aa r X i v : . [ m a t h . R T ] D ec EXCELLENT EXTENSIONS AND HOMOLOGICALCONJECTURES
YINGYING ZHANG
Abstract.
In this paper, we introduce the notion of excellent extension of rings.Let Γ be an excellent extension of an artin algebra Λ, we prove that Λ satisfies theGorenstein symmetry conjecture (resp. finitistic dimension conjecture, Auslander-Gorenstein conjecture, Nakayama conjecture) if and only if so does Γ. As a specialcase of excellent extensions, if G is a finite group whose order is invertible in Λacting on Λ and Λ is G -stable, we prove that if the skew group algebras Λ G satisfiesstrong Nakayama conjecture (resp. generalized Nakayama conjecture), then so doesΛ. Introduction
Let Λ be an artin algebra and all modules are finitely generated unless statedotherwise. Denote by modΛ the category of finitely generated left Λ-modules. For amodule M ∈ modΛ, pd Λ M and id Λ M are the projective and injective dimensions of M respectively.The following homological conjectures are very important in the representationtheory of artin algebras. Auslander-Reiten Conjecture (ARC)
Any module M ∈ modΛ satisfyingExt ≥ ( M, M ⊕ Λ) = 0 implies that M is projective. Gorenstein Projective Conjecture (GPC) If M is a Gorenstein projectiveΛ-module such that Ext ≥ ( M, M ) = 0, then M is projective. Strong Nakayama Conjecture (SNC)
Any module M ∈ modΛ satisfyingExt ≥ ( M, Λ) = 0 implies M = 0. Generalized Nakayama Conjecture (GNC)
For any simple module S ∈ modΛ, there exists i ≥ i Λ ( S, Λ) = 0.Let 0 → Λ → I → I → · · · be a minimal injective resolution of the Λ-module Λ. Auslander-Gorenstein Conjecture (AGC)
If pd Λ I i ≤ i for any i ≥
0, then Λis Gorenstein (that is, the left and right self-injective dimensions of Λ are finite).
MSC 2010: 16E10, 16E30.Key words: Excellent extension, Skew group algebra, Homological conjecture.
Nakayama Conjecture (NC) If I i is projective for any i ≥
0, then Λ is self-injective.
Wakamatsu Tilting Conjecture (WTC)
Let T Λ be a Wakamatsu tilting mod-ule with Γ = End( T Λ ). Then id T Λ = id Γ T . Gorenstein Symmetric Conjecture (GSC) id Λ Λ = id Λ Λ. Finitistic Dimension Conjecture (FDC) findim Λ := sup { pd Λ M | M ∈ mod Λwith pd Λ M < ∞} < ∞ .These conjectures remain still open now, and there are close relations among themas follows which means the implications hold true for each artin algebra. F DC + (cid:11) (cid:19) SN C + GN C + K S (cid:11) (cid:19) AGC + N CW T C (cid:11) (cid:19) ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ARC (cid:11) (cid:19)
GSC GP C.
We refer to [AR, BFS, CT, FZ, LH, LJ, W1, W2, W3, X1, X2, X3, Y, Z] for details.The notion of excellent extension of rings was introduced by Passman in [P] whichis important in studying the algebraic structure of group rings. We will give somecommon examples of excellent extension of rings in this paper (see Example 2.2 fordetails). Many algebraists have studied the invariant properties of artin algebrasunder excellent extensions such as the projectivity, injectivity, finite representationtype, CM-finite, CM-free and representation dimension (see [Bo, HS, L, P, PS] andso on). As a special case of excellent extensions, Reiten and Riedtmann introducedthe notion of skew group algebras in [RR]. In this paper, we will connect excellentextensions with homological conjectures. The outline of this article is as follows.In Section 2, we give some terminology and some known results that will be usedin the later part.In Section 3, we aim to prove the following
Theorem 1.1. ([Theorem 3.2 and 3.3]) If Γ is an excellent extension of an artinalgebra Λ , then Λ satisfies AGC (resp. NC, GSC, FDC) if and only if so does Γ . Theorem 1.2. ([Theorem 3.6])
Let Λ be an artin algebra and G a finite group whoseorder is invertible in Λ acting on Λ . If Λ is G -stable and the skew group algebra Λ G satisfies SNC (resp. GNC), then so does Λ . Preliminaries
In this section, we give some terminology and preliminary results.
Excellent extensions.
First we recall the notion of weak excellent extensions ofrings as a generalization of that of excellent extensions of rings.
Definition 2.1.
Let Λ be a subring of a ring Γ such that Λ and Γ have the sameidentity. Then Γ is called a ring extension of Λ, and denoted by Γ ≥ Λ. A ringextension Γ ≥ Λ is called a weak excellent extension if:(1) Γ is right Λ-projective [P, p. 273], that is N Γ is a submodule of M Γ and if N Λ is a direct summand of M Λ , denote by N Λ | M Λ , then N Γ | M Γ .(2) Γ is finite extension of Λ, that is, there exist γ , . . . , γ n ∈ Γ such that Γ = P ni =1 γ i Λ.(3) Γ Λ is flat and Λ Γ is projective.Recall from [Bo, P] that a ring extension Γ ≥ Λ is called an excellent extension ifit is weak excellent and Γ Λ and Λ Γ are free with a common basis γ , . . . , γ n , such thatΛ γ i = γ i Λ for any 1 ≤ i ≤ n . Here we list some examples of excellent extensions. Example . [ARS, Bo, P, RR](1) For a ring Λ, M n (Λ) (the matrix ring of Λ of degree n ) is an excellent extensionof Λ.(2) Let Λ be a ring and G a finite group. If | G | − ∈ Λ, then the skew group ringΛ G is an excellent extension of Λ.(3) Let A be a finite-dimensional algebra over a field K , and let F be a finiteseparable field extension of K . Then A ⊗ K F is an excellent extension of A .(4) Let K be a field, and let G be a group and H a normal subgroup of G . If[ G : H ] is finite and is not zero in K , then KG is an excellent extension of KH .(5) Let K be a field of characteristic p , and let G be a finite group and H anormal subgroup of G . If H contains a Sylow p -subgroup of G , then KG isan excellent extension of KH .(6) Let K be a field and G a finite group. If G acts on K (as field automorphisms)with kernel H , then the skew group ring K ∗ G is an excellent extension of thegroup ring KH , and the center Z ( K ∗ G ) of K ∗ G is an excellent extensionof the center Z ( KH ) of KH . Proposition 2.3. [HS, Lemma 3.5]
Let Γ ≥ Λ be a weak excellent extension. If Λ is an artin algebra, then so is Γ . From now on, let Λ be an artin algebra and Γ ≥ Λ be an excellent extension.Then by Proposition 2.3 it follows that Γ is also an artin algebra. By the adjointisomorphism theorem we have the following adjoint pair ( F , H ): F := Γ Λ ⊗ − : mod Λ → mod Γ ,H := Hom Γ (Γ , − ) : mod Γ → mod Λ . Then by [HS, Lemma 4.7] we have
YINGYING ZHANG
Lemma 2.4.
Both ( F, H ) and ( H, F ) are adjoint pairs. So it follows that F and H are both exact functors therefore preserve projectiveand injective modules. Skew group algebras.
Let Λ be an algebra and G be a group with identity 1 actingon Λ, that is, a map G × Λ −→ Λ via ( σ, λ ) σ ( λ ) such that(a) The map σ : Λ → Λ is an algebra automorphism for each σ in G .(b) ( σ σ )( λ ) = σ ( σ ( λ )) for all σ , σ ∈ G and λ ∈ Λ.(c) 1( λ )= λ for all λ ∈ Λ.Let Λ be an artin algebra, G a finite group whose order is invertible in Λ and G −→ Aut
Λ a group homomorphism. The data involved in defining a new categoryof Λ-modules in terms of G :(1) For any X ∈ modΛ and σ ∈ G , let σ X be the Λ-module as follows: as a k -vector space σ X = X , the action on σ X is given by λ · x = σ − ( λ ) x for all λ ∈ Λ and x ∈ X .(2) Given a morphism of Λ-modules f : X −→ Y , define σ f : σ X −→ σ Y by σ f ( x ) = f ( x ) for each x ∈ σ X .Then σ f is also a Λ-homomorphism. Indeed, for x ∈ X and λ ∈ Λ we have σ f ( λ · x ) = f ( σ − ( λ ) x ) = σ − ( λ ) f ( x ) = λ · σ f ( x ). Using the above setup, we can define a functor F σ by F σ ( X )= σ X and F σ ( f )= σ f for X, Y ∈ modΛ and homomorphism f : X −→ Y .Then one can get the following observation immediately. Proposition 2.5. F σ : modΛ −→ modΛ is an automorphism and the inverse is F σ − . To state our main results in this paper, we need the following definition from [RR].
Definition 2.6.
The skew group algebra Λ G that G acts on Λ is given by the followingdata:(a) As an abelian group, Λ G is the free left Λ-module with the elements of G asa basis.(b) For all λ σ and λ τ in Λ and σ and τ in G , the multiplication in Λ G is definedby the rule ( λ σ σ )( λ τ τ ) = ( λ σ σ ( λ τ )) στ .In particular, when G is a finite group whose order is invertible in Λ, the naturalinclusion Λ ֒ → Λ G induces the restriction functor H : modΛ G −→ modΛ and theinduction functor F : modΛ −→ modΛ G which are the same as above when Γ = Λ G .We recall the following facts from [RR, pp.227, 235]. Proposition 2.7. (a)
Let M ∈ modΛ and σ ∈ G . We have isomorphisms of F M ∼ = L σ ∈ G ( σ ⊗ M ) ∼ = L σ ∈ Gσ M as Λ -modules. Then HF M ∼ = L σ ∈ G ( σ ⊗ M ) ∼ = L σ ∈ Gσ M . (b) The natural morphism I → HF is a split monomorphism of functions, where I : modΛ → modΛ is the identity functor. Dually, the natural morphism F H → J is a split epimorphism of functions, where J : modΛ G → modΛ G is the identity functor. For the convenience of the readers, we give an easy example to understand skewgroup algebras. We refer to [RR] for more information.
Example . Let Λ be the path algebra of the quiver Q. The cyclic group G = Z / Z acts on Λ by switching 2 and 2 ′ , α and β and fixing the vertex 1. Then the quiversof Λ and Λ G are as follows:2 [ [ G (cid:4) (cid:4) C C G (cid:26) (cid:26) γ (cid:30) (cid:30) ❂❂❂❂❂❂❂❂ Q = 1 α @ @ ✁✁✁✁✁✁✁✁ β (cid:30) (cid:30) ❂❂❂❂❂❂❂❂ Q ′ = 22 ′ ′ δ @ @ ✁✁✁✁✁✁✁✁ Homological conjectures
Let Λ be an artin algebra and Γ be its excellent extension. Since F and H preserveinjective modules, we have the following result which states that Λ and Λ G have thesame self-injective dimension. Lemma 3.1. id Λ Λ = id Γ Γ and id Λ Λ = id Γ Γ . In particular, Λ is self-injective(resp.Gorenstein) if and only if Γ is self-injective(resp. Gorenstein).Proof. Since F and H preserve injective modules, it follows thatid Γ Γ = id F Λ Λ ≤ id Λ Λ ≤ id HF Λ Λ = id H Γ Γ ≤ id Γ Γ . Thus id Λ Λ = id Γ Γ . Similarly, we have id Λ Λ = id Γ Γ . (cid:3) Theorem 3.2. (1) Λ satisfies GSC if and only if so does Γ . (2) Λ satisfies FDC if and only if so does Γ .Proof. (1) By Lemma 3.1, it follows that Λ satisfies GSC if and only if id Λ Λ = id Λ Λif and only if id Γ Γ = id Γ Γ if and only if Γ satisfies GSC.(2) By [HS, Proposition 3.6(1)], we get the assertion. (cid:3)
Theorem 3.3. (1) Λ satisfies AGC if and only if so does Γ . (2) Λ satisfies NC if and only if so does Γ .Proof. (1) Assume that Γ satisfies AGC. Let0 → Λ → I → I → · · · (3.1) YINGYING ZHANG be a minimal injective resolution of the Λ-module Λ with pd I i ≤ i for any i ≥ F to (3.1) we have an injective resolution of Γ as a Γ-module:0 → Γ → F I → F I → · · · . Take a minimal injective resolution of Γ as a Γ-module:0 → Γ → J → J → · · · . Then we have that J i is a direct summand of F I i for any i ≥
0. Therefore pd J i ≤ pd F I i ≤ pd I i ≤ i for any i ≥
0. Then Γ is Gorenstein since Γ satisfies AGC. ByLemma 3.1 we know that Λ is Gorenstein.Conversely, assume Λ satisfies AGC. Take a minimal injective resolution of Γ as aΓ-module: 0 → Γ → J → J → · · · (3.2)with pd J i ≤ i for any i ≥
0. Applying the functor H to (3.2) we get an injectiveresolution of H Γ as a Λ-module:0 → H Γ → HJ → HJ → · · · . If 0 → Λ → I → I → · · · is a minimal injective resolution of Λ-module Λ, then I i is a direct summand of HJ i for any i ≥ H Γ = Λ Γ is free. It follows that pd I i ≤ pd HJ i ≤ pd J i ≤ i forany i ≥
0. Then Λ is Gorenstein since Λ satisfies AGC. By Lemma 3.1 we have thatΓ is Gorenstein.(2) Assume that Γ satisfies NC. Let0 → Λ → I → I → · · · (3.3)be a minimal injective resolution of the Λ-module Λ with I i is projective for any i ≥
0. Applying the functor F to (3.3) we have an injective resolution of Γ as aΓ-module: 0 → Γ → F I → F I → · · · . Take a minimal injective resolution of Γ as a Γ-module:0 → Γ → J → J → · · · . Then we have that J i is a direct summand of F I i for any i ≥
0. Therefore J i isprojective for any i ≥
0. Thus Γ is self-injective since Γ satisfies NC. By Lemma 3.1we have that Λ is self-injective.Conversely, assume Λ satisfies NC. Take a minimal injective resolution of Γ as aΓ-module: 0 → Γ → J → J → · · · (3.4)with J i is projective for any i ≥
0. Applying the functor H to (3.4) we get aninjective resolution of H Γ as a Λ-module:0 → H Γ → HJ → HJ → · · · . If 0 → Λ → I → I → · · · is a minimal injective resolution of Λ-module Λ, then I i is a direct summand of HJ i for any i ≥ H Γ = Λ Γ is free. It follows that I i is projective for any i ≥
0. Then Λ is self-injective since Λ satisfies NC. By Lemma 3.1 we have that Γ isself-injective. (cid:3)
In particular, let Λ be an artin algebra and G be a finite group whose order n isinvertible in Λ acting on Λ, now we connect skew group algebras with homologicalconjectures. Before doing this, we introduce the notion of G -stable. Definition 3.4.
A Λ-module X is called G -stable if σ X ∼ = X for any σ ∈ G . Λ iscalled G -stable if it is G -stable as a left Λ-module.The following result plays a crucial role in the sequel. Proposition 3.5.
Let
M, N ∈ mod Λ satisfying Ext i Λ ( M, N ) = 0 with i ≥ . If N is G -stable, then Ext i Λ G ( F M, F N ) = 0 .Proof. If i = 0, then from the adjoint isomorphism theorem and Proposition 2.7(a)it follows thatHom Λ G ( F M, F N ) ∼ = Hom Λ ( M, HF N ) ∼ = M σ ∈ G Hom Λ ( M, σ N ) ∼ = (Hom Λ ( M, N )) n . We have finished to prove that Hom Λ G ( F M, F N ) = 0.If i ≥
1, taking a projective resolution of M in modΛ: · · · → P → P → M → F M by applying the functor F : · · · → F P → F P → F M → . Set P • = ( · · · → P → P → F P • = ( · · · → F P → F P → i Λ ( M, N ) = H i (Hom Λ ( P • , N ))is i th-homology of the complex Hom Λ ( P • , N ) andExt i Λ G ( F M, F N ) = H i (Hom Λ G ( F P • , F N ))is i th-homology of the complex Hom Λ G ( F P • , F N ). By the adjoint isomorphism the-orem and Proposition 2.7(a) we have Hom Λ G ( F P • , F N ) ∼ = L σ ∈ G Hom Λ ( P • , σ N ) ∼ =(Hom Λ ( P • , N )) n . It follows that Ext i Λ G ( F M, F N ) = 0. (cid:3)
In the following we give a connection between SNC and GNC for Λ and that forΛ G . Theorem 3.6. If Λ is G -stable and Λ G satisfies SNC (resp. GNC), then so does Λ . YINGYING ZHANG
Proof. (1) Assume Λ G satisfies SNC. Let M ∈ mod Λ with Ext ≥ ( M, Λ) = 0. ByProposition 3.5 we have Ext ≥ G ( F M, Λ G ) = 0. Then F M = 0 since Λ G satisfies SNC.It follows from Proposition 2.7(b) that M = 0 as a direct summand of HF M = 0.(2) Assume Λ G satisfies GNC. Let S ∈ mod Λ be a simple module with Ext ≥ ( S, Λ) =0. By Proposition 3.5 we have Ext ≥ G ( F S, Λ G ) = 0. By Propositions 2.7(a) andProposition 2.5, we have that HF S ∼ = L σ ∈ G σ S is a semisimple Λ-module. From[FJ, Theorem 4] we know that F S is a semisimple Λ G -module. Set F S := L j ∈ J ′ S ′ j ,where S ′ j is simple for any j ∈ J ′ . Then Ext ≥ G ( S ′ j , Λ G ) = 0 for any j ∈ J ′ . SinceΛ G satisfies GNC, S ′ j = 0 for any j ∈ J ′ . So F S = 0. By Proposition 2.7(b), wehave S = 0 as a direct summand of HF S = 0. (cid:3)
We end this article with the following interesting question:If Γ is an excellent extension of an artin algebra Λ, does Λ satisfy WTC (resp.SNC, GNC, ARC, GPC) if and only if so does Γ?
Acknowledgement.
The author would like to thank Prof. Osamu Iyama forhis hospitality during her stay in Nagoya with the support of CSC Fellowship. Herspecial thanks are due to Prof. Zhaoyong Huang and Xiaojin Zhang for helpful con-versations on the subject. She also thanks the referee for the useful suggestions. Thiswork was partially supported by NSFC (Grant No. 11571164), the program B forOutstanding PhD candidate of Nanjing University(201602B043) and also supportedby the Fundamental Research Funds for the Central Universities(2017B07314).
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Department of Mathematics, Hohai University, Nanjing 210098, Jiangsu Province,P.R. China
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