aa r X i v : . [ m a t h . K T ] J u l Gorenstein pro jective modules and Frobenius extensions
Ren Wei
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, ChinaSchool of Mathematical Sciences, Fudan University, Shanghai 200433, China
Abstract
We prove that for a Frobenius extension, if a module over the extension ring isGorenstein projective, then its underlying module over the the base ring is Gorensteinprojective; the converse holds if the Frobenius extension is either left-Gorenstein orseparable (e.g. the integral group ring extension Z ⊂ Z G ).Moreover, for the Frobenius extension R ⊂ A = R [ x ] / ( x ), we show that: a graded A -module is Gorenstein projective in GrMod( A ), if and only if its ungraded A -moduleis Gorenstein projective, if and only if its underlying R -module is Gorenstein projec-tive. It immediately follows that an R -complex is Gorenstein projective if and only ifall its items are Gorenstein projective R -modules. Key Words:
Gorenstein projective module; Frobenius extension; graded module
Introduction
A module M is said to be Gorenstein projective [6] if there exists a totally acyclic complexof projective modules P := · · · → P → P → P − → · · · such that M = Ker( P → P − ). Thestudy of Gorenstein projective modules plays an important role in some areas such as represen-tation theory of Artin algebras, the theory of stable and singularity categories, and cohomologytheory of commutative rings. Especially, for finitely generated Gorenstein projective modules,there are several different terminologies in the literature, such as modules of G-dimension zero,maximal Cohen-Macaulay modules and totally reflexive modules.For a given ring R , it is important to find a “well-behaved” extension ring A in the sense thatsome useful information can transfer between R and A . In this paper, we intend to study relationsof Gorenstein projective modules along Frobenius extensions of rings. The theory of Frobeniusextensions was developed by Kasch [15] as a generalization of Frobenius algebras, and was furtherstudied by Nakayama-Tsuzuku [19] and Morita [18]. A classical example of Frobenius extensionis the integral group ring extension Z ⊂ Z G for a finite group G . Other examples include Hopf E-mail address: [email protected]. ubalgebras [22], finite extensions of enveloping algebras of Lie super-algebras [3], envelopingalgebras of Lie coloralgebras [9]. We refer to a lecture due to Kadison [14].We are partly inspired by an observation of Buchweitz [4, Section 8.2]: for a finite group G ,a Z G -module, or equivalently an integral representation of G , is maximal Cohen-Macaulay over Z G if and only if the underlying Z -module is maximal Cohen-Macaulay, or equivalently, theunderlying Z -module is free. In [5], Chen introduces a generalization of Frobenius extension,called the totally reflexive extension of rings, and proves that totally reflexive modules transferalong such extension. However, is this true for not necessarily finitely generated Gorensteinprojective modules? As it is pointed out at the end of [5], a different argument is needed.The first main result gives a partial answer to the above question; see Theorems 2.5 and 2.11. Theorem A.
Let R ⊂ A be a Frobenius extension, M a left A -module. If M is Gorensteinprojective in Mod( A ) , then the underlying R -module M is Gorenstein projective; the converseholds if R ⊂ A is either a left-Gorenstein or a separable Frobenius extension. We remark that Z ⊂ Z G is both a left-Gorenstein and a separable Frobenius extension,so Buchweitz’s observation is true for not necessarily finitely generated Gorenstein projectivemodules. In order to prove Theorem 2.5, we need a fact that over a left-Gorenstein ring, ( GP , W )is a cotorsion pair [2]. We use GP to denote the class of Gorenstein projective modules, and W to denote the class of modules with finite projective dimension. However, we further showin Theorem 2.7 that the cotorsion pair ( GP , W ) is cogenerated by a set. This result generalizes[12, Theorem 8.3] from Iwanaga-Gorenstein rings to left-Gorenstein rings. It seems to be ofparticular interest, since this will induce a cofibrantly generated model structure on the categoryof modules by applying Hovey’s correspondence [12, Theorem 2.2], such that the associatedhomotopy category is exactly the stable category GP .The second inspirational example of this paper is the ring extension R ⊂ A = R [ x ] / ( x ). Onecan also view A as a graded ring with a copy of R (generated by 1) in degree 0 and a copy of R (generated by x ) in degree 1. It is shown in Theorem 3.2 that: Theorem B.
A graded A -module is Gorenstein projective in GrMod( A ) , if and only if its un-graded module is Gorenstein projective in Mod( A ) , if and only if its underlying module is Goren-stein projective in Mod( R ) . For the graded ring A = R [ x ] / ( x ), there is an observation that the category GrMod( A ) isautomatically isomorphic to the category Ch( R ) of R -complexes; see for example [11]. So aGorenstein projective graded A -module is precisely the Gorenstein projective R -complex intro-duced by Enochs and Garc´ıa Rozas [7]. It is immediate that (Corollary 3.3): an R -complex isGorenstein projective if and only if all its items are Gorenstein projective R -modules; see also [25,Theorem 1]. This generalizes [7, Theorem 4.5] and [17, Theorem 3.1] by removing the conditionsthat the base ring R is Iwanaga-Gorenstein and is right coherent and left perfect, respectively. he paper is organized as follows. In Section 2, we introduce the notion of left-GorensteinFrobenius extensions, and it is shown that over left-Gorenstein rings, ( GP , W ) is a cotorsionpair cogenerated by a set. We study the separable Frobenius extensions. Then, Theorem A isproved. In Section 3, we focus on Gorenstein projective graded R [ x ] / ( x )-modules, and we provethe result in Theorem B.2. Gorenstein projective modules over Frobenius extensions
Throughout, all rings are associative with a unit. Homomorphisms of rings are required tosend the unit to the unit. Let R be a ring. A left R -module M is sometimes written as R M .For two left R -modules M and N , denote by Hom R ( M, N ) the abelian group consisting of left R -homomorphisms between them. A right R -module M is sometimes written as M R . We identifyright R -modules with left R op -modules, where R op is the opposite ring of R . For two right R -modules M and N , the abelian group of right R -homomorphisms is denoted by Hom R op ( M, N ).We denote by Mod( R ) the category of left R -modules, and Mod( R op ) the category of right R -modules. Let S be another ring. An R - S -bimodule M is written as R M S .We always denote a ring extension ι : R ֒ → A by R ⊂ A . The natural bimodule R A R isgiven by rar ′ := ι ( r ) · a · ι ( r ′ ). Similarly, we consider R A and R A A etc. For a ring extension R ⊂ A , there is a restricted functor Res : Mod( A ) → Mod( R ) sends A M to R M , given by rm := ι ( r ) m . The structure map ι is usually suppressed. In the opposite direction, there arefunctors T = A ⊗ R − : Mod( R ) → Mod( A ) and H = Hom R ( A, − ) : Mod( R ) → Mod( A ). It isclear that ( T, Res ) and (
Res, H ) are adjoint pairs.
We refer to [14, Definition 1.1, Theorem 1.2] for the definition of Frobenius extensions.
Definition 2.1.
A ring extension R ⊂ A is a Frobenius extension, provided that one of thefollowing equivalent conditions holds: (1) The functors T = A ⊗ R − and H = Hom R ( A, − ) are naturally equivalent. (2) R A is finite generated projective and A A R ∼ = ( R A A ) ∗ = Hom R ( R A A , R ) . (3) A R is finite generated projective and R A A ∼ = ( A A R ) ∗ = Hom R op ( A A R , R ) . (4) There exists an R - R -homomorphism τ : A → R and elements x i , y i in A , such that forany a ∈ A , one has P i x i τ ( y i a ) = a and P i τ ( ax i ) y i = a . Lemma 2.2.
Let R ⊂ A be a Frobenius extension of rings, M a left A -module. If A M isGorenstein projective, then the underlying left R -module R M is also Gorenstein projective.Proof. Let M be a Gorenstein projective left A -module. There exists a totally acyclic complex, i.e.an acyclic complex of projective A -modules P := · · · → P → P → P − → · · · with Hom A ( P , P )being an acyclic complex for each projective A -module P , such that M = Ker( P → P − ). Note hat each P i is a projective left R -module. Then by restricting P one gets an acyclic complex ofprojective R -modules.Let Q be a projective left R -module. It follows from isomorphisms Hom R ( A, Q ) ∼ = A ⊗ R Q thatHom R ( A, Q ) is a projective left A -modules. Then the complex Hom A ( P , Hom R ( A, Q )) is acyclic.Moreover, there are isomorphisms Hom R ( P , Q ) ∼ = Hom R ( A ⊗ A P , Q ) ∼ = Hom A ( P , Hom R ( A, Q )).This implies that the complex Hom R ( P , Q ) is acyclic, and hence the underlying R -module M isGorenstein projective. (cid:3) Lemma 2.3.
Let R ⊂ A be a Frobenius extension of rings, M a left A -module. If the underlyingmodule R M is Gorenstein projective, then the following hold: (1) For any projective A -module P and any i > , Ext iA ( M, P ) = 0 . (2) A ⊗ R M is a Gorenstein projective left A -module.Proof. (1) For any left A -module M and any left R -module N , there are isomorphismsHom A ( M, A ⊗ R N ) ∼ = Hom A ( M, Hom R ( A, N )) ∼ = Hom R ( A ⊗ A M, N ) ∼ = Hom R ( M, N ) . Moreover, by replacing A M with an A -projective resolution P • of M and observing that P • is alsoan R -projective resolution of R M , we have an isomorphism of cohomology Ext iA ( M, A ⊗ R N ) ∼ =Ext iR ( M, N ) for any i > P be a projective left A -module. There is a split epimorphism θ : A ⊗ R P → P of A -modules given by θ ( a ⊗ R x ) = ax for any a ∈ A and x ∈ P , and then P is a direct summand of A ⊗ R P . Since P is projective as a left R -module, and R M is Gorenstein projective by assumption,we have Ext iA ( M, A ⊗ R P ) ∼ = Ext iR ( M, P ) = 0, and then Ext iA ( M, P ) = 0 as desired.(2) Let P := · · · → P → P → P − → · · · be a totally acyclic complex of projective R -modulessuch that R M = Ker( P → P − ). It is easy to see that A ⊗ R P is an acyclic complex of projective A -modules, and A ⊗ R M = Ker( A ⊗ R P → A ⊗ R P − ). Moreover, for any projective A -module P ,the complex Hom A ( A ⊗ R P , P ) ∼ = Hom R ( P , P ) is acyclic. So A ⊗ R M is a Gorenstein projectiveleft A -module. (cid:3) Following [2, Theorem VII2.5], a ring Λ is called left-Gorenstein provided the category Mod(Λ)of left Λ-modules is a Gorenstein category. This is equivalent to the condition that the globalGorenstein projective dimension of Λ is finite. By [6, Theorem 10.2.14], each Iwanaga-Gorensteinring (i.e. two-sided noetherian ring with left and right self-injective dimension) is left-Gorenstein.The converse is not true in general. For example, let S n = S [ x , x , · · · , x n ] be the polynomialring in n indeterminates over a non-noetherian hereditary ring S . Let R i = S i − ⊗ S i − be thetrivial extension of S i − by S i − for i ≥ S = S ). Then R i is a left-Gorenstein ring forevery i ≥
1, whereas R i is non-noetherian, and hence is not Iwanaga-Gorenstein. efinition 2.4. Let R ⊂ A be a Frobenius extension. Then R ⊂ A is called a left-GorensteinFrobenius extension provided in addition that A is left-Gorenstein. Theorem 2.5.
Let R ⊂ A be a left-Gorenstein Frobenius extension of rings, M a left A -module.Then M is a Gorenstein projective left A -module if and only if the underlying left R -module M is Gorenstein projective.Proof. By Lemma 2.2, it suffices to prove that when the underlying module R M is Gorensteinprojective, M is a Gorenstein projective left A -module.Note that over a left-Gorenstein ring A , a module M is Gorenstein projective if and only ifExt iA ( M, N ) = 0 for any module N of finite projective dimension and any i >
0; see [2] orTheorem 2.7 below. Assume that N is an A -module with projective dimension n . Then thereis an exact sequence 0 → K → P → N → A -modules, where P is projective and K is ofprojective dimension n −
1. By induction on the projective dimension of modules, it is deducedfrom Lemma 2.3(1) that Ext iA ( M, N ) ∼ = Ext i +1 A ( M, K ) = 0. The assertion follows. (cid:3)
For a finite group G , it is easy to see that the integral group ring Z G is Iwanaga-Gorenstein,since there is an exact sequence 0 → Z G → Q G → Q / Z G → Z G -modules,where Q G = Hom Z ( Z G, Q ) is an injective Z G -module, and similarly Q / Z G is injective. Corollary 2.6.
Let G be a finite group, M a left Z G -module. Then M is a Gorenstein projectiveleft Z G -module if and only if the underlying left Z -module M is Gorenstein projective. Recall that a pair of classes ( X , Y ) of modules is a cotorsion pair provided that X = ⊥ Y and Y = X ⊥ , where ⊥ Y = { X | Ext ( X, Y ) = 0 , ∀ Y ∈ Y } and X ⊥ = { Y | Ext ( X, Y ) =0 , ∀ X ∈ X } . The cotorsion pair ( X , Y ) is said to be cogenerated by a set S if S ⊥ = Y . Over anIwanaga-Gorenstein ring A , it follows from [12, Theorem 8.3] that ( GP , W ) is a cotorsion paircogenerated by a set, where GP is the class of Gorenstein projective modules, and W is the classof modules with finite projective dimension.It follows from [2] that over a left-Gorenstein ring, ( GP , W ) is a cotorsion pair. We havemore in the next result, which also generalizes [12, Theorem 8.3] from Iwanaga-Gorenstein ringsto left-Gorenstein rings. It seems to be of particular interest, since by Hovey’s correspondence[12, Theorem 2.2] between cotorsion pairs and model structures, we get a cofibrantly generatedGorenstein projective model structure on the category of modules. Moreover, the homotopycategory associated with the model structure is exactly the stable category GP . Theorem 2.7.
Let A be a left-Gorenstein ring. The cotorsion pair ( GP , W ) is cogenerated by aset.Proof. Note that over a left-Gorenstein ring, a module is Gorenstein projective if and only if itis a syzygy of an acyclic complex of projectives. We denote by ac e P ( A ) the class of all acycliccomplexes of projective A -modules. For a module M , we use M to denote the complex with M oncentrated in degree zero. The cardinal of a complex C := · · · → C i +1 → C i → C i − → · · · isdefined to be | C | = | L i ∈ Z C i | . Claim 1.
Let ℵ > | A | + ℵ be an infinite cardinal, P := · · · → P ∂ → P ∂ → P − ∂ − → · · · be a complex in ac e P ( A ). Let C = M be a subcomplex of P , where M ≤ P is a submodulewith | M | ≤ ℵ . There exists a subcomplex D ∈ ac e P ( A ), such that | D | ≤ ℵ , C ≤ D and D / C ∈ ac e P ( A ).It follows from the Kaplansky theorem that every projective module is a direct sum of countablygenerated projective modules. Then P n = L i ∈ I n P n,i with each P n,i countably generated. Let S = L i ∈ J P ,i , where J = { i ∈ I | M ∩ P ,i = 0 } . Then M ≤ S , | S | ≤ ℵ , S and P /S areprojective modules. We can now consider the acyclic complex · · · / / L ∂ / / L ∂ / / L ∂ / / L ∂ / / S ∂ / / ∂ ( S ) / / , ( S L i is a submodule of P i of cardinality less than or equal to ℵ such that ∂ i ( L i ) =Ker( ∂ i − | L i − ) for all i > L = S ). Now, we can embed ∂ ( S ) into a projectivesubmodule S − ≤ P − , such that | S − | ≤ ℵ and P − /S − being a projective module. Thenconsider the acyclic complex · · · / / L ∂ / / L ∂ / / L ∂ / / L ∂ / / S − ∂ − / / ∂ − ( S − ) / / , ( S L i is taken as before. If we embed L into a projective submodule S of P andconstruct L i as before, we then get a complex which is also acyclic: · · · / / L ∂ / / L ∂ / / S ∂ / / S − + ∂ ( S ) ∂ − / / ∂ − ( S − ) / / . ( S S ≤ P with | S | ≤ ℵ , which contains L , such that P /S is a projective module. We then get an acyclic complex · · · / / L ∂ / / L ∂ / / S ∂ / / S + ∂ ( S ) ∂ / / S − + ∂ ( S ) ∂ − / / ∂ − ( S − ) / / . ( S · · · / / L ∂ / / L ∂ / / L ∂ / / S ∂ / / S − + ∂ ( S ) ∂ − / / ∂ − ( S − ) / / , ( S · · · / / L ∂ / / L ∂ / / L ∂ / / L ∂ / / S − ∂ − / / ∂ − ( S − ) / / , ( S · · · / / L ∂ / / L ∂ / / L ∂ / / L − ∂ − / / S − ∂ − / / ∂ − ( S − ) / / , ( S · · · / / L ∂ / / L ∂ / / L ∂ / / S − ∂ − / / S − + ∂ − ( S − ) ∂ − / / ∂ − ( S − ) / / , ( S · · · / / L ∂ / / L ∂ / / S ∂ / / S − + ∂ ( S ) ∂ − / / S − + ∂ − ( S − ) ∂ − / / ∂ − ( S − ) / / , ( S S ki are projective submodules of P i , such that | S ki | ≤ ℵ and P i /S ki being projective. f we continue this zig-zag procedure, we then find acyclic complexes ( Sn ) for all n , in sucha way that there are infinitely many n with ( Sn ) i a projective submodule of P i for each i ∈ Z .Furthermore, we have M ≤ ( Sn ) and | ( Sn ) | ≤ ℵ · ℵ ≤ ℵ for any n . Let D be the direct limitof ( Sn ), n ∈ Z . Then D is the desired acyclic complex of projective modules. Claim 2.
Let ℵ > | A | + ℵ be an infinite cardinal, and M a Gorenstein projective A -module.Then for any submodule K ≤ M with | K | ≤ ℵ , there exists a submodule N of M , such that K ≤ N , N and M/N are Gorenstein projective modules, and | N | ≤ ℵ .There exists an acyclic complex P := · · · → P → P → P − → · · · of projective A -modules,such that M = Ker( P → P − ). By the above argument, for complex C = K , there is an acyclicsubcomplex D := · · · → D → D → D − → · · · of projective A -modules, such that | D | ≤ ℵ , C ≤ D and D / C ∈ ac e P ( A ). Thus, N = Ker( D → D − ) is the desired submodule of M . Claim 3. ( GP , W ) is a cotorsion pair cogenerated by a set.Let M ∈ GP . By transfinite induction we can find a continuous chain of submodules of M ,say { M α ; α < λ } , for some ordinal number λ such that M = ∪ α<λ M α ; M , M α +1 /M α are in GP ,and | M | ≤ ℵ , | M α +1 /M α | ≤ ℵ for any α < λ . But since GP is closed under extensions anddirect limits, in fact each M α belongs to GP , and so every module in GP is the direct union of acontinuous chain of submodules in GP with cardinality less than or equal to ℵ . Note that GP isa Kaplansky class (see [8, 10]), or equivalently, a deconstructible class (see [23]).Thus, if we let S be a representative set of modules M ∈ GP with | M | ≤ ℵ , then a module N ∈ GP ⊥ if and only if Ext A ( M, N ) = 0 for any M ∈ S , that is, ( GP , GP ⊥ ) is cogenerated bythe set S (see e.g. [6, Theorem 7.3.4]). The equality GP ⊥ = W follows by a standard argument,so we omit it. This completes the proof. (cid:3) The separable algebra enjoys some of the attractive properties of semisimple algebras. Theseparability of rings and algebras has been concerned by many authors, for example, Azumaya,Auslander and Goldman. We refer to [20, Charpter 10] and [14, Section 2.4] for separable rings(algebras).
Definition 2.8.
A ring extension R ⊂ A is separable provided the multiplication map ϕ : A ⊗ R A → A ( a ⊗ R b → ab ) is a split epimorphism of A -bimodules. If R ⊂ A is simultaneously aFrobenius extension and a separable extension, then it is called a separable Frobenius extension. Note that for any left A -module M , there is a natural map θ : A ⊗ R M → M given by θ ( a ⊗ R m ) = am for any a ∈ A and m ∈ M . It is easy to check that θ is surjective, and asan R -homomorphism it is split. However, in general θ is not split as an A -homomorphism. Thefollowing is analogous to the results in [20] for separable algebras over commutative rings. emma 2.9. The following are equivalent: (1) R ⊂ A is a separable extension. (2) For any A -bimodule M , θ : A ⊗ R M → M is a split epimorphism of A -bimodules. (3) There exists an element e ∈ A ⊗ R A , such that ϕ ( e ) = 1 A and ae = ea for any a ∈ A .Proof. (1) is a special case of (2) by letting M = A . Now assume (1) holds. For an A -bimodule M , we have the following diagram( A ⊗ R A ) ⊗ A M ϕ ⊗ id M / / µ (cid:15) (cid:15) A ⊗ A M π (cid:15) (cid:15) A ⊗ R M θ / / M where π is a natural isomorphism, and µ is the composition( A ⊗ R A ) ⊗ A M −→ A ⊗ R ( A ⊗ A M ) id A ⊗ π −→ A ⊗ R M. An easy calculation shows that the diagram commutes. Let ψ : A → A ⊗ R A be a homomorphismof A -bimodules such that ϕψ = id A . If we define χ = µ ( ψ ⊗ id M ) π − , then χ is an A -bimodulehomomorphism such that θχ = id M . Hence, the epimorphism of A -bimodules θ : A ⊗ R M → M is split.It remains to prove the equivalence of (1) and (3). If ϕ : A ⊗ R A → A is split, then e = ψ (1 A ) ∈ A ⊗ R A , such that ϕ ( e ) = ϕ ( ψ (1 A )) = 1 A , and ae = ψ ( a A ) = ψ (1 A a ) = ea for any a ∈ A . Conversely, if there is an element e ∈ A ⊗ R A satisfying (3), and ψ : A → A ⊗ R A is definedby ψ ( a ) = ae , then ϕψ ( a ) = ϕ ( ae ) = aϕ ( e ) = a . Moreover, ψ ( ab ) = ( ab ) e = a ( be ) = aψ ( b ),and ψ ( ab ) = a ( be ) = a ( eb ) = ( ae ) b = ψ ( a ) b , that is, ψ is an A -bimodule homomorphism. Thus, R ⊂ A is separable. (cid:3) Example 2.10. (1)
For a finite group G , Z ⊂ Z G is a separable Frobenius extension. Indeed,let e = | G | P g ∈ G g ⊗ Z g − ∈ Z G ⊗ Z Z G , where | G | is the order of G . It is direct to check that e satisfies the condition (3) of the above lemma. (2) ([14, Example 2.7]) Let F be a field and set A = M ( F ) . Let R be the subalgebra of A with F -basis consisting of the idempotents and matrix units: e = e + e , e = e + e , e , e , e , e , e . Then R ⊂ A is a separable Frobenius extension. If R ⊂ A is a separable extension, it follows from the above argument that as left A -modules, M is a direct summand of A ⊗ R M . The following is immediate from Lemma 2.2 and Lemma2.3(2). Theorem 2.11.
Let R ⊂ A be a separable Frobenius extension, M a left A -module. Then M is a Gorenstein projective A -module if and only if the underlying R -module M is Gorensteinprojective. e note that relationship between Gorenstein projective modules over ring extensions areconsidered in other conditions, for example, in [13] for excellent extensions of rings, and in [16]for cross product of Hopf algebras.3. Gorenstein projective graded R [ x ] / ( x ) -modules Throughout this section, R is an arbitrary ring, A = R [ x ] / ( x ) is the quotient of the polynomialring, where x is a variable which is supposed to commute with all the elements of R . Lemma 3.1.
The extension of rings R ⊂ A is a Frobenius extension.Proof. It is clear that A R is a finitely generated projective module. There is an R - A -homomorphism ϕ : A → Hom R op ( A A R , R ) given by ϕ ( r + r x )( s + s x ) = r s + r s + r s for any r + r x and s + s x in A , and a homorphism ψ : Hom R op ( A A R , R ) → A which maps any f ∈ Hom R op ( A A R , R )to an element f ( x ) + ( f (1) − f ( x )) x in A . It is direct to check that ϕψ = id and ψϕ = id. Theassertion follows. (cid:3) One can view A as a graded ring with a copy of R (generated by 1) in degree 0 and a copyof R (generated by x ) in degree 1, and 0 otherwise. A graded A -module M is an A -modulewith a additive subgroup decomposition M = L i ∈ Z M i , such that A i M j ⊂ M i + j for all i and j . Consider graded A -modules M and N . An A -linear map f : M → N has degree d if f ( M i ) ⊂ N i + d . The set of all degree d maps from M to N is denoted by Hom A ( M, N ) d . We defineHom Gr ( M, N ) := Hom A ( M, N ) . The category GrMod( A ) consists of graded left A -modules andthe morphisms are taken to be the graded morphism of degree zero. Note that by forgetting thegrading on a module, there is naturally a functor GrMod( A ) → Mod( A ).There is an observation that the category GrMod( A ) is isomorphic to the category Ch( R ) of R -complexes, where M = L i ∈ Z M i corresponds to the cochain complex · · · → M i − → M i → M i +1 → · · · of R -modules, with the differential corresponding to multiplication by x ; see forexample [11]. It is clear that the isomorphism of categories between GrMod( A ) and Ch( R )automatically preserves projectives.Let C be an abelian category with enough projectives. An object M ∈ C is said to be Gorensteinprojective if it is a syzygy of a totally acyclic complex of projectives. The notion of Gorensteinprojective complexes is introduced by Enochs and Garc´ıa Rozas [7, Definition 4.1] as Gorensteinprojective objects in Ch( R ). We call the Gorenstein projective objects in GrMod( A ) to beGorenstein projective graded A -modules. Observation.
Let M = L i ∈ Z M i ∈ GrMod( A ) . Then M is a Gorenstein projective graded A -module if and only if · · · → M i − → M i → M i +1 → · · · is a Gorenstein projective R -complex. The main result of this section is stated as follows.
Theorem 3.2.
Let M ∈ GrMod( A ) be a graded A -module. The following are equivalent: (1) M is Gorenstein projective in GrMod( A ) . M is Gorenstein projective in Mod( A ) . (3) M is Gorenstein projective in Mod( R ) . The next result is immediate, which generalizes [7, Theorem 4.5] by removing the prerequisitethat the base ring is Iwanaga-Gorenstein, and generalizes [17, Theorem 3.1] by removing thecondition that the base ring is right coherent and left perfect; see also [25, Theorem 1].
Corollary 3.3.
Let M be an R -complex. Then M is Gorenstein projective in Ch( R ) if and onlyif each item M i is Gorenstein projective in Mod( R ) . There is a result due to Gillespie and Hovey [11, Proposition 3.8]: every dg-projective complexover R is a Gorenstein projective A -module, and the converse holds if R is left and right noetherianand of finite global dimension. It is well-known that the projective dimension of a Gorensteinprojective module is either zero or infinity, see for example [6, Proposition 10.2.3]. If R is a ringof finite global dimension, then dg-projective R -complex and Gorenstein projective R -complexcoincide. So the assumption of noetherian ring in [11, Proposition 3.8] is not needed.In the rest of this section, we are devoted to prove Theorem 3.2. For any graded A -module M and d ∈ Z , we define M [ d ] to be a shift of M , which is equal to M as an ungraded A -module buthas grading M [ d ] i = M i + d . For any R -module N , we denote by N the graded A -module with N in degree -1 and 0; the differential corresponding to multiplication by x is exactly the identity of N . The next result is well-known. Lemma 3.4.
Let N be a graded A -module. Then N is projective in GrMod( A ) if and only if N is projective in Mod( A ) . If we consider N as an R -complex, then N is projective in Ch( R ) , andthere is a family of projective R -modules { P i } i ∈ Z such that N = Q i ∈ Z P i [ − i ] . Lemma 3.5.
Let M be a graded A -module. If M is Gorenstein projective in GrMod( A ) , thenthe ungraded module M is Gorenstein projective in Mod( A ) .Proof. Let M ∈ GrMod( A ). Assume that there is a totally acyclic complexes of projectives P := · · · → P → P → P − → · · · in GrMod( A ), such that M = Ker( P → P − ). Note thatevery item P j = L i ∈ Z P ij is a projective module in Mod( A ), and then P is also an exact sequenceof projective modules in Mod( A ).Let D be a projective left R -module. Then D [ − i ] is projective in GrMod( A ) for any i ∈ Z . Notethat for any N ∈ GrMod( A ), we have Hom Gr ( N, D [ − i ]) ∼ = Hom Ch( R ) ( N, D [ − i ]) ∼ = Hom R ( N i , D ).Then, the complex Hom Gr ( P , D [ − i ]) ∼ = Hom R ( P i , D ) is acyclic, where P i := · · · → P i → P i → P i − → · · · . Moreover, the complex Hom R ( P , D ) is acyclic for any projective R -module D .Let Q be a projective left A -module. Then Q is a projective left R -module, and A ⊗ R Q is aprojective A -module. The canonical epimorphism θ : A ⊗ R Q → Q of A -modules is split. More-over, by the argument in Lemma 2.3, there is an isomorphism Hom A ( P , A ⊗ R Q ) ∼ = Hom R ( P , Q ).This implies that the complex Hom A ( P , A ⊗ R Q ) is acyclic. Hence, Hom A ( P , Q ) is acyclic. It ields that P is a totally acyclic complex of projective A -modules, and M is Gorenstein projectivein Mod( A ). (cid:3) Lemma 3.6.
Let M ∈ GrMod( A ) . If M is Gorenstein projective in Mod( A ) , then there is anexact sequence → M → N → L → in GrMod( A ) with N projective, L Gorenstein projectivein
Mod( A ) ; and moreover, it also remains exact after applying Hom Gr ( − , P ) for any projectivemodule P ∈ GrMod( A ) .Proof. We consider the graded A -module M = L i ∈ Z M i as an R -complex with differential δ of degree 1. Since M is Gorenstein projective in Mod( A ), each M i is a Gorenstein projective A -module. By Lemma 2.2, M is a Gorenstein projective R -module, and so is M i for any i ∈ Z .Then there exists an exact sequence 0 → M i f i → G i → H i → R ) with G i projectiveand H i Gorenstein projective. Let D be any projective R -module. For any g i : M i → D , thereexists an R -homomorphism h i : G i → D such that g i = h i f i .Consider the following commutative diagram... (cid:15) (cid:15) ... (cid:15) (cid:15) M i − δ (cid:15) (cid:15) g i δ ●●●●●●●●●● (cid:18) f i − f i δ (cid:19) / / N i − = G i − ⊕ G i (0 h i ) t t ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ( ) (cid:15) (cid:15) DM i (cid:15) (cid:15) g i " " ❋❋❋❋❋❋❋❋❋❋ (cid:18) f i f i +1 δ (cid:19) / / N i = G i ⊕ G i +1( h i u u ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ (cid:15) (cid:15) ... D ...This implies that there exists an exact sequences 0 → M → N → L → A ) with N projective, such that the induced sequence 0 → Hom Gr ( L, D [ − i ]) → Hom Gr ( N, D [ − i ]) → Hom Gr ( M, D [ − i ]) → → Hom R ( L i , D ) → Hom R ( N i , D ) → Hom R ( M i , D ) → Ext R ( L i , D ) → . So Ext R ( L i , D ) = 0. Specifically, Ext R ( L i , G i ) = 0, and then we get the following commutativediagram: 0 / / M i / / N i / / (cid:15) (cid:15) L i / / (cid:15) (cid:15) / / M i / / G i / / H i / / L i ⊕ G i = H i ⊕ N i , and then L i is Gorensteinprojective in Mod( R ). So L = L i ∈ Z L i is also a Gorenstein projective R -module. et Q be a projective module in Mod( A ). Then Ext A ( L, A ⊗ R Q ) ∼ = Ext R ( L, Q ) = 0. Since Q is a direct summand of A ⊗ R Q , Ext A ( L, Q ) = 0, and then it yields from the exact sequence0 → M → N → L → A ) that L is a Gorenstein projective A -module.Let P ∈ GrMod( A ) be projective. Then P = Q i ∈ Z P i [ − i ] for a family of projective R -modules { P i } i ∈ Z . Note that for any graded A -module M , Hom Gr ( M, P ) ∼ = Q i ∈ Z Hom R ( M i , P i ). Then,from the exact sequence0 → Y i ∈ Z Hom R ( L i , P i ) → Y i ∈ Z Hom R ( N i , P i ) → Y i ∈ Z Hom R ( M i , P i ) → , we deduce the desired exact sequence0 −→ Hom Gr ( L, P ) −→ Hom Gr ( N, P ) −→ Hom Gr ( M, P ) −→ . (cid:3) Lemma 3.7.
Let M ∈ GrMod( A ) . If M is Gorenstein projective in Mod( A ) , then there isan exact sequence → K → N → M → in GrMod( A ) , where N is projective and K isGorenstein projective in Mod( A ) . Moreover, it also remains exact after applying Hom Gr ( − , P ) for any projective module P ∈ GrMod( A ) .Proof. Let M = L i ∈ Z M i ∈ GrMod( A ), P a projective module in GrMod( A ). Then P = Q i ∈ Z P i [ − i ], where P i are projective R -modules. Moreover, Hom Gr ( M, P ) ∼ = Q i ∈ Z Hom R ( M i , P i ).Since the category GrMod( A ) has enough projectives, there exists an exact sequence 0 → K → N → M → A ) with N projective. Considered as an exact sequence in Mod( A ), ityields that K is Gorenstein projective in Mod( A ) since the class of Gorenstein projective modulesis closed under taking kernel of epimorphisms.Since M i is Gorenstein projective in Mod( A ), it follows from Lemma 2.2 that M i is alsoGorenstein projective as an R -module. Then the sequence0 → Y i ∈ Z Hom R ( M i , P i ) → Y i ∈ Z Hom R ( N i , P i ) → Y i ∈ Z Hom R ( K i , P i ) → Y i ∈ Z Ext R ( M i , P i ) = 0is exact. This yields the desired exact sequence0 −→ Hom Gr ( M, P ) −→ Hom Gr ( N, P ) −→ Hom Gr ( K, P ) −→ . (cid:3) Proof of Theorem 3.2. (1) ⇒ (2) is precisely the result of Lemma 3.5. (2) ⇒ (3) follows fromLemma 2.2 since A = R [ x ] / ( x ) is a Frobenius extension of R .(2) ⇒ (1). Let M ∈ GrMod( A ), and M is Gorenstein projective in Mod( A ). By Lemma 3.7,there is an exact sequence 0 → K → P → M → A ), where P is projectiveand K is Gorenstein projective in Mod( A ), which is also Hom Gr ( − , P )-exact for any projectivemodule P ∈ GrMod( A ). Repeat this procedure, we get a Hom Gr ( − , P )-exact exact sequence · · · → P → P → M → A ) with P i projective. Similarly, by applying Lemma 3.6, e have a Hom Gr ( − , P )-exact exact sequence 0 → M → P → P − → · · · in GrMod( A ) with P i projective. Splice this two sequences together, and then we obtain a totally acyclic complex ofprojectives in GrMod( A ), such that M is Gorenstein projective in GrMod( A ).(3) ⇒ (2). By Lemma 2.3(1), it suffices to construct the right part of the totally acyclic complexof projective A -modules. Since M is a Gorenstein projective R -module, the argument in Lemma3.6 works, that is, there is an exact sequence 0 → M → P → L → A ), where P isprojective and L is Gorenstein projective in Mod( R ). Moreover, the sequence is Hom R ( − , D )-exact for any projective R -module D . Let P be any projective A -module. Thus, the abovesequence is Hom A ( − , A ⊗ R P )-exact, and furthermore, Hom A ( − , P )-exact. Successively, we builda Hom A ( − , P )-exact exact sequence 0 → M → P → P − → · · · with P i being projective A -modules. This completes the proof. (cid:3) Finally, let us mention recent works on R [ x ] / ( x )-modules. Note that A = R [ x ] / ( x ) is thering of dual numbers over R , and differential R -modules (i.e. modules equipped with an R -endomorphism of square zero) are just A -modules. Avramov, Buchweitz and Iyengar [1] introduceprojective, free and flat classes for differential modules and give some inequalities. These resultsspecialize to basic theorems in commutative algebra and algebraic topology. Ringel and Zhang[21] investigate representations of quivers over the algebra of dual numbers; for a hereditaryArtin algebra R , a bijective correspondence between the stable category of finitely generatedGorenstein projective differential R -modules and the category of finitely generated R -modules isgiven. Wei [24] shows that for any ring, a differential module is Gorenstein projective if and onlyif its underlying module is Gorenstein projective. ACKNOWLEDGEMENTS.
This work was supported by National Natural Science Founda-tion of China (Grant No. 11401476) and China Postdoctoral Science Foundation (Grant No.2016M591592). The author thanks Professor Chen Xiao-Wu for sharing his thoughts on thistopic. This research was completed when author was a postdoctor at Fudan University supervisedby Professor Wu Quan-Shui. The author thanks the referee for helpful comments and suggestions.
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