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A construction of Gorenstein-projective modules
Journal of Algebra 323 (2010) 1802–1812
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Journal of Algebra www.elsevier.com/locate/jalgebra
A construction of Gorenstein-projective modules Zhi-Wei Li, Pu Zhang ∗,1 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, PR China
a r t i c l e
i n f o
Article history: Received 30 September 2009 Available online 8 January 2010 Communicated by Luchezar L. Avramov Keywords: Gorenstein-projective module Finite Cohen–Macaulay type Gorenstein algebra
a b s t r a c t We determine all the Gorenstein-projective modules over the T 2 A M extension of a Gorenstein algebra, and over , where A and 0 B
B are self-injective algebras, and M is an A-B bimodule with A M and M B projective. Using this, we obtain a criterion for the CMfiniteness of the T 2 -extensions of CM-finite Gorenstein algebras. As an application we get non-trivial examples of CM-finite Gorenstein algebras. © 2010 Elsevier Inc. All rights reserved.
1. Introduction and main results 1.1. Throughout A is an Artin algebra, and all modules are finitely generated. Let A-mod be the category of left A-modules, and P ( A ) the full subcategory of A-mod of projective A-modules. An d0
A-module M is Gorenstein-projective, if there is an exact sequence · · · −→ P −1 −→ P 0 −−→ P 1 −→ · · · in P ( A ), which stays exact under Hom A (−, A ), and such that M ∼ = Ker d0 . Denote by A-G proj the full subcategory of A-mod of Gorenstein-projective modules. Auslander and Bridger [AB] introduced these modules under the name modules of G-dimension zero; Enochs and Jenda [EJ1] defined them for arbitrary modules (not necessarily finitely generated) over a ring. They are also called totally reflexive modules (see Avramov and Martsinkovsky [AM]); in a commutative noetherian Gorenstein local ring, they are exactly the maximal Cohen–Macaulay modules. Gorenstein-projective modules are particular important to the Gorenstein algebras (see [EJ2]). An Iwanaga–Gorenstein ring, or simply a Gorenstein ring R, is a left and right noetherian ring with inj.dim R R < ∞ and inj.dim R R < ∞. A Gorenstein ring R is n-Gorenstein if inj.dim R R n. In this case inj.dim R R n (see [I]). An algebra A is Gorenstein if it is a Gorenstein ring. An additive full subcategory X of A-mod, which is closed under direct summands, is of finite type if X has only finitely many isomorphism classes of indecomposable objects. In particular, if A-G proj
* 1
Corresponding author. E-mail addresses: [email protected] (Z.-W. Li), [email protected] (P. Zhang). Supported by the NSF of China (10725104), and STCSM (09XD1402500).
0021-8693/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2009.12.030
Z.-W. Li, P. Zhang / Journal of Algebra 323 (2010) 1802–1812
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is of finite type, then A is said to be of finite Cohen–Macaulay type, or simply, CM-finite. Clearly, A is CM-finite if and only if there is a module E such that A-G proj = add( E ), where add( E ) is the full subcategory of A-mod of direct summands of finite direct sums of copies of E. 1.2. Gorenstein-projective modules receive a lot of attention: they form the basis of Gorenstein homological algebra (see e.g. [CFH,EJ2,Hol]), they play an important role in the Tate cohomology of algebras (see e.g. [AM,J]), they are widely used in the theory of singularities, as well as stable categories (see e.g. [B,BGS,BR,CPST,Hap,K]). We are interested in constructing explicitly all Gorenstein-projective modules, and in investigating criteria for the CM-finiteness of a given algebra. Such information will be helpful to the understanding of these modules, and to their various applications. We also believe that in the future work the representation theory of Artin algebra and Gorenstein homological algebra may have more interactions. 1.3. Given an A-B-bimodule M, consider the upper triangular matrix Artin algebra Λ = with multiplication given by the one of matrices. A Λ-module is identified with a triple
X
A
A MB , 0 B X , or Y φ
if φ is clear, where X is an A-module, Y is a B-module, and φ : M ⊗ B Y −→ X is an simply Y A-map. We refer to Auslander, the representation theory of Λ. In A A Reiten and Smalø [ARS, p. 73] for X particular, let T 2 ( A ) = . Then a T 2 ( A )-module is a triple , where φ : Y −→ X is in A-mod. φ 0 A
Y
Note that A is n-Gorenstein if and only if T 2 ( A ) is (n + 1)-Gorenstein ([FGR]; also [Hap]). The following result inductively determines all the Gorenstein-projective modules over the T 2 extension of a Gorenstein algebra, and over the upper triangular matrix Artin algebra Λ as above where A and B are self-injective algebras, and M is an A-B bimodule with A M and M B projective. Theorem 1.1. (i) Let A be a Gorenstein Artin algebra. Then a T 2 ( A )-module
X Y φ
is Gorenstein-projective if and only if X ,
Y and Coker φ are Gorenstein-projective A-modules and φ : Y −→ X is injective. A A MB X with A and B self-injective, and A M and M B projective. Then a Λ-module is (ii) Let Λ = φ 0
B
Gorenstein-projective if and only if φ : M ⊗ B Y −→ X is injective.
Y
1.4. Let Γ be an algebra. Denote by Ω(Γ ) the full subcategory of Γ -mod of torsionless Γ -modules (i.e., submodules of projective modules), and by P 1 Ω(Γ ) the full subcategory of Γ -mod consisting of torsionless Γ -modules of projective dimension at most one. Inspired by Auslander’s idea on the representation type of T 2 -extensions [A, Chapter IV, 3], and using Theorem 1.1(i) we get the following criterion for the CM-finiteness of the T 2 -extensions of CMfinite Gorenstein algebras. Theorem 1.2. Let A be a CM-finite Gorenstein algebra, with A-G proj = add( E ) and Γ = End A ( E )op . If P 1 Ω(Γ ) is of finite type then T 2 ( A ) is CM-finite. Furthermore, if inj.dim A A 1, then T 2 ( A ) is CM-finite if and only if Ω(Γ ) is of finite type. For general descriptions of CM-finiteness of Gorenstein algebras we refer to [C] and [LZ]; for the geometric motivations of CM-finiteness in commutative local case, we refer to [K,BGS,CPST]; one can find various properties of CM-finite Gorenstein algebras in [B] and [GZ]. As an application, by using results of Dlab and Ringel in [DR], we get “non-trivial” examples of CM-finite Gorenstein algebras A. Here “non-trivial” means A itself is representation-infinite and gl.dim A = ∞ (otherwise, A is of course CM-finite). Corollary-Example 1.3. Let k be an algebraically closed field, and n = 4 or 5. Then T 2 (k[x]/xn ) is a representation-infinite, CM-finite Gorenstein algebra of infinite global dimension.
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Proof. R n = k[x]/xn is a representation-finite self-injective algebra, so it is CM-finite Gorenstein with R n -G proj = R n -mod. Let Γ be the Auslander algebra of R n . By Lemma 7.1 and Proposition 7.2 in [DR] Ω(Γ ) is of finite type if and only if n 5; and Γ is representation-finite if and only if n 3. So, if n = 4 or 5, then T 2 (k[x]/xn ) is CM-finite by Theorem 1.2; and it is representation-infinite by Proposition 5.8 in [ARS, p. 216]. 2 2. Proof of Theorem 1.1 2.1. By the definition of a Gorenstein-projective module, we know A-G proj ⊆ ⊥ A, where ⊥ A = { M ∈ A-mod | ExtiA ( M , A ) = 0, ∀i 1}. One has Lemma 2.1. (See Enochs and Jenda [EJ2, Corollary 11.5.3].) If A is a Gorenstein algebra then A-G proj = ⊥ A. In the rest of this section, A and B are algebras, M = A M B is an A-B-bimodule, such that Λ = A2.2. M is an Artin algebra. In particular, if A = B and A M B = A A A , then we get Λ = T 2 ( A ). Note that a 0 B f Λ-map is identified with a pair g : YX φ −→ X φ , where f : X −→ X is an A-map, g : Y −→ Y Y P is a B-map, such that f φ = φ (id M ⊗ g ). All of the indecomposable projective Λ-modules are M⊗B Q 00 I and , and all of the indecomposable injective Λ-modules are Hom ( M , I ) ϕ and J , id Q M⊗B Q
A
I
where P and Q range over indecomposable projective A-modules and indecomposable projective Bmodules, respectively, I and J range over indecomposable injective A-modules and indecomposable injective B-modules, respectively, and ϕ I (m ⊗ f ) = f (m), ∀m ∈ M, f ∈ Hom A ( M , I ). We will identify A ⊗ A Q with Q , and Hom A ( A , I ) with I . 2.3. Let A be a Gorenstein algebra with a minimal injective resolution of d r −1
d0
AA
0 −→ A − → I 0 −→ I 1 −→ · · · −−→ I r −→ 0. Then the projective T 2 ( A )-module
A A
(1)
has a minimal injective resolution d r −1
d0
I d0 I 1 d r −1 0 −→ A −−→ I 0 − −−→ I −→ · · · −−−−→ II r −→ 0. A 0
r
1
Using this minimal injective resolution, by a routine computation and Lemma 2.1, one sees Lemma 2.2. Let A be a Gorenstein algebra. Then
ExtiT 2 ( A ) In particular, if
X Y φ
X A , A = ExtiA ( X , A ), Y φ
∀ i 0.
is a Gorenstein-projective T 2 ( A )-module, then X is a Gorenstein-projective A-module.
2.4. Let A be a Gorenstein algebra with a minimal injective resolution (1) of that (see also Happel [Hap])
0 −→
A 0
0 −− →
∂r −1
−−→
I0 I0
∂
0 −→
∂1 I1 −→ I 1 ⊕ I 0 (1,0)
Ir I r ⊕ I r −1 (1,0)
0
(1,dr −1 )
−−−−−−→
I2 I 2 ⊕ I 1 (1,0)
0 Ir
−→ 0
A A.
It is easy to verify
−→ · · · (2)
Z.-W. Li, P. Zhang / Journal of Algebra 323 (2010) 1802–1812
is a minimal injective resolution of projective T 2 ( A )-module
∂0 =
−d0
(−1d0 )
∂i =
,
−di −di 0
A 0
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, where
,
1 d i −1
i = 1, . . . , r − 1.
We simply denote this resolution as
0 −→
A 0
−→ Q • .
(2)
2.5. For simplicity, we write Hom A ( M , N ) as ( M , N ) for any A-modules M and N. For an A-map f : M −→ N and an A-module L, denote the induced map Hom A ( L , f ) : Hom A ( L , M ) −→ Hom A ( L , M ) by f L ,∗ , or simply by f ∗ ; denote the induced map Hom A ( f , L ) : Hom A ( N , L ) −→ Hom A ( M , L ) by f L∗ , or simply by f ∗ . X For a T 2 ( A )-module , where φ : Y −→ X is an A-map, consider the complex φ Y
∗ σ
d0∗
d1∗
dr −1∗
X • : 0 −→ Ker φ ∗A −−→ ( X , I 0 ) − −→ ( X , I 1 ) −−→ · · · −−−→ ( X , I r ) −→ 0 where
σ : Ker φ ∗A −→ Hom A ( X , A ) is the embedding, and the complex d0∗
dr −1∗
Y • : 0 −→ 0 −→ (Y , I 0 ) − −→ (Y , I 1 ) −−→ · · · −−−→ (Y , I r ) −→ 0. d1∗
We emphasize that Ker φ ∗A is in the 0-th position, and (Y , I 0 ) is in the 1-st position. Then Φ • : X • −→ Y • is a chain map, where
Φ•
0
:= 0,
Φ • i := Hom A (φ, I i −1 ),
i = 1, . . . , r + 1.
Denote the mapping cone of Φ • by C (Φ • ), that is,
C Φ
•
−d0∗ φ∗
∗ −∗ σ
−d1∗ 0 φ ∗ d0∗
: 0 −→ Ker φ A −−−→ ( X , I 0 ) −−−−→ ( X , I 1 ) ⊕ (Y , I 0 ) −−−−−−→ ( X , I 2 ) ⊕ (Y , I 1 ) −dr −1∗ 0 φ ∗ dr −2∗
(φ ∗ ,dr −1∗ )
−→ · · · −−−−−−−−−→ ( X , I r ) ⊕ (Y , I r −1 ) −−−−−−→ (Y , I r ) −→ 0. (So, Ker φ ∗A is in the (−1)-st position.) It is clear that H −1 (C (Φ • )) = 0 = H 0 (C (Φ • )). Lemma 2.3. Let A be a Gorenstein algebra. With the notation above we have is acyclic, if and only if Φ •
:
Proof. Applying Hom T 2 ( A )
X•
−→
X Y φ
Y • is a quasi-isomorphism.
X Y φ
∈⊥
A 0
if and only if C (Φ • )
, − to (2), we know that YX φ ∈ ⊥ 0A if and only if the complex
0 −→ Hom T 2 ( A )
X
X A , 0 −→ Hom T 2 ( A ) YX φ , Q • Y φ
A
(3)
is acyclic (note that Hom T 2 ( A ) , 0 is in the (−1)-st position). By routine computations one Y φ can see that this complex is isomorphic to C (Φ • ), and then the assertion follows. For convenience we include the main details. We have isomorphisms
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Hom T 2 ( A )
X A , 0 ∼ = Ker φ ∗A ; Y φ
Hom T 2 ( A )
I = Hom T 2 ( A ) YX φ , I ⊕ Ii i i −1 (1,0)
f g fφ
X I 0 ∼ , I = Hom A ( X , I 0 ); Y φ 0
f ∈ Hom A ( X , I i ), g ∈ Hom A (Y , I i −1 )
∼ = Hom A ( X , I i ) ⊕ Hom A (Y , I i −1 ),
i = 1, . . . , r ;
and
Hom T 2 ( A )
X 0 , I ∼ = Hom A (Y , I r ). Y φ r
These isomorphisms form a chain isomorphism between the complexes (3) and C (Φ • ).
2
Lemma 2.4. Let A be a Gorenstein algebra. Assume that X is a Gorenstein-projective A-module, and φ : Y −→ X is an arbitrary A-map. Then C (Φ • ) is acyclic if and only if Y is a Gorenstein-projective A-module and φ ∗A : Hom A ( X , A ) −→ Hom A (Y , A ) is surjective.
(10)
(1,0)
Proof. By the short exact sequence of complexes 0 −→ Y • −−→ C (Φ • ) −−−→ X • [1] −→ 0 we have the long exact sequence
−→ H 0 X • −→ H 0 Y • −→ H 0 C Φ • −→ H 1 X • −→ H 1 Y • −→ H 1 C Φ • −→ H 2 X • −→ · · · −→ H r X • −→ H r Y • −→ H r C Φ • −→ H r +1 X • −→ H r +1 Y • −→ H r +1 C Φ • −→ 0 = H r +2 X • .
0 −→ H −1 C Φ •
Recall that H −1 (C (Φ • )) = 0 = H 0 (C (Φ • )). Since X is Gorenstein-projective, it follows that the complex ∗
d0∗
d1∗
dr −1∗
0 −→ ( X , A ) − → ( X , I 0 ) −−→ ( X , I 1 ) −−→ · · · −−−→ ( X , I r ) −→ 0 is acyclic. This means that H i ( X • ) = 0 for i = 2, . . . , r + 1 (note that ( X , I 1 ) is in the 2-nd position of X • ). So
H i C Φ•
∼ = Hi Y • ,
i = 2, . . . , r + 1;
(4)
and
0 −→ H 1 X • −→ H 1 Y • −→ H 1 C Φ • ∗
−→ 0
d0∗
is exact. Since 0 −→ Hom A (Y , A ) − −→ Hom A (Y , I 0 ) −−→ Hom A (Y , I 1 ) is exact, it follows that H 1 (Y • ) = Hom A (Y , A ). By construction
H 1 X • = Ker d0∗ / Im(∗ σ ) = Hom A ( X , A )/ Ker φ ∗A ∼ = Im φ ∗A . So we have the exact sequence
0 −→ Im φ ∗A −→ Hom A (Y , A ) −→ H 1 C Φ •
−→ 0.
(5)
Now, by (4) and (5) we see that C (Φ • ) is acyclic if and only if φ ∗A is surjective and H i (Y • ) = 0, i = 2, . . . , r + 1; if and only if φ ∗A is surjective and Y is a Gorenstein-projective A-module. 2
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2.6. Let A be an Artin algebra. Lemma 2.5. (See Beligiannis [B, Proposition 3.4].) The functor Hom A (−, A ) induces an equivalence A-G proj ∼ = A op -G proj with a quasi-inverse Hom A op (−, A ).
2.7. Proof of Theorem 1.1(i). Since T 2 ( A ) =
X
A
A ⊥ A
⊕
A X0
X
, by Lemma 2.1
is a Gorenstein-
Y φ
⊥ A
∈ A and Y φ ∈ 0 ; if and only if X is a GorensteinY φ projective A-module (by Lemma 2.2), and C (Φ • ) is acyclic (by Lemma 2.3); if and only if X and Y are Gorenstein-projective A-modules and φ ∗A = Hom(φ, A ) : Hom A ( X , A ) −→ Hom A (Y , A ) is surjective
projective T 2 ( A )-module if and only if
(by Lemma 2.4). Assume that X and Y are Gorenstein-projective A-modules and φ ∗A = Hom(φ, A ) : Hom A ( X , A ) −→ Hom A (Y , A ) is surjective. Then we have the injective A-map
Hom A op φ ∗A , A : Hom A op Hom A (Y , A ), A −→ Hom A op Hom A ( X , A ), A , by Lemma 2.5 this implies that φ : Y −→ X is injective. Applying Hom A (−, A ) to the exact sequence φ
0 −→ Y −→ X −→ Coker φ −→ 0 we see that Coker φ ∈ ⊥ A, i.e., Coker φ is a Gorenstein-projective A-module, by Lemma 2.1. Conversely, if X , Y and Coker φ are Gorenstein-projective A-modules and φ is injective, then φ
Ext1A (Coker φ, A ) = 0. Applying Hom A (−, A ) to the exact sequence 0 −→ Y −→ X −→ Coker φ −→ 0 we see that φ ∗A is surjective. This completes the proof of Theorem 1.1(i). 2 If A is a self-injective algebra, then every A-module is Gorenstein-projective. So we have Corollary 2.6. Let A be a self-injective algebra. Then
X
if φ : Y −→ X is injective.
Y φ
is a Gorenstein-projective T 2 ( A )-module if and only
The modules in Corollary 2.6 have been extensively studied in [RS]. See also [BR, p. 101]. Example 2.7. Let B = T 2 ( A ) with A a self-injective algebra. Consider
A
A A A 0 A 0 A 0 0 A A 0 0 0 A
T 2(B) =
.
By 1.3 a T 2 ( B )-module M can be identified with a commutative square of A-maps
Y
φ
X (6)
α
β φ
Y
X
By Theorem 1.1(i) M is a Gorenstein-projective T 2 ( B )-module if and only if
α
Coker β
=
Coker α Coker β
are Gorenstein-projective T 2 ( A )-modules, and
α β
:
X
X Y φ
Y φ
,
−→
X
φ
YX
Y φ
, and is a
monomorphism; if and only if φ, φ , α , β are injective, and the canonical map Coker β −→ Coker α is injective (by Corollary 2.6). So, we conclude that if we identify a T 2 ( B )-module M with a commutative square (6), then M is a Gorenstein-projective T 2 ( B )-module if and only if φ, φ , α , β are injective, and the commutative square (6) is the pull-back of (α , φ).
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2.8. Proof of Theorem 1.1(ii). Since by assumptions M B is a projective module and are injective modules, it follows from the adjoint isomorphisms
AA
and
AM
Hom B −, Hom A ( M , A ) ∼ = Hom A ( M ⊗ B −, A ) and
Hom B −, Hom A ( M , M ) ∼ = Hom A ( M ⊗ B −, M ) that Hom A ( M , A ) and Hom A ( M , M ) are injective left B-modules. Thus
0 Hom A ( M , A )
and
are injective Λ-modules. It follows that we get a minimal injective resolution
0 −→
A 0
id 0
−−→
A Hom A ( M , A )ϕ
A
0 id −− →
0 Hom A ( M , A )
−→ 0,
0 Hom A ( M , M )
(7)
and a minimal injective resolution 0 id
0 −→
M B
id
−−−−→
0 B
⊕
M Hom A ( M , M ) ϕ M
0 0 , − id
−−−−−−−→
0 Hom A ( M , M )
−→ 0,
(8)
where : B −→ Hom A ( M , M ) is the left B-map sending 1 to id M . Using (7) and (8) we see that Λ is a Gorenstein algebra (one can similarly X construct a minimal injective resolution of ΛΛ ), with inj.dimΛ Λ = 1. For any Λ-module , recall that for an φ Y
∗ is the map Hom ( X , N ) −→ Hom ( M ⊗ Y , N ) induced by φ : M ⊗ Y −→ X . UsA-module N, φ N A A B B ing (7) and (8) we see that there hold
Ext1Λ
X A , 0 = Hom B Y , Hom A ( M , A ) /Υ1 Y φ
with Υ1 = { g ∈ Hom B (Y , Hom A ( M , A )) | ϕ A (id M ⊗ g ) ∈ φ ∗A (Hom A ( X , A ))}; and
Ext1Λ
X M , B = Hom B Y , Hom A ( M , M ) /Υ2 Y φ
with
Υ2 = − g 1 + g 2 g 1 ∈ Hom B (Y , B ), g 2 ∈ Hom B Y , Hom A ( M , M ) , ∗ ϕM (idM ⊗ g2 ) ∈ φM Hom A ( X , M ) . ⊕ M , by Lemma 2.1 YX φ is a Gorenstein-projective Λ-module if and only if X B ⊥ M , and ∈ ; if and only if Y φ B
Since Λ =
X
Y φ
∈
⊥ A 0
A 0
Υ1 = Hom B Y , Hom A ( M , A ) ,
and
Υ2 = Hom B Y , Hom A ( M , M ) .
If φ : M ⊗ B Y −→ X is injective, then φ ∗A : Hom A ( X , A ) −→ Hom A ( M ⊗ B Y , A ) is surjective since A A ∗ : Hom ( X , M ) −→ Hom ( M ⊗ Y , M ) is injective, and hence Υ1 = Hom B (Y , Hom A ( M , A )). Also, φ M A A B is surjective since A M is injective, it follows that
Z.-W. Li, P. Zhang / Journal of Algebra 323 (2010) 1802–1812
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Υ2 = − g 1 + g 2 g 1 ∈ Hom B (Y , B ), g 2 ∈ Hom B Y , Hom A ( M , M ) = Hom B Y , Hom A ( M , M ) . Conversely, if Υ1 = Hom B (Y , Hom A ( M , A )), then for any B-map g : Y −→ Hom A ( M , A ), there exists an A-map f : X −→ A such that
M ⊗B Y
id M ⊗ g
M ⊗ B Hom A ( M , A ) ϕA
φ f
X
A
commutes. Note that the adjoint isomorphism ψ : Hom B (Y , Hom A ( M , A )) ∼ = Hom A ( M ⊗ B Y , A ) is given by g −→ “m ⊗ y −→ g ( y )(m)”, and that
ϕ A (idM ⊗ g )(m ⊗ y ) = g ( y )(m) = ψ( g )(m ⊗ y ), it follows that the condition Υ1 = Hom B (Y , Hom A ( M , A )) just says that φ ∗A : Hom A ( X , A ) −→ Hom A ( M ⊗ B Y , A ) is surjective. So we have the injective A-map
Hom A op φ ∗A , A : Hom A op Hom A ( M ⊗ B Y , A ), A −→ Hom A op Hom A ( X , A ), A . Since A is self-injective, it follows that the functor Hom A (−, A ) induces a duality between A-mod and A op -mod with dual inverse Hom A op (−, A ) (see Proposition 3.4(iii) in p. 124 of [ARS]). So we see that φ : M ⊗ B Y −→ X is injective. This completes the proof of Theorem 1.1(ii). 2 3. Proof of Theorem 1.2 3.1.
The following facts are well known: their proofs are via a standard use of Lemma 2.1.
Lemma 3.1. (i) If A is a 1-Gorenstein algebra, then A-G proj is closed under taking submodules. (ii) If A is an n-Gorenstein algebra (n 2), and f n −1
f1
G n−1 −−→ G n−2 −→ · · · −→ G 1 −→ G 0 is an exact sequence in A-G proj, then Ker f n−1 ∈ A-G proj. In particular, if A is 2-Gorenstein, then A-G proj is closed under taking kernels. 3.2. Let G be an A-module, and Γ = End A (G )op . We will use the following philosophy of Yoneda. Consider the functor Hom A (G , −) : A-mod −→ Γ -mod. Then Hom A (G , −) induces an equivalence add(G ) ∼ = P (Γ ); and if G ∈ add(G ) then f −→ f ∗ = Hom A (G , f ) gives an isomorphism Hom A (G , X ) ∼ = HomΓ (Hom A (G , G ), Hom A (G , X )) (see e.g. [ARS, p. 33]). f
g
Let G be a generator, i.e., P ( A ) ⊆ add(G ). If 0 −→ X −→ Y −→ Z is a sequence in A-mod such that f∗
g∗
f
g
0 −→ Hom A (G , X ) −−→ Hom A (G , Y ) −−→ Hom A (G , Z ) is exact in Γ -mod, then 0 −→ X −→ Y −→ Z is exact (see e.g. [R]).
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3.3. For an algebra Γ , objects of the category Morph P (Γ ) are Γ -maps f : P 2 −→ P 1 in P (Γ ), and morphisms from f to f are pairs ( g 2 , g 1 ) of Γ -maps, such that the diagram
P2
f
P1
g2
g1
P 2
f
P 1
commutes. Let R( f , f ) be the subgroup of HomMorph P (Γ ) ( f , f ) consisting of the morphisms ( g 2 , g 1 ) with the property that there is some h : P 1 −→ P 2 such that g 1 = f h. Then R gives a relation on Morph P (Γ ). Denote by Morph P (Γ )/R the corresponding factor category by HomMorph P (Γ )/R ( f , f ) = HomMorph P (Γ ) ( f , f )/R( f , f ). See [ARS, p. 102]. We consider the full subcategory of Morph P (Γ ) given by
C = f : P 2 −→ P 1 in P (Γ ) f is injective and Coker f ∈ P 1 Ω(Γ ) , where Ω(Γ ) is the full subcategory of Γ -mod of torsionless Γ -modules (i.e., submodules of projective modules), and P 1 Ω(Γ ) is the full subcategory of Γ -mod consisting of torsionless Γ -modules of projective dimension at most one. We also consider the full subcategory C /R of Morph P (Γ )/R. Note that Morph P (Γ ), and hence Morph P (Γ )/R, are a Krull–Schmidt categories [A, p. 571]. Since C is closed under direct summands in Morph P (Γ ), it follows that C and hence C /R are Krull–Schmidt. 3.4. We need the following lemmas. Lemma 3.2. Let A be a CM-finite Gorenstein algebra, with A-G proj = add( E ) and Γ = End A ( E )op . Then T 2 ( A )-G proj can be embedded into C , and an object in T 2 ( A )-G proj is indecomposable if and only if it is indecomposable as an object in C . Furthermore, if inj.dim A A 1, then T 2 ( A )-G proj ∼ = C. Proof. For a Gorenstein-projective T 2 ( A )-module
X
φ
Y φ
, by Theorem 1.1(i) we get an exact sequence
0 −→ Y −→ X −→ Coker φ −→ 0 of Gorenstein-projective A-modules, and hence an exact sequence φ∗
0 −→ Hom A ( E , Y ) −−→ Hom A ( E , X ) −→ Hom A ( E , Coker φ) of projective Γ -modules. Since Coker φ∗ is a submodule of Hom A ( E , Coker φ), it follows that Coker φ∗ ∈ P 1 Ω(Γ ). Define F : T 2 ( A )-G proj −→ C by
F
X Y φ
φ
∗ = Hom A ( E , Y ) −→ Hom A ( E , X );
and define F on morphisms in the natural way. Then F is clearly a fully faithful functor. If inj.dim A A 1, then we claim that F is dense. In fact, for any object f : P 2 −→ P 1 in C , there exist a unique X ∈ add( E ), a unique Y ∈ add( E ), and an A-map φ : Y −→ X , such that P 1 = Hom A ( E , X ), P 2 = Hom A ( E , Y ), and f = φ∗ . Since by assumption Coker f ∈ P 1 Ω(Γ ), there is Z ∈ add( E ) such that Coker f is a submodule of Hom A ( E , Z ). Thus we have an exact sequence f =φ∗
φ
0 −→ Hom A ( E , Y ) −−−−→ Hom A ( E , X ) −→ Hom A ( E , Z ). Since E is a generator, 0 −→ Y −→ X −→ Z is also exact (see 3.2), and hence Coker φ is a submodule of Gorenstein-projective module Z . Since inj.dim A A 1, it follows from Lemma 3.1(i) that Coker φ is Gorenstein-projective. By Theorem 1.1(i)
X
Y φ
is a Gorenstein-projective T 2 ( A )-module, and by construction F
f : P 2 −→ P 1 . This proves that F is an equivalence.
2
X Y φ
is exactly the object
Lemma 3.3. Let A be a CM-finite algebra, with A-G proj = add( E ) and Γ = End A ( E )op . Then there is an equivalence of categories C /R ∼ = P 1 Ω(Γ ) sending indecomposable objects to indecomposable objects.
Z.-W. Li, P. Zhang / Journal of Algebra 323 (2010) 1802–1812
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Proof. For any object f : P 2 −→ P 1 in C /R, define H : C /R −→ P 1 Ω(Γ ) by
H ( f : P 2 −→ P 1 ) = Coker f . By the argument on p. 102 of [ARS], H is fully faithful. It is easy to see that H is dense: for any Y ∈ f
P 1 Ω(Γ ), by definition we have an exact sequence 0 −→ P 2 −→ P 1 −→ Y −→ 0, hence f : P 2 −→ P 1 is an object of C /R, and by construction we have H ( f : P 2 −→ P 1 ) = Y . By construction H and hence a quasi-inverse of H are additive functors, it follows that H sends indecomposable objects to indecomposable objects. 2 Lemma 3.4. For any Artin algebra, C is of finite type if and only if C /R is of finite type. Proof. This is essentially proved in the second paragraph of p. 216 in [ARS]. In fact (non-zero) indecomposable objects in C /R are exactly the indecomposable objects in C which are not isomorphic to an object id : P −→ P with P indecomposable in P (Γ ). Then the assertion follows from the fact that C has only finitely many indecomposable objects of the form id : P −→ P with P indecomposable in P (Γ ). 2 Lemma 3.5. If A is a CM-finite 2-Gorenstein algebra with A-G proj = add( E ) and Γ = End A ( E )op , then P 1 Ω(Γ ) = Ω(Γ ). f
Proof. Let Y be an arbitrary Γ -module. Take a projective presentation P 1 −→ P 0 −→ Y −→ 0 of Y . Then there is an A-map h : E 1 −→ E 0 with E 0 , E 1 ∈ add( E ) such that f = Hom A ( E , h). Put X = Ker h. It follows from Lemma 3.1(ii) that X ∈ add( E ). By the exact sequence of Γ -modules
0 −→ Hom A ( E , X ) −→ Hom A ( E , E 1 ) −→ Hom A ( E , E 0 ) −→ Y −→ 0 we see that proj.dim Y 2. Now we prove P 1 Ω(Γ ) = Ω(Γ ). In fact, for any Z ∈ Ω(Γ ), say, Z is a submodule of projective g Γ -module P ; taking a minimal projective presentation Q 2 −→ Q 1 −→ Z −→ 0, we get an exact seg quence Q 2 −→ Q 1 −→ P −→ P / Z −→ 0. By the argument above we have proj.dim P / Z 2, so g is injective, and hence Z ∈ P 1 Ω(Γ ). 2
3.5. Proof of Theorem 1.2. The assertion follows from Lemmas 3.2–3.4. If in addition inj.dim A A 1, then A is a CM-finite 2-Gorenstein algebra, and then the corresponding assertion follows from Lemmas 3.2–3.5. 2 Acknowledgment We sincerely thank the referee for his or her valuable suggestions and comments. References [A]
M. Auslander, Representation Dimension of Artin Algebras, Queen Mary College Math. Notes, Queen Mary College, London, 1971, also in: I. Reiten, S. Smalø, Ø. Solberg (Eds.), Selected Works of Maurice Auslander. Part 1 (II), Amer. Math. Soc., 1999, pp. 505–574. [AB] M. Auslander, M. Bridger, Stable Module Theory, Mem. Amer. Math. Soc., vol. 94, Amer. Math. Soc., Providence, RI, 1969. [ARS] M. Auslander, I. Reiten, S.O. Smalø, Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math., vol. 36, Cambridge Univ. Press, 1995. [AM] L.L. Avramov, A. Martsinkovsky, Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. 85 (3) (2002) 393–440. [B] A. Beligiannis, Cohen–Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras, J. Algebra 288 (1) (2005) 137–211.
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[EJ1] [EJ2] [FGR] [GZ] [LZ] [Hap] [Hol] [I] [J] [K] [R] [RS]
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A. Beligiannis, I. Reiten, Homological and Homotopical Aspects of Torsion Theories, Mem. Amer. Math. Soc., vol. 188, Amer. Math. Soc., Providence, RI, 2007. R.-O. Buchweitz, G.-M. Greuel, F.-O. Schreyer, Cohen–Macaulay modules on hypersurface singularities II, Invent. Math. 88 (1) (1987) 165–182. X.W. Chen, An Auslander-type result for Gorenstein-projective modules, Adv. Math. 218 (2008) 2043–2050. L.W. Christensen, A. Frankild, H. Holm, On Gorenstein projective, injective and flat dimensions—a functorial description with applications, J. Algebra 302 (1) (2006) 231–279. L.W. Christensen, G. Piepmeyer, J. Striuli, R. Takahashi, Finite Gorenstein representation type implies simple singularity, Adv. Math. 218 (2008) 1012–1026. V. Dlab, C.M. Ringel, The module theoretical approach to quasi-hereditary algebras, in: Representations of Algebras and Related Topics, Kyoto, 1990, in: London Math. Soc. Lecture Note Ser., vol. 168, Cambridge Univ. Press, Cambridge, 1992, pp. 200–224. E.E. Enochs, O.M.G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (4) (1995) 611–633. E.E. Enochs, O.M.G. Jenda, Relative Homological Algebra, de Gruyter Exp. Math., vol. 30, Walter de Gruyter Co., 2000. R. Fossum, P. Griffith, I. Reiten, Trivial Extensions of Abelian Categories, Lecture Notes in Math., vol. 456, Springer-Verlag, 1975. N. Gao, P. Zhang, Gorenstein derived categories, J. Algebra, in press. Z.W. Li, P. Zhang, Gorenstein algebras of finite Cohen–Macaulay type, Adv. Math. 223 (2010) 728–734. D. Happel, On Gorenstein algebras, in: Representation Theory of Finite Groups and Finite-Dimensional Algebras, in: Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 389–404. H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (1–3) (2004) 167–193. Y. Iwanaga, On rings with finite self-injective dimension II, Tsukuba J. Math. 4 (1980) 107–113. P. Jørgensen, Existence of Gorenstein projective resolutions and Tate cohomology, J. Eur. Math. Soc. 9 (2007) 59–76. H. Knörrer, Cohen–Macaulay modules on hypersurface singularities I, Invent. Math. 88 (1) (1987) 153–164. C.M. Ringel, An introduction to the representation dimension of Artin algebras, 2008, unpublished. C.M. Ringel, M. Schmidmeier, The Auslander–Reiten translation in submodule categories, Trans. Amer. Math. Soc. 360 (2) (2008) 691–716.
Contents lists available at ScienceDirect
Journal of Algebra www.elsevier.com/locate/jalgebra
A construction of Gorenstein-projective modules Zhi-Wei Li, Pu Zhang ∗,1 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, PR China
a r t i c l e
i n f o
Article history: Received 30 September 2009 Available online 8 January 2010 Communicated by Luchezar L. Avramov Keywords: Gorenstein-projective module Finite Cohen–Macaulay type Gorenstein algebra
a b s t r a c t We determine all the Gorenstein-projective modules over the T 2 A M extension of a Gorenstein algebra, and over , where A and 0 B
B are self-injective algebras, and M is an A-B bimodule with A M and M B projective. Using this, we obtain a criterion for the CMfiniteness of the T 2 -extensions of CM-finite Gorenstein algebras. As an application we get non-trivial examples of CM-finite Gorenstein algebras. © 2010 Elsevier Inc. All rights reserved.
1. Introduction and main results 1.1. Throughout A is an Artin algebra, and all modules are finitely generated. Let A-mod be the category of left A-modules, and P ( A ) the full subcategory of A-mod of projective A-modules. An d0
A-module M is Gorenstein-projective, if there is an exact sequence · · · −→ P −1 −→ P 0 −−→ P 1 −→ · · · in P ( A ), which stays exact under Hom A (−, A ), and such that M ∼ = Ker d0 . Denote by A-G proj the full subcategory of A-mod of Gorenstein-projective modules. Auslander and Bridger [AB] introduced these modules under the name modules of G-dimension zero; Enochs and Jenda [EJ1] defined them for arbitrary modules (not necessarily finitely generated) over a ring. They are also called totally reflexive modules (see Avramov and Martsinkovsky [AM]); in a commutative noetherian Gorenstein local ring, they are exactly the maximal Cohen–Macaulay modules. Gorenstein-projective modules are particular important to the Gorenstein algebras (see [EJ2]). An Iwanaga–Gorenstein ring, or simply a Gorenstein ring R, is a left and right noetherian ring with inj.dim R R < ∞ and inj.dim R R < ∞. A Gorenstein ring R is n-Gorenstein if inj.dim R R n. In this case inj.dim R R n (see [I]). An algebra A is Gorenstein if it is a Gorenstein ring. An additive full subcategory X of A-mod, which is closed under direct summands, is of finite type if X has only finitely many isomorphism classes of indecomposable objects. In particular, if A-G proj
* 1
Corresponding author. E-mail addresses: [email protected] (Z.-W. Li), [email protected] (P. Zhang). Supported by the NSF of China (10725104), and STCSM (09XD1402500).
0021-8693/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2009.12.030
Z.-W. Li, P. Zhang / Journal of Algebra 323 (2010) 1802–1812
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is of finite type, then A is said to be of finite Cohen–Macaulay type, or simply, CM-finite. Clearly, A is CM-finite if and only if there is a module E such that A-G proj = add( E ), where add( E ) is the full subcategory of A-mod of direct summands of finite direct sums of copies of E. 1.2. Gorenstein-projective modules receive a lot of attention: they form the basis of Gorenstein homological algebra (see e.g. [CFH,EJ2,Hol]), they play an important role in the Tate cohomology of algebras (see e.g. [AM,J]), they are widely used in the theory of singularities, as well as stable categories (see e.g. [B,BGS,BR,CPST,Hap,K]). We are interested in constructing explicitly all Gorenstein-projective modules, and in investigating criteria for the CM-finiteness of a given algebra. Such information will be helpful to the understanding of these modules, and to their various applications. We also believe that in the future work the representation theory of Artin algebra and Gorenstein homological algebra may have more interactions. 1.3. Given an A-B-bimodule M, consider the upper triangular matrix Artin algebra Λ = with multiplication given by the one of matrices. A Λ-module is identified with a triple
X
A
A MB , 0 B X , or Y φ
if φ is clear, where X is an A-module, Y is a B-module, and φ : M ⊗ B Y −→ X is an simply Y A-map. We refer to Auslander, the representation theory of Λ. In A A Reiten and Smalø [ARS, p. 73] for X particular, let T 2 ( A ) = . Then a T 2 ( A )-module is a triple , where φ : Y −→ X is in A-mod. φ 0 A
Y
Note that A is n-Gorenstein if and only if T 2 ( A ) is (n + 1)-Gorenstein ([FGR]; also [Hap]). The following result inductively determines all the Gorenstein-projective modules over the T 2 extension of a Gorenstein algebra, and over the upper triangular matrix Artin algebra Λ as above where A and B are self-injective algebras, and M is an A-B bimodule with A M and M B projective. Theorem 1.1. (i) Let A be a Gorenstein Artin algebra. Then a T 2 ( A )-module
X Y φ
is Gorenstein-projective if and only if X ,
Y and Coker φ are Gorenstein-projective A-modules and φ : Y −→ X is injective. A A MB X with A and B self-injective, and A M and M B projective. Then a Λ-module is (ii) Let Λ = φ 0
B
Gorenstein-projective if and only if φ : M ⊗ B Y −→ X is injective.
Y
1.4. Let Γ be an algebra. Denote by Ω(Γ ) the full subcategory of Γ -mod of torsionless Γ -modules (i.e., submodules of projective modules), and by P 1 Ω(Γ ) the full subcategory of Γ -mod consisting of torsionless Γ -modules of projective dimension at most one. Inspired by Auslander’s idea on the representation type of T 2 -extensions [A, Chapter IV, 3], and using Theorem 1.1(i) we get the following criterion for the CM-finiteness of the T 2 -extensions of CMfinite Gorenstein algebras. Theorem 1.2. Let A be a CM-finite Gorenstein algebra, with A-G proj = add( E ) and Γ = End A ( E )op . If P 1 Ω(Γ ) is of finite type then T 2 ( A ) is CM-finite. Furthermore, if inj.dim A A 1, then T 2 ( A ) is CM-finite if and only if Ω(Γ ) is of finite type. For general descriptions of CM-finiteness of Gorenstein algebras we refer to [C] and [LZ]; for the geometric motivations of CM-finiteness in commutative local case, we refer to [K,BGS,CPST]; one can find various properties of CM-finite Gorenstein algebras in [B] and [GZ]. As an application, by using results of Dlab and Ringel in [DR], we get “non-trivial” examples of CM-finite Gorenstein algebras A. Here “non-trivial” means A itself is representation-infinite and gl.dim A = ∞ (otherwise, A is of course CM-finite). Corollary-Example 1.3. Let k be an algebraically closed field, and n = 4 or 5. Then T 2 (k[x]/xn ) is a representation-infinite, CM-finite Gorenstein algebra of infinite global dimension.
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Proof. R n = k[x]/xn is a representation-finite self-injective algebra, so it is CM-finite Gorenstein with R n -G proj = R n -mod. Let Γ be the Auslander algebra of R n . By Lemma 7.1 and Proposition 7.2 in [DR] Ω(Γ ) is of finite type if and only if n 5; and Γ is representation-finite if and only if n 3. So, if n = 4 or 5, then T 2 (k[x]/xn ) is CM-finite by Theorem 1.2; and it is representation-infinite by Proposition 5.8 in [ARS, p. 216]. 2 2. Proof of Theorem 1.1 2.1. By the definition of a Gorenstein-projective module, we know A-G proj ⊆ ⊥ A, where ⊥ A = { M ∈ A-mod | ExtiA ( M , A ) = 0, ∀i 1}. One has Lemma 2.1. (See Enochs and Jenda [EJ2, Corollary 11.5.3].) If A is a Gorenstein algebra then A-G proj = ⊥ A. In the rest of this section, A and B are algebras, M = A M B is an A-B-bimodule, such that Λ = A2.2. M is an Artin algebra. In particular, if A = B and A M B = A A A , then we get Λ = T 2 ( A ). Note that a 0 B f Λ-map is identified with a pair g : YX φ −→ X φ , where f : X −→ X is an A-map, g : Y −→ Y Y P is a B-map, such that f φ = φ (id M ⊗ g ). All of the indecomposable projective Λ-modules are M⊗B Q 00 I and , and all of the indecomposable injective Λ-modules are Hom ( M , I ) ϕ and J , id Q M⊗B Q
A
I
where P and Q range over indecomposable projective A-modules and indecomposable projective Bmodules, respectively, I and J range over indecomposable injective A-modules and indecomposable injective B-modules, respectively, and ϕ I (m ⊗ f ) = f (m), ∀m ∈ M, f ∈ Hom A ( M , I ). We will identify A ⊗ A Q with Q , and Hom A ( A , I ) with I . 2.3. Let A be a Gorenstein algebra with a minimal injective resolution of d r −1
d0
AA
0 −→ A − → I 0 −→ I 1 −→ · · · −−→ I r −→ 0. Then the projective T 2 ( A )-module
A A
(1)
has a minimal injective resolution d r −1
d0
I d0 I 1 d r −1 0 −→ A −−→ I 0 − −−→ I −→ · · · −−−−→ II r −→ 0. A 0
r
1
Using this minimal injective resolution, by a routine computation and Lemma 2.1, one sees Lemma 2.2. Let A be a Gorenstein algebra. Then
ExtiT 2 ( A ) In particular, if
X Y φ
X A , A = ExtiA ( X , A ), Y φ
∀ i 0.
is a Gorenstein-projective T 2 ( A )-module, then X is a Gorenstein-projective A-module.
2.4. Let A be a Gorenstein algebra with a minimal injective resolution (1) of that (see also Happel [Hap])
0 −→
A 0
0 −− →
∂r −1
−−→
I0 I0
∂
0 −→
∂1 I1 −→ I 1 ⊕ I 0 (1,0)
Ir I r ⊕ I r −1 (1,0)
0
(1,dr −1 )
−−−−−−→
I2 I 2 ⊕ I 1 (1,0)
0 Ir
−→ 0
A A.
It is easy to verify
−→ · · · (2)
Z.-W. Li, P. Zhang / Journal of Algebra 323 (2010) 1802–1812
is a minimal injective resolution of projective T 2 ( A )-module
∂0 =
−d0
(−1d0 )
∂i =
,
−di −di 0
A 0
1805
, where
,
1 d i −1
i = 1, . . . , r − 1.
We simply denote this resolution as
0 −→
A 0
−→ Q • .
(2)
2.5. For simplicity, we write Hom A ( M , N ) as ( M , N ) for any A-modules M and N. For an A-map f : M −→ N and an A-module L, denote the induced map Hom A ( L , f ) : Hom A ( L , M ) −→ Hom A ( L , M ) by f L ,∗ , or simply by f ∗ ; denote the induced map Hom A ( f , L ) : Hom A ( N , L ) −→ Hom A ( M , L ) by f L∗ , or simply by f ∗ . X For a T 2 ( A )-module , where φ : Y −→ X is an A-map, consider the complex φ Y
∗ σ
d0∗
d1∗
dr −1∗
X • : 0 −→ Ker φ ∗A −−→ ( X , I 0 ) − −→ ( X , I 1 ) −−→ · · · −−−→ ( X , I r ) −→ 0 where
σ : Ker φ ∗A −→ Hom A ( X , A ) is the embedding, and the complex d0∗
dr −1∗
Y • : 0 −→ 0 −→ (Y , I 0 ) − −→ (Y , I 1 ) −−→ · · · −−−→ (Y , I r ) −→ 0. d1∗
We emphasize that Ker φ ∗A is in the 0-th position, and (Y , I 0 ) is in the 1-st position. Then Φ • : X • −→ Y • is a chain map, where
Φ•
0
:= 0,
Φ • i := Hom A (φ, I i −1 ),
i = 1, . . . , r + 1.
Denote the mapping cone of Φ • by C (Φ • ), that is,
C Φ
•
−d0∗ φ∗
∗ −∗ σ
−d1∗ 0 φ ∗ d0∗
: 0 −→ Ker φ A −−−→ ( X , I 0 ) −−−−→ ( X , I 1 ) ⊕ (Y , I 0 ) −−−−−−→ ( X , I 2 ) ⊕ (Y , I 1 ) −dr −1∗ 0 φ ∗ dr −2∗
(φ ∗ ,dr −1∗ )
−→ · · · −−−−−−−−−→ ( X , I r ) ⊕ (Y , I r −1 ) −−−−−−→ (Y , I r ) −→ 0. (So, Ker φ ∗A is in the (−1)-st position.) It is clear that H −1 (C (Φ • )) = 0 = H 0 (C (Φ • )). Lemma 2.3. Let A be a Gorenstein algebra. With the notation above we have is acyclic, if and only if Φ •
:
Proof. Applying Hom T 2 ( A )
X•
−→
X Y φ
Y • is a quasi-isomorphism.
X Y φ
∈⊥
A 0
if and only if C (Φ • )
, − to (2), we know that YX φ ∈ ⊥ 0A if and only if the complex
0 −→ Hom T 2 ( A )
X
X A , 0 −→ Hom T 2 ( A ) YX φ , Q • Y φ
A
(3)
is acyclic (note that Hom T 2 ( A ) , 0 is in the (−1)-st position). By routine computations one Y φ can see that this complex is isomorphic to C (Φ • ), and then the assertion follows. For convenience we include the main details. We have isomorphisms
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Hom T 2 ( A )
X A , 0 ∼ = Ker φ ∗A ; Y φ
Hom T 2 ( A )
I = Hom T 2 ( A ) YX φ , I ⊕ Ii i i −1 (1,0)
f g fφ
X I 0 ∼ , I = Hom A ( X , I 0 ); Y φ 0
f ∈ Hom A ( X , I i ), g ∈ Hom A (Y , I i −1 )
∼ = Hom A ( X , I i ) ⊕ Hom A (Y , I i −1 ),
i = 1, . . . , r ;
and
Hom T 2 ( A )
X 0 , I ∼ = Hom A (Y , I r ). Y φ r
These isomorphisms form a chain isomorphism between the complexes (3) and C (Φ • ).
2
Lemma 2.4. Let A be a Gorenstein algebra. Assume that X is a Gorenstein-projective A-module, and φ : Y −→ X is an arbitrary A-map. Then C (Φ • ) is acyclic if and only if Y is a Gorenstein-projective A-module and φ ∗A : Hom A ( X , A ) −→ Hom A (Y , A ) is surjective.
(10)
(1,0)
Proof. By the short exact sequence of complexes 0 −→ Y • −−→ C (Φ • ) −−−→ X • [1] −→ 0 we have the long exact sequence
−→ H 0 X • −→ H 0 Y • −→ H 0 C Φ • −→ H 1 X • −→ H 1 Y • −→ H 1 C Φ • −→ H 2 X • −→ · · · −→ H r X • −→ H r Y • −→ H r C Φ • −→ H r +1 X • −→ H r +1 Y • −→ H r +1 C Φ • −→ 0 = H r +2 X • .
0 −→ H −1 C Φ •
Recall that H −1 (C (Φ • )) = 0 = H 0 (C (Φ • )). Since X is Gorenstein-projective, it follows that the complex ∗
d0∗
d1∗
dr −1∗
0 −→ ( X , A ) − → ( X , I 0 ) −−→ ( X , I 1 ) −−→ · · · −−−→ ( X , I r ) −→ 0 is acyclic. This means that H i ( X • ) = 0 for i = 2, . . . , r + 1 (note that ( X , I 1 ) is in the 2-nd position of X • ). So
H i C Φ•
∼ = Hi Y • ,
i = 2, . . . , r + 1;
(4)
and
0 −→ H 1 X • −→ H 1 Y • −→ H 1 C Φ • ∗
−→ 0
d0∗
is exact. Since 0 −→ Hom A (Y , A ) − −→ Hom A (Y , I 0 ) −−→ Hom A (Y , I 1 ) is exact, it follows that H 1 (Y • ) = Hom A (Y , A ). By construction
H 1 X • = Ker d0∗ / Im(∗ σ ) = Hom A ( X , A )/ Ker φ ∗A ∼ = Im φ ∗A . So we have the exact sequence
0 −→ Im φ ∗A −→ Hom A (Y , A ) −→ H 1 C Φ •
−→ 0.
(5)
Now, by (4) and (5) we see that C (Φ • ) is acyclic if and only if φ ∗A is surjective and H i (Y • ) = 0, i = 2, . . . , r + 1; if and only if φ ∗A is surjective and Y is a Gorenstein-projective A-module. 2
Z.-W. Li, P. Zhang / Journal of Algebra 323 (2010) 1802–1812
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2.6. Let A be an Artin algebra. Lemma 2.5. (See Beligiannis [B, Proposition 3.4].) The functor Hom A (−, A ) induces an equivalence A-G proj ∼ = A op -G proj with a quasi-inverse Hom A op (−, A ).
2.7. Proof of Theorem 1.1(i). Since T 2 ( A ) =
X
A
A ⊥ A
⊕
A X0
X
, by Lemma 2.1
is a Gorenstein-
Y φ
⊥ A
∈ A and Y φ ∈ 0 ; if and only if X is a GorensteinY φ projective A-module (by Lemma 2.2), and C (Φ • ) is acyclic (by Lemma 2.3); if and only if X and Y are Gorenstein-projective A-modules and φ ∗A = Hom(φ, A ) : Hom A ( X , A ) −→ Hom A (Y , A ) is surjective
projective T 2 ( A )-module if and only if
(by Lemma 2.4). Assume that X and Y are Gorenstein-projective A-modules and φ ∗A = Hom(φ, A ) : Hom A ( X , A ) −→ Hom A (Y , A ) is surjective. Then we have the injective A-map
Hom A op φ ∗A , A : Hom A op Hom A (Y , A ), A −→ Hom A op Hom A ( X , A ), A , by Lemma 2.5 this implies that φ : Y −→ X is injective. Applying Hom A (−, A ) to the exact sequence φ
0 −→ Y −→ X −→ Coker φ −→ 0 we see that Coker φ ∈ ⊥ A, i.e., Coker φ is a Gorenstein-projective A-module, by Lemma 2.1. Conversely, if X , Y and Coker φ are Gorenstein-projective A-modules and φ is injective, then φ
Ext1A (Coker φ, A ) = 0. Applying Hom A (−, A ) to the exact sequence 0 −→ Y −→ X −→ Coker φ −→ 0 we see that φ ∗A is surjective. This completes the proof of Theorem 1.1(i). 2 If A is a self-injective algebra, then every A-module is Gorenstein-projective. So we have Corollary 2.6. Let A be a self-injective algebra. Then
X
if φ : Y −→ X is injective.
Y φ
is a Gorenstein-projective T 2 ( A )-module if and only
The modules in Corollary 2.6 have been extensively studied in [RS]. See also [BR, p. 101]. Example 2.7. Let B = T 2 ( A ) with A a self-injective algebra. Consider
A
A A A 0 A 0 A 0 0 A A 0 0 0 A
T 2(B) =
.
By 1.3 a T 2 ( B )-module M can be identified with a commutative square of A-maps
Y
φ
X (6)
α
β φ
Y
X
By Theorem 1.1(i) M is a Gorenstein-projective T 2 ( B )-module if and only if
α
Coker β
=
Coker α Coker β
are Gorenstein-projective T 2 ( A )-modules, and
α β
:
X
X Y φ
Y φ
,
−→
X
φ
YX
Y φ
, and is a
monomorphism; if and only if φ, φ , α , β are injective, and the canonical map Coker β −→ Coker α is injective (by Corollary 2.6). So, we conclude that if we identify a T 2 ( B )-module M with a commutative square (6), then M is a Gorenstein-projective T 2 ( B )-module if and only if φ, φ , α , β are injective, and the commutative square (6) is the pull-back of (α , φ).
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2.8. Proof of Theorem 1.1(ii). Since by assumptions M B is a projective module and are injective modules, it follows from the adjoint isomorphisms
AA
and
AM
Hom B −, Hom A ( M , A ) ∼ = Hom A ( M ⊗ B −, A ) and
Hom B −, Hom A ( M , M ) ∼ = Hom A ( M ⊗ B −, M ) that Hom A ( M , A ) and Hom A ( M , M ) are injective left B-modules. Thus
0 Hom A ( M , A )
and
are injective Λ-modules. It follows that we get a minimal injective resolution
0 −→
A 0
id 0
−−→
A Hom A ( M , A )ϕ
A
0 id −− →
0 Hom A ( M , A )
−→ 0,
0 Hom A ( M , M )
(7)
and a minimal injective resolution 0 id
0 −→
M B
id
−−−−→
0 B
⊕
M Hom A ( M , M ) ϕ M
0 0 , − id
−−−−−−−→
0 Hom A ( M , M )
−→ 0,
(8)
where : B −→ Hom A ( M , M ) is the left B-map sending 1 to id M . Using (7) and (8) we see that Λ is a Gorenstein algebra (one can similarly X construct a minimal injective resolution of ΛΛ ), with inj.dimΛ Λ = 1. For any Λ-module , recall that for an φ Y
∗ is the map Hom ( X , N ) −→ Hom ( M ⊗ Y , N ) induced by φ : M ⊗ Y −→ X . UsA-module N, φ N A A B B ing (7) and (8) we see that there hold
Ext1Λ
X A , 0 = Hom B Y , Hom A ( M , A ) /Υ1 Y φ
with Υ1 = { g ∈ Hom B (Y , Hom A ( M , A )) | ϕ A (id M ⊗ g ) ∈ φ ∗A (Hom A ( X , A ))}; and
Ext1Λ
X M , B = Hom B Y , Hom A ( M , M ) /Υ2 Y φ
with
Υ2 = − g 1 + g 2 g 1 ∈ Hom B (Y , B ), g 2 ∈ Hom B Y , Hom A ( M , M ) , ∗ ϕM (idM ⊗ g2 ) ∈ φM Hom A ( X , M ) . ⊕ M , by Lemma 2.1 YX φ is a Gorenstein-projective Λ-module if and only if X B ⊥ M , and ∈ ; if and only if Y φ B
Since Λ =
X
Y φ
∈
⊥ A 0
A 0
Υ1 = Hom B Y , Hom A ( M , A ) ,
and
Υ2 = Hom B Y , Hom A ( M , M ) .
If φ : M ⊗ B Y −→ X is injective, then φ ∗A : Hom A ( X , A ) −→ Hom A ( M ⊗ B Y , A ) is surjective since A A ∗ : Hom ( X , M ) −→ Hom ( M ⊗ Y , M ) is injective, and hence Υ1 = Hom B (Y , Hom A ( M , A )). Also, φ M A A B is surjective since A M is injective, it follows that
Z.-W. Li, P. Zhang / Journal of Algebra 323 (2010) 1802–1812
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Υ2 = − g 1 + g 2 g 1 ∈ Hom B (Y , B ), g 2 ∈ Hom B Y , Hom A ( M , M ) = Hom B Y , Hom A ( M , M ) . Conversely, if Υ1 = Hom B (Y , Hom A ( M , A )), then for any B-map g : Y −→ Hom A ( M , A ), there exists an A-map f : X −→ A such that
M ⊗B Y
id M ⊗ g
M ⊗ B Hom A ( M , A ) ϕA
φ f
X
A
commutes. Note that the adjoint isomorphism ψ : Hom B (Y , Hom A ( M , A )) ∼ = Hom A ( M ⊗ B Y , A ) is given by g −→ “m ⊗ y −→ g ( y )(m)”, and that
ϕ A (idM ⊗ g )(m ⊗ y ) = g ( y )(m) = ψ( g )(m ⊗ y ), it follows that the condition Υ1 = Hom B (Y , Hom A ( M , A )) just says that φ ∗A : Hom A ( X , A ) −→ Hom A ( M ⊗ B Y , A ) is surjective. So we have the injective A-map
Hom A op φ ∗A , A : Hom A op Hom A ( M ⊗ B Y , A ), A −→ Hom A op Hom A ( X , A ), A . Since A is self-injective, it follows that the functor Hom A (−, A ) induces a duality between A-mod and A op -mod with dual inverse Hom A op (−, A ) (see Proposition 3.4(iii) in p. 124 of [ARS]). So we see that φ : M ⊗ B Y −→ X is injective. This completes the proof of Theorem 1.1(ii). 2 3. Proof of Theorem 1.2 3.1.
The following facts are well known: their proofs are via a standard use of Lemma 2.1.
Lemma 3.1. (i) If A is a 1-Gorenstein algebra, then A-G proj is closed under taking submodules. (ii) If A is an n-Gorenstein algebra (n 2), and f n −1
f1
G n−1 −−→ G n−2 −→ · · · −→ G 1 −→ G 0 is an exact sequence in A-G proj, then Ker f n−1 ∈ A-G proj. In particular, if A is 2-Gorenstein, then A-G proj is closed under taking kernels. 3.2. Let G be an A-module, and Γ = End A (G )op . We will use the following philosophy of Yoneda. Consider the functor Hom A (G , −) : A-mod −→ Γ -mod. Then Hom A (G , −) induces an equivalence add(G ) ∼ = P (Γ ); and if G ∈ add(G ) then f −→ f ∗ = Hom A (G , f ) gives an isomorphism Hom A (G , X ) ∼ = HomΓ (Hom A (G , G ), Hom A (G , X )) (see e.g. [ARS, p. 33]). f
g
Let G be a generator, i.e., P ( A ) ⊆ add(G ). If 0 −→ X −→ Y −→ Z is a sequence in A-mod such that f∗
g∗
f
g
0 −→ Hom A (G , X ) −−→ Hom A (G , Y ) −−→ Hom A (G , Z ) is exact in Γ -mod, then 0 −→ X −→ Y −→ Z is exact (see e.g. [R]).
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3.3. For an algebra Γ , objects of the category Morph P (Γ ) are Γ -maps f : P 2 −→ P 1 in P (Γ ), and morphisms from f to f are pairs ( g 2 , g 1 ) of Γ -maps, such that the diagram
P2
f
P1
g2
g1
P 2
f
P 1
commutes. Let R( f , f ) be the subgroup of HomMorph P (Γ ) ( f , f ) consisting of the morphisms ( g 2 , g 1 ) with the property that there is some h : P 1 −→ P 2 such that g 1 = f h. Then R gives a relation on Morph P (Γ ). Denote by Morph P (Γ )/R the corresponding factor category by HomMorph P (Γ )/R ( f , f ) = HomMorph P (Γ ) ( f , f )/R( f , f ). See [ARS, p. 102]. We consider the full subcategory of Morph P (Γ ) given by
C = f : P 2 −→ P 1 in P (Γ ) f is injective and Coker f ∈ P 1 Ω(Γ ) , where Ω(Γ ) is the full subcategory of Γ -mod of torsionless Γ -modules (i.e., submodules of projective modules), and P 1 Ω(Γ ) is the full subcategory of Γ -mod consisting of torsionless Γ -modules of projective dimension at most one. We also consider the full subcategory C /R of Morph P (Γ )/R. Note that Morph P (Γ ), and hence Morph P (Γ )/R, are a Krull–Schmidt categories [A, p. 571]. Since C is closed under direct summands in Morph P (Γ ), it follows that C and hence C /R are Krull–Schmidt. 3.4. We need the following lemmas. Lemma 3.2. Let A be a CM-finite Gorenstein algebra, with A-G proj = add( E ) and Γ = End A ( E )op . Then T 2 ( A )-G proj can be embedded into C , and an object in T 2 ( A )-G proj is indecomposable if and only if it is indecomposable as an object in C . Furthermore, if inj.dim A A 1, then T 2 ( A )-G proj ∼ = C. Proof. For a Gorenstein-projective T 2 ( A )-module
X
φ
Y φ
, by Theorem 1.1(i) we get an exact sequence
0 −→ Y −→ X −→ Coker φ −→ 0 of Gorenstein-projective A-modules, and hence an exact sequence φ∗
0 −→ Hom A ( E , Y ) −−→ Hom A ( E , X ) −→ Hom A ( E , Coker φ) of projective Γ -modules. Since Coker φ∗ is a submodule of Hom A ( E , Coker φ), it follows that Coker φ∗ ∈ P 1 Ω(Γ ). Define F : T 2 ( A )-G proj −→ C by
F
X Y φ
φ
∗ = Hom A ( E , Y ) −→ Hom A ( E , X );
and define F on morphisms in the natural way. Then F is clearly a fully faithful functor. If inj.dim A A 1, then we claim that F is dense. In fact, for any object f : P 2 −→ P 1 in C , there exist a unique X ∈ add( E ), a unique Y ∈ add( E ), and an A-map φ : Y −→ X , such that P 1 = Hom A ( E , X ), P 2 = Hom A ( E , Y ), and f = φ∗ . Since by assumption Coker f ∈ P 1 Ω(Γ ), there is Z ∈ add( E ) such that Coker f is a submodule of Hom A ( E , Z ). Thus we have an exact sequence f =φ∗
φ
0 −→ Hom A ( E , Y ) −−−−→ Hom A ( E , X ) −→ Hom A ( E , Z ). Since E is a generator, 0 −→ Y −→ X −→ Z is also exact (see 3.2), and hence Coker φ is a submodule of Gorenstein-projective module Z . Since inj.dim A A 1, it follows from Lemma 3.1(i) that Coker φ is Gorenstein-projective. By Theorem 1.1(i)
X
Y φ
is a Gorenstein-projective T 2 ( A )-module, and by construction F
f : P 2 −→ P 1 . This proves that F is an equivalence.
2
X Y φ
is exactly the object
Lemma 3.3. Let A be a CM-finite algebra, with A-G proj = add( E ) and Γ = End A ( E )op . Then there is an equivalence of categories C /R ∼ = P 1 Ω(Γ ) sending indecomposable objects to indecomposable objects.
Z.-W. Li, P. Zhang / Journal of Algebra 323 (2010) 1802–1812
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Proof. For any object f : P 2 −→ P 1 in C /R, define H : C /R −→ P 1 Ω(Γ ) by
H ( f : P 2 −→ P 1 ) = Coker f . By the argument on p. 102 of [ARS], H is fully faithful. It is easy to see that H is dense: for any Y ∈ f
P 1 Ω(Γ ), by definition we have an exact sequence 0 −→ P 2 −→ P 1 −→ Y −→ 0, hence f : P 2 −→ P 1 is an object of C /R, and by construction we have H ( f : P 2 −→ P 1 ) = Y . By construction H and hence a quasi-inverse of H are additive functors, it follows that H sends indecomposable objects to indecomposable objects. 2 Lemma 3.4. For any Artin algebra, C is of finite type if and only if C /R is of finite type. Proof. This is essentially proved in the second paragraph of p. 216 in [ARS]. In fact (non-zero) indecomposable objects in C /R are exactly the indecomposable objects in C which are not isomorphic to an object id : P −→ P with P indecomposable in P (Γ ). Then the assertion follows from the fact that C has only finitely many indecomposable objects of the form id : P −→ P with P indecomposable in P (Γ ). 2 Lemma 3.5. If A is a CM-finite 2-Gorenstein algebra with A-G proj = add( E ) and Γ = End A ( E )op , then P 1 Ω(Γ ) = Ω(Γ ). f
Proof. Let Y be an arbitrary Γ -module. Take a projective presentation P 1 −→ P 0 −→ Y −→ 0 of Y . Then there is an A-map h : E 1 −→ E 0 with E 0 , E 1 ∈ add( E ) such that f = Hom A ( E , h). Put X = Ker h. It follows from Lemma 3.1(ii) that X ∈ add( E ). By the exact sequence of Γ -modules
0 −→ Hom A ( E , X ) −→ Hom A ( E , E 1 ) −→ Hom A ( E , E 0 ) −→ Y −→ 0 we see that proj.dim Y 2. Now we prove P 1 Ω(Γ ) = Ω(Γ ). In fact, for any Z ∈ Ω(Γ ), say, Z is a submodule of projective g Γ -module P ; taking a minimal projective presentation Q 2 −→ Q 1 −→ Z −→ 0, we get an exact seg quence Q 2 −→ Q 1 −→ P −→ P / Z −→ 0. By the argument above we have proj.dim P / Z 2, so g is injective, and hence Z ∈ P 1 Ω(Γ ). 2
3.5. Proof of Theorem 1.2. The assertion follows from Lemmas 3.2–3.4. If in addition inj.dim A A 1, then A is a CM-finite 2-Gorenstein algebra, and then the corresponding assertion follows from Lemmas 3.2–3.5. 2 Acknowledgment We sincerely thank the referee for his or her valuable suggestions and comments. References [A]
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