ Mr−1 ≥ Mr−2 ≥ . . . ≥ M0 ≥ M−1 = 0 with the filtration factors Mi /Mi−1 isomorphic to T(i) H (i) M , i ≥ 0. Proof. By hypothesis and Lemma 1.7, Diagram B becomes (Diagram (C)) 0O T(r) H (r) M o
0o
T(r−2) Br−1 (M ) o O
T(r−1) H (r−1) M o
0
ηr−2
0O ... o
0O
η3
T(1) H (1) M / Ker ξ O
O
T(3) H (3) M o
η2
O
0
T(2) B3 (M ) o
η
/M
/ T(1) B2 (M ) O
T HM O
T(2) H (2) M
0
0
/0
O
Let us consider now Mr−1 := (ηr−2 ◦ . . . ◦ η2 ◦ η)−1 (T(r−1) H (r−1) M ) Mr−2 := (ηr−3 ◦ . . . ◦ η2 ◦ η)−1 (T(r−2) H (r−2) M ) ... M2 := η
−1
(T(2) H (2) M )
M1 := Ker ξ M0 := T HM. It is easy to prove that in such a way we have constructed a filtration of M satisfying the thesis. Analogously we get
0
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A. TONOLO
Theorem 1.9. Let N be a left B-module such that H (j+1) T(j) N = 0 = H (i) T(i+1) N for each j ≥ 0 and i ≥ 0. Then there exists a filtration N = N−1 ≥ N0 ≥ N1 ≥ . . . ≥ Nr−1 ≥ Nr = 0 with the filtration factors Ni−1 /Ni isomorphic to H (i) T(i) N , i ≥ 0. The above results can be specialized in our main theorems. Theorem 1.10. Let A V be a tilting module of projective dimension ≤ r, and B = End A V . Then a left A-module M is sequentially static if and only if there exist a filtration M = Mr ≥ Mr−1 ≥ Mr−2 ≥ . . . ≥ M0 ≥ M−1 = 0 with e-static filtration factors Me /Me−1 , e ≥ 0. In such a case the filtration factor Me /Me−1 is isomorphic to T(e) H (e) M for each e ≥ 0. Proof. Let M be a sequentially static module. Applying Theorem 1.8 we construct the filtration. Moreover, by hypothesis, Mi /Mi−1 ∼ = T(i) H (i) M belongs to KEi (V ). Conversely, assume there exists a filtration M = Mr ≥ Mr−1 ≥ Mr−2 ≥ . . . ≥ M0 ≥ M−1 = 0 with Mi /Mi−1 in KEi (V ) for each i ≥ 0. Let us consider the short exact sequences 0 → Mi−1 → Mi → Mi /Mi−1 → 0, i ≥ 1. Fix j ≥ 0. First, if j ≥ 1, H (j) Ml = 0 for each l < j: we have H (j) M0 = H (j) T HM = H (j) (M0 /M1 ) = 0. Assume by induction 0 = H (j) Ml−1 ; then we have 0 = H (j) Ml−1 → H (j) Ml → H (j) (Ml /Ml−1 ) = 0, and hence H (j) Ml = 0. Second, applying H (j) we get H (j−1) (Mi /Mi−1 ) → H (j) Mi−1 → H (j) Mi → H (j) (Mi /Mi−1 ) → H (j+1) Mi−1 ; if j = i we obtain H (j) Mj ∼ = H (j) (Mj /Mj−1 ), if j < i we obtain H (j) Mi−1 ∼ = H (j) Mi . Therefore, for each j ≥ 0, we have H (j) M = H (j) Mr ∼ = H (j) (Mj /Mj−1 ). = ... ∼ = H (j) Mj ∼ Since Mj /Mj−1 belongs to KEj (V ), by [5, Theorem 1.14] the module H (j) M ∼ = (j) (j) H (Mj /Mj−1 ) belongs to KTj (V ) and hence T(i) H M = 0 if i 6= j: then M is sequentially static. Analogously we can prove
SEQUENTIALLY STATIC AND COSTATIC MODULES
11
Theorem 1.11. Let A V be a tilting module of projective dimension ≤ r, and B = End A V . Then a left B-module N is sequentially costatic if and only if there exist a filtration N = N−1 ≥ N0 ≥ N1 ≥ . . . ≥ Nr−1 ≥ Nr = 0 with e-costatic filtration factors Ne−1 /Ne , e ≥ 0. In such a case the filtration factor Ne−1 /Ne is isomorphic to H (e) T(e) N for each e ≥ 0. There is a striking analogy between our Theorems 1.10, 1.11 and the characterization of sequentially Cohen-Macaulay modules over a commutative Gorenstein ring due to Christian Peskine (see [6, Ch. III, Theorem 2.11], [4, Theorem 1.4]): in fact they are its covariant version in the more general setting of modules over an arbitrary associative ring, with the tilting module playing the role of the Gorenstein ring and the Cohen-Macaulay condition substituted by its homological characterization (see e.g. [6, Theorems 6.3, 12.3]). References [1] I. Assem. Tilting theory – an introduction, Topics in algebra, Part 1. Banach Center Publ. 26, Part 1. PWN, Warsaw 1990. [2] S. Brenner and M. Butler. Generalizations of the Bernstein-Gelfand-Ponomariev reflection functors, in “Proc. ICRA II, Ottawa 1979,” pp.103–169, LNM 832, Springer Verlag, Berlin, 1981. [3] R. R. Colby and K. R. Fuller. Tilting, cotilting and serially tilted rings, Comm. Algebra 18, (1990), 1585–1615. [4] J. Herzog and E. Sbarra. Sequentially Cohen-Macaulay modules and local cohomology, preprint. [5] Y. Miyashita. Tilting modules of finite projective dimension, Math. Z. 193, (1986), 113–146. [6] R. P. Stanley. Combinatorics and Commutative Algebra, 2nd ed., Progress in Mathematics 41, Boston, Birkh¨ auser, 1996. [7] A. Tonolo. Weakly cotilting modules of finite injective dimension, preprint. ` (Alberto Tonolo) Dipartimento di Matematica Pura ed Applicata, Universita di Padova, via Belzoni 7, I-35131 Padova - Italy E-mail address, Alberto Tonolo:
[email protected]
