Gorenstein projective modules and Frobenius extensions

Abstract

We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective, then its underlying module over the the base ring is Gorenstein projective; the converse holds if the Frobenius extension is either left-Gorenstein or separable (e.g. the integral group ring extension $\mathbb{Z}\subset \mathbb{Z}G$). Moreover, for the Frobenius extension $R\subset A=R[x]/(x^2)$, we show that: a graded $A$-module is Gorenstein projective in $\mathrm{GrMod}(A)$, if and only if its ungraded $A$-module is Gorenstein projective, if and only if its underlying $R$-module is Gorenstein projective. It immediately follows that an $R$-complex is Gorenstein projective if and only if all its items are Gorenstein projective $R$-modules.

Full text

arXiv:1707.05885v1 [math.KT] 18 Jul 2017
Gorenstein projective modules and Frobenius extensions
Ren Wei
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
Abstract
We prove that for a Frobenius extension, if a module over the extension ring is
Gorenstein projective, then its underlying module over the the base ring is Gorenstein
projective; the converse holds if the Frobenius extension is either left-Gorenstein or
separable (e.g. the integral group ring extension Z⊂ZG).
Moreover, for the Frobenius extension R⊂A=R[x]/(x2), we show that: a graded
A-module is Gorenstein projective in GrMod(A), if and only if its ungraded A-module
is 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 if
all its items are Gorenstein projective R-modules.
Key Words: Gorenstein projective module; Frobenius extension; graded module
2010 MSC: 16G50, 13B02, 16W50.
1. Introduction
A module Mis said to be Gorenstein projective [6] if there exists a totally acyclic complex
of projective modules P:= · · · → P1→P0→P−1→ · · · such that M= Ker(P0→P−1). The
study 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 cohomology
theory 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 Ain the sense that
some useful information can transfer between Rand A. In this paper, we intend to study relations
of Gorenstein projective modules along Frobenius extensions of rings. The theory of Frobenius
extensions was developed by Kasch [15] as a generalization of Frobenius algebras, and was further
studied by Nakayama-Tsuzuku [19] and Morita [18]. A classical example of Frobenius extension
is the integral group ring extension Z⊂ZGfor a finite group G. Other examples include Hopf
E-mail address: renwei@fudan.edu.cn.
1

subalgebras [22], finite extensions of enveloping algebras of Lie super-algebras [3], enveloping
algebras 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,
aZG-module, or equivalently an integral representation of G, is maximal Cohen-Macaulay over
ZGif and only if the underlying Z-module is maximal Cohen-Macaulay, or equivalently, the
underlying 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 transfer
along such extension. However, is this true for not necessarily finitely generated Gorenstein
projective 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⊂Abe a Frobenius extension, Ma left A-module. If Mis Gorenstein
projective in Mod(A), then the underlying R-module Mis Gorenstein projective; the converse
holds if R⊂Ais either a left-Gorenstein or a separable Frobenius extension.
We remark that Z⊂ZGis both a left-Gorenstein and a separable Frobenius extension,
so Buchweitz’s observation is true for not necessarily finitely generated Gorenstein projective
modules. 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
Wto denote the class of modules with finite projective dimension. However, we further show
in 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 of
particular interest, since this will induce a cofibrantly generated model structure on the category
of modules by applying Hovey’s correspondence [12, Theorem 2.2], such that the associated
homotopy category is exactly the stable category GP .
The second inspirational example of this paper is the ring extension R⊂A=R[x]/(x2). One
can also view Aas 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]/(x2), there is an observation that the category GrMod(A) is
automatically isomorphic to the category Ch(R) of R-complexes; see for example [11]. So a
Gorenstein 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 is
Gorenstein 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 conditions
that the base ring Ris Iwanaga-Gorenstein and is right coherent and left perfect, respectively.
2

The paper is organized as follows. In Section 2, we introduce the notion of left-Gorenstein
Frobenius extensions, and it is shown that over left-Gorenstein rings, (GP ,W) is a cotorsion
pair cogenerated by a set. We study the separable Frobenius extensions. Then, Theorem A is
proved. In Section 3, we focus on Gorenstein projective graded R[x]/(x2)-modules, and we prove
the result in Theorem B.
2. Gorenstein projective modules over Frobenius extensions
Throughout, all rings are associative with a unit. Homomorphisms of rings are required to
send the unit to the unit. Let Rbe a ring. A left R-module Mis sometimes written as RM.
For two left R-modules Mand N, denote by HomR(M, N) the abelian group consisting of left
R-homomorphisms between them. A right R-module Mis sometimes written as MR. We identify
right R-modules with left Rop-modules, where Rop is the opposite ring of R. For two right R-
modules Mand N, the abelian group of right R-homomorphisms is denoted by HomRop (M, N ).
We denote by Mod(R) the category of left R-modules, and Mod(Rop) the category of right R-
modules. Let Sbe another ring. An R-S-bimodule Mis written as RMS.
We always denote a ring extension ι:R ֒→Aby R⊂A. The natural bimodule RARis
given by rar′:= ι(r)·a·ι(r′). Similarly, we consider RAand RAAetc. For a ring extension
R⊂A, there is a restricted functor Res : Mod(A)→Mod(R) sends AMto RM, given by
rm := ι(r)m. The structure map ιis usually suppressed. In the opposite direction, there are
functors T=A⊗R−: Mod(R)→Mod(A) and H= HomR(A, −) : Mod(R)→Mod(A). It is
clear that (T, Res) and (Res, H ) are adjoint pairs.
2.1 Frobenius extensions
We refer to [14, Definition 1.1, Theorem 1.2] for the definition of Frobenius extensions.
Definition 2.1. A ring extension R⊂Ais a Frobenius extension, provided that one of the
following equivalent conditions holds:
(1) The functors T=A⊗R−and H= HomR(A, −)are naturally equivalent.
(2) RAis finite generated projective and AAR∼
=(RAA)∗= HomR(RAA, R).
(3) ARis finite generated projective and RAA∼
=(AAR)∗= HomRop (AAR, R).
(4) There exists an R-R-homomorphism τ:A→Rand elements xi,yiin A, such that for
any a∈A, one has P
i
xiτ(yia) = aand P
i
τ(axi)yi=a.
Lemma 2.2. Let R⊂Abe a Frobenius extension of rings, Ma left A-module. If AMis
Gorenstein projective, then the underlying left R-module RMis also Gorenstein projective.
Proof. Let Mbe a Gorenstein projective left A-module. There exists a totally acyclic complex, i.e.
an acyclic complex of projective A-modules P:= · · · → P1→P0→P−1→ · · · with HomA(P, P )
being an acyclic complex for each projective A-module P, such that M= Ker(P0→P−1). Note
3

that each Piis a projective left R-module. Then by restricting Pone gets an acyclic complex of
projective R-modules.
Let Qbe a projective left R-module. It follows from isomorphisms HomR(A, Q)∼
=A⊗RQthat
HomR(A, Q) is a projective left A-modules. Then the complex HomA(P,HomR(A, Q)) is acyclic.
Moreover, there are isomorphisms HomR(P, Q)∼
=HomR(A⊗AP, Q)∼
=HomA(P,HomR(A, Q)).
This implies that the complex HomR(P, Q) is acyclic, and hence the underlying R-module Mis
Gorenstein projective. 
Lemma 2.3. Let R⊂Abe a Frobenius extension of rings, Ma left A-module. If the underlying
module RMis Gorenstein projective, then the following hold:
(1) For any projective A-module Pand any i > 0,Exti
A(M, P ) = 0.
(2) A⊗RMis a Gorenstein projective left A-module.
Proof. (1) For any left A-module Mand any left R-module N, there are isomorphisms
HomA(M, A ⊗RN)∼
=HomA(M, HomR(A, N )) ∼
=HomR(A⊗AM, N )∼
=HomR(M, N ).
Moreover, by replacing AMwith an A-projective resolution P•of Mand observing that P•is also
an R-projective resolution of RM, we have an isomorphism of cohomology Exti
A(M, A ⊗RN)∼
=
Exti
R(M, N ) for any i > 0.
Let Pbe a projective left A-module. There is a split epimorphism θ:A⊗RP→Pof A-
modules given by θ(a⊗Rx) = ax for any a∈Aand x∈P, and then Pis a direct summand of
A⊗RP. Since Pis projective as a left R-module, and RMis Gorenstein projective by assumption,
we have Exti
A(M, A ⊗RP)∼
=Exti
R(M, P ) = 0, and then Exti
A(M, P ) = 0 as desired.
(2) Let P:= · · · → P1→P0→P−1→ · · · be a totally acyclic complex of projective R-modules
such that RM= Ker(P0→P−1). It is easy to see that A⊗RPis an acyclic complex of projective
A-modules, and A⊗RM= Ker(A⊗RP0→A⊗RP−1). Moreover, for any projective A-module P,
the complex HomA(A⊗RP, P )∼
=HomR(P, P ) is acyclic. So A⊗RMis a Gorenstein projective
left A-module. 
2.2 Left-Gorenstein Frobenius extensions
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 global
Gorenstein projective dimension of Λ is finite. By [6, Theorem 10.2.14], each Iwanaga-Gorenstein
ring (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 Sn=S[x1, x2,··· , xn] be the polynomial
ring in nindeterminates over a non-noetherian hereditary ring S. Let Ri=Si−1⊗Si−1be the
trivial extension of Si−1by Si−1for i≥1 (set S0=S). Then Riis a left-Gorenstein ring for
every i≥1, whereas Riis non-noetherian, and hence is not Iwanaga-Gorenstein.
4

Definition 2.4. Let R⊂Abe a Frobenius extension. Then R⊂Ais called a left-Gorenstein
Frobenius extension provided in addition that Ais left-Gorenstein.
Theorem 2.5. Let R⊂Abe a left-Gorenstein Frobenius extension of rings, Ma left A-module.
Then Mis 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 RMis Gorenstein
projective, Mis a Gorenstein projective left A-module.
Note that over a left-Gorenstein ring A, a module Mis Gorenstein projective if and only if
Exti
A(M, N ) = 0 for any module Nof finite projective dimension and any i > 0; see [2] or
Theorem 2.7 below. Assume that Nis an A-module with projective dimension n. Then there
is an exact sequence 0 →K→P→N→0 of A-modules, where Pis projective and Kis of
projective dimension n−1. By induction on the projective dimension of modules, it is deduced
from Lemma 2.3(1) that Exti
A(M, N )∼
=Exti+1
A(M, K ) = 0. The assertion follows. 
For a finite group G, it is easy to see that the integral group ring ZGis Iwanaga-Gorenstein,
since there is an exact sequence 0 →ZG→QG→Q/ZG→0 of left or right ZG-modules,
where QG= HomZ(ZG, Q) is an injective ZG-module, and similarly Q/ZGis injective.
Corollary 2.6. Let Gbe a finite group, Ma left ZG-module. Then Mis a Gorenstein projective
left ZG-module if and only if the underlying left Z-module Mis 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|Ext1(X, Y ) = 0,∀Y∈ Y} and X⊥={Y|Ext1(X, Y ) =
0,∀X∈ X }. The cotorsion pair (X,Y) is said to be cogenerated by a set Sif S⊥=Y. Over an
Iwanaga-Gorenstein ring A, it follows from [12, Theorem 8.3] that (GP ,W) is a cotorsion pair
cogenerated by a set, where GP is the class of Gorenstein projective modules, and Wis the class
of modules with finite projective dimension.
It follows from [2] that over a left-Gorenstein ring, (GP,W) is a cotorsion pair. We have
more in the next result, which also generalizes [12, Theorem 8.3] from Iwanaga-Gorenstein rings
to 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 generated
Gorenstein projective model structure on the category of modules. Moreover, the homotopy
category associated with the model structure is exactly the stable category GP.
Theorem 2.7. Let Abe a left-Gorenstein ring. The cotorsion pair (GP ,W)is cogenerated by a
set.
Proof. Note that over a left-Gorenstein ring, a module is Gorenstein projective if and only if it
is a syzygy of an acyclic complex of projectives. We denote by ac e
P(A) the class of all acyclic
complexes of projective A-modules. For a module M, we use Mto denote the complex with M
5

concentrated in degree zero. The cardinal of a complex C:= · · · → Ci+1 →Ci→Ci−1→ · · · is
defined to be |C|=|Li∈ZCi|.
Claim 1. Let ℵ>|A|+ℵ0be an infinite cardinal, P:= · · · → P1
∂1
→P0
∂0
→P−1
∂−1
→ · · ·
be a complex in ac e
P(A). Let C=Mbe a subcomplex of P, where M≤P0is a submodule
with |M| ≤ ℵ. There exists a subcomplex D∈ac e
P(A), such that |D| ≤ ℵ,C≤Dand
D/C∈ac e
P(A).
It follows from the Kaplansky theorem that every projective module is a direct sum of countably
generated projective modules. Then Pn=Li∈InPn,i with each Pn,i countably generated. Let
S1
0=Li∈J0P0,i, where J0={i∈I0|M∩P0,i 6= 0}. Then M≤S1
0,|S1
0| ≤ ℵ,S1
0and P0/S1
0are
projective modules. We can now consider the acyclic complex
··· //L1
4
∂4//L1
3
∂3//L1
2
∂2//L1
1
∂1//S1
0
∂0//∂0(S1
0)//0,(S1)
where L1
iis a submodule of Piof cardinality less than or equal to ℵsuch that ∂i(L1
i) =
Ker(∂i−1|L1
i−1) for all i > 0 (we let L1
0=S1
0). Now, we can embed ∂0(S1
0) into a projective
submodule S2
−1≤P−1, such that |S2
−1| ≤ ℵ and P−1/S2
−1being a projective module. Then
consider the acyclic complex
··· //L2
3
∂3//L2
2
∂2//L2
1
∂1//L2
0
∂0//S2
−1
∂−1//∂−1(S2
−1)//0,(S2)
where each L2
iis taken as before. If we embed L2
0into a projective submodule S3
0of P0and
construct L3
ias before, we then get a complex which is also acyclic:
··· //L3
2
∂2//L3
1
∂1//S3
0
∂0//S2
−1+∂0(S3
0)∂−1//∂−1(S2
−1)//0.(S3)
Now choose a projective submodule S4
1≤P1with |S4
1| ≤ ℵ, which contains L3
1, such that P1/S4
1
is a projective module. We then get an acyclic complex
··· //L4
3
∂3//L4
2
∂2//S4
1
∂1//S3
0+∂1(S4
1)∂0//S2
−1+∂0(S3
0)∂−1//∂−1(S2
−1)//0.(S4)
Now we turn over and get the following acyclic complexes
··· //L5
3
∂3//L5
2
∂2//L5
1
∂1//S5
0
∂0//S2
−1+∂0(S5
0)∂−1//∂−1(S2
−1)//0,(S5)
··· //L6
3
∂3//L6
2
∂2//L6
1
∂1//L6
0
∂0//S6
−1
∂−1//∂−1(S6
−1)//0,(S6)
··· //L7
2
∂2//L7
1
∂1//L7
0
∂0//L7
−1
∂−1//S7
−2
∂−2//∂−2(S7
−2)//0,(S7)
··· //L8
2
∂2//L8
1
∂1//L8
0
∂0//S8
−1
∂−1
//S7
−2+∂−1(S8
−1)∂−2//∂−2(S7
−2)//0,(S8)
··· //L9
2
∂2//L9
1
∂1//S9
0
∂0//S8
−1+∂0(S9
0)∂−1//S7
−2+∂−1(S8
−1)∂−2//∂−2(S7
−2)//0,(S9)
where Sk
iare projective submodules of Pi, such that |Sk
i| ≤ ℵ and Pi/Sk
ibeing projective.
6

If we continue this zig-zag procedure, we then find acyclic complexes (Sn) for all n, in such
a way that there are infinitely many nwith (Sn)ia projective submodule of Pifor each i∈Z.
Furthermore, we have M≤(Sn)0and |(Sn)| ≤ ℵ0· ℵ ≤ ℵ for any n. Let Dbe the direct limit
of (Sn), n∈Z. Then Dis the desired acyclic complex of projective modules.
Claim 2. Let ℵ>|A|+ℵ0be an infinite cardinal, and Ma Gorenstein projective A-module.
Then for any submodule K≤Mwith |K| ≤ ℵ, there exists a submodule Nof M, such that
K≤N,Nand M/N are Gorenstein projective modules, and |N| ≤ ℵ.
There exists an acyclic complex P:= · · · → P1→P0→P−1→ · · · of projective A-modules,
such that M= Ker(P0→P−1). By the above argument, for complex C=K, there is an acyclic
subcomplex D:= · · · → D1→D0→D−1→ · · · of projective A-modules, such that |D| ≤ ℵ,
C≤Dand D/C∈ac e
P(A). Thus, N= Ker(D0→D−1) 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 {Mα;α < λ}, for some ordinal number λsuch that M=∪α<λMα;M0,Mα+1/Mαare in GP ,
and |M0| ≤ ℵ,|Mα+1/Mα| ≤ ℵ for any α < λ. But since GP is closed under extensions and
direct limits, in fact each Mαbelongs to GP , and so every module in GP is the direct union of a
continuous chain of submodules in GP with cardinality less than or equal to ℵ. Note that GP is
a Kaplansky class (see [8, 10]), or equivalently, a deconstructible class (see [23]).
Thus, if we let Sbe a representative set of modules M∈ GP with |M| ≤ ℵ, then a module
N∈ GP ⊥if and only if Ext1
A(M, N ) = 0 for any M∈ S, that is, (GP ,GP ⊥) is cogenerated by
the set S(see e.g. [6, Theorem 7.3.4]). The equality GP⊥=Wfollows by a standard argument,
so we omit it. This completes the proof. 
2.3 Separable Frobenius extensions
The separable algebra enjoys some of the attractive properties of semisimple algebras. The
separability 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⊂Ais separable provided the multiplication map ϕ:A⊗R
A→A(a⊗Rb→ab) is a split epimorphism of A-bimodules. If R⊂Ais simultaneously a
Frobenius 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⊗RM→Mgiven by
θ(a⊗Rm) = am for any a∈Aand m∈M. It is easy to check that θis surjective, and as
an R-homomorphism it is split. However, in general θis not split as an A-homomorphism. The
following is analogous to the results in [20] for separable algebras over commutative rings.
7

Lemma 2.9. The following are equivalent:
(1) R⊂Ais a separable extension.
(2) For any A-bimodule M,θ:A⊗RM→Mis a split epimorphism of A-bimodules.
(3) There exists an element e∈A⊗RA, such that ϕ(e) = 1Aand 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⊗RA)⊗AMϕ⊗idM//
µ

A⊗AM
π

A⊗RMθ//M
where πis a natural isomorphism, and µis the composition
(A⊗RA)⊗AM−→ A⊗R(A⊗AM)idA⊗π
−→ A⊗RM.
An easy calculation shows that the diagram commutes. Let ψ:A→A⊗RAbe a homomorphism
of A-bimodules such that ϕψ = idA. If we define χ=µ(ψ⊗idM)π−1, then χis an A-bimodule
homomorphism such that θχ = idM. Hence, the epimorphism of A-bimodules θ:A⊗RM→M
is split.
It remains to prove the equivalence of (1) and (3). If ϕ:A⊗RA→Ais split, then e=
ψ(1A)∈A⊗RA, such that ϕ(e) = ϕ(ψ(1A)) = 1A, and ae =ψ(a1A) = ψ(1Aa) = ea for any
a∈A. Conversely, if there is an element e∈A⊗RAsatisfying (3), and ψ:A→A⊗RAis defined
by 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, ψis an A-bimodule homomorphism. Thus,
R⊂Ais separable. 
Example 2.10. (1) For a finite group G,Z⊂ZGis a separable Frobenius extension. Indeed,
let e=1
|G|Pg∈Gg⊗Zg−1∈ZG⊗ZZG, 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 Fbe a field and set A=M4(F). Let Rbe the subalgebra of Awith
F-basis consisting of the idempotents and matrix units:
e1=e11 +e44, e2=e22 +e33, e21, e31 , e41, e42, e43 .
Then R⊂Ais a separable Frobenius extension.
If R⊂Ais a separable extension, it follows from the above argument that as left A-modules,
Mis a direct summand of A⊗RM. The following is immediate from Lemma 2.2 and Lemma
2.3(2).
Theorem 2.11. Let R⊂Abe a separable Frobenius extension, Ma left A-module. Then M
is a Gorenstein projective A-module if and only if the underlying R-module Mis Gorenstein
projective.
8

We note that relationship between Gorenstein projective modules over ring extensions are
considered 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]/(x2)-modules
Throughout this section, Ris an arbitrary ring, A=R[x]/(x2) is the quotient of the polynomial
ring, where xis a variable which is supposed to commute with all the elements of R.
Lemma 3.1. The extension of rings R⊂Ais a Frobenius extension.
Proof. It is clear that ARis a finitely generated projective module. There is an R-A-homomorphism
ϕ:A→HomRop (AAR, R) given by ϕ(r0+r1x)(s0+s1x) = r0s0+r0s1+r1s0for any r0+r1xand
s0+s1xin A, and a homorphism ψ: HomRop (AAR, R)→Awhich maps any f∈HomRop (AAR, R)
to an element f(x) + (f(1) −f(x))xin A. It is direct to check that ϕψ = id and ψϕ = id. The
assertion follows. 
One can view Aas 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, and 0 otherwise. A graded A-module Mis an A-module
with a additive subgroup decomposition M=Li∈ZMi, such that AiMj⊂Mi+jfor all i
and j. Consider graded A-modules Mand N. An A-linear map f:M→Nhas degree dif
f(Mi)⊂Ni+d. The set of all degree dmaps from Mto Nis denoted by HomA(M, N )d. We define
HomGr(M, N ) := HomA(M, N)0. The category GrMod(A) consists of graded left A-modules and
the morphisms are taken to be the graded morphism of degree zero. Note that by forgetting the
grading 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=Li∈ZMicorresponds to the cochain complex · · · → Mi−1→Mi→
Mi+1 → · · · of R-modules, with the differential corresponding to multiplication by x; see for
example [11]. It is clear that the isomorphism of categories between GrMod(A) and Ch(R)
automatically preserves projectives.
Let Cbe an abelian category with enough projectives. An object M∈ C is said to be Gorenstein
projective if it is a syzygy of a totally acyclic complex of projectives. The notion of Gorenstein
projective complexes is introduced by Enochs and Garc´ıa Rozas [7, Definition 4.1] as Gorenstein
projective objects in Ch(R). We call the Gorenstein projective objects in GrMod(A) to be
Gorenstein projective graded A-modules.
Observation. Let M=Li∈ZMi∈GrMod(A). Then Mis a Gorenstein projective graded
A-module if and only if · · · → Mi−1→Mi→Mi+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) Mis Gorenstein projective in GrMod(A).
9

(2) Mis Gorenstein projective in Mod(A).
(3) Mis Gorenstein projective in Mod(R).
The next result is immediate, which generalizes [7, Theorem 4.5] by removing the prerequisite
that the base ring is Iwanaga-Gorenstein, and generalizes [17, Theorem 3.1] by removing the
condition that the base ring is right coherent and left perfect; see also [25, Theorem 1].
Corollary 3.3. Let Mbe an R-complex. Then Mis Gorenstein projective in Ch(R)if and only
if each item Miis Gorenstein projective in Mod(R).
There is a result due to Gillespie and Hovey [11, Proposition 3.8]: every dg-projective complex
over Ris a Gorenstein projective A-module, and the converse holds if Ris left and right noetherian
and of finite global dimension. It is well-known that the projective dimension of a Gorenstein
projective module is either zero or infinity, see for example [6, Proposition 10.2.3]. If Ris a ring
of finite global dimension, then dg-projective R-complex and Gorenstein projective R-complex
coincide. 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 Mas an ungraded A-module but
has grading M[d]i=Mi+d. For any R-module N, we denote by Nthe graded A-module with N
in degree -1 and 0; the differential corresponding to multiplication by xis exactly the identity of
N. The next result is well-known.
Lemma 3.4. Let Nbe a graded A-module. Then Nis projective in GrMod(A)if and only if N
is projective in Mod(A). If we consider Nas an R-complex, then Nis projective in Ch(R), and
there is a family of projective R-modules {Pi}i∈Zsuch that N=Qi∈ZPi[−i].
Lemma 3.5. Let Mbe a graded A-module. If Mis Gorenstein projective in GrMod(A), then
the ungraded module Mis Gorenstein projective in Mod(A).
Proof. Let M∈GrMod(A). Assume that there is a totally acyclic complexes of projectives
P:= · · · → P1→P0→P−1→ · · · in GrMod(A), such that M= Ker(P0→P−1). Note that
every item Pj=Li∈ZPi
jis a projective module in Mod(A), and then Pis also an exact sequence
of projective modules in Mod(A).
Let Dbe a projective left R-module. Then D[−i] is projective in GrMod(A) for any i∈Z. Note
that for any N∈GrMod(A), we have HomGr(N, D[−i]) ∼
=HomCh(R)(N, D[−i]) ∼
=HomR(Ni, D).
Then, the complex HomGr(P,D[−i]) ∼
=HomR(Pi, D) is acyclic, where Pi:= · · · → Pi
1→Pi
0→
Pi
−1→ · · · . Moreover, the complex HomR(P, D) is acyclic for any projective R-module D.
Let Qbe a projective left A-module. Then Qis a projective left R-module, and A⊗RQis a
projective A-module. The canonical epimorphism θ:A⊗RQ→Qof A-modules is split. More-
over, by the argument in Lemma 2.3, there is an isomorphism HomA(P, A ⊗RQ)∼
=HomR(P, Q).
This implies that the complex HomA(P, A ⊗RQ) is acyclic. Hence, HomA(P, Q) is acyclic. It
10

yields that Pis a totally acyclic complex of projective A-modules, and Mis Gorenstein projective
in Mod(A). 
Lemma 3.6. Let M∈GrMod(A). If Mis Gorenstein projective in Mod(A), then there is an
exact sequence 0→M→N→L→0in GrMod(A)with Nprojective, LGorenstein projective
in Mod(A); and moreover, it also remains exact after applying HomGr(−, P )for any projective
module P∈GrMod(A).
Proof. We consider the graded A-module M=Li∈ZMias an R-complex with differential δ
of degree 1. Since Mis Gorenstein projective in Mod(A), each Miis a Gorenstein projective
A-module. By Lemma 2.2, Mis a Gorenstein projective R-module, and so is Mifor any i∈Z.
Then there exists an exact sequence 0 →Mifi
→Gi→Hi→0 in Mod(R) with Giprojective
and HiGorenstein projective. Let Dbe any projective R-module. For any gi:Mi→D, there
exists an R-homomorphism hi:Gi→Dsuch that gi=hifi.
Consider the following commutative diagram
.
.
.

.
.
.

Mi−1
δ

giδ##
●
●
●
●
●
●
●
●
●
●
fi−1
fiδ//Ni−1=Gi−1⊕Gi
(0 hi)
tt❥❥❥❥❥❥❥❥❥❥
(0 1
0 0 )

D
Mi
gi
""
❋
❋
❋
❋
❋
❋
❋
❋
❋
❋
fi
fi+1δ//Ni=Gi⊕Gi+1
(hi0)
uu❥❥❥❥❥❥❥❥❥❥❥

.
.
.D.
.
.
This implies that there exists an exact sequences 0 →M→N→L→0 in GrMod(A) with
Nprojective, such that the induced sequence 0 →HomGr(L, D[−i]) →HomGr (N, D[−i]) →
HomGr(M, D[−i]) →0 is still exact. Moreover, we have an exact sequence
0→HomR(Li, D)→HomR(Ni, D)→HomR(Mi, D)→Ext1
R(Li, D)→0.
So Ext1
R(Li, D) = 0. Specifically, Ext1
R(Li, Gi) = 0, and then we get the following commutative
diagram:
0//Mi//Ni//

Li//

0
0//Mi//Gi//Hi//0
By a version of Schanuel’s Lemma, we have Li⊕Gi=Hi⊕Ni, and then Liis Gorenstein
projective in Mod(R). So L=Li∈ZLiis also a Gorenstein projective R-module.
11

Let Qbe a projective module in Mod(A). Then Ext1
A(L, A ⊗RQ)∼
=Ext1
R(L, Q) = 0. Since
Qis a direct summand of A⊗RQ, Ext1
A(L, Q) = 0, and then it yields from the exact sequence
0→M→N→L→0 in Mod(A) that Lis a Gorenstein projective A-module.
Let P∈GrMod(A) be projective. Then P=Qi∈ZPi[−i] for a family of projective R-modules
{Pi}i∈Z. Note that for any graded A-module M, HomGr(M , P )∼
=Qi∈ZHomR(Mi, P i). Then,
from the exact sequence
0→Y
i∈Z
HomR(Li, P i)→Y
i∈Z
HomR(Ni, P i)→Y
i∈Z
HomR(Mi, P i)→0,
we deduce the desired exact sequence
0−→ HomGr(L, P )−→ HomGr(N , P )−→ HomGr(M, P )−→ 0.

Lemma 3.7. Let M∈GrMod(A). If Mis Gorenstein projective in Mod(A), then there is
an exact sequence 0→K→N→M→0in GrMod(A), where Nis projective and Kis
Gorenstein projective in Mod(A). Moreover, it also remains exact after applying HomGr(−, P )
for any projective module P∈GrMod(A).
Proof. Let M=Li∈ZMi∈GrMod(A), Pa projective module in GrMod(A). Then P=
Qi∈ZPi[−i], where Piare projective R-modules. Moreover, HomGr(M, P )∼
=Qi∈ZHomR(Mi, P i).
Since the category GrMod(A) has enough projectives, there exists an exact sequence 0 →K→
N→M→0 in GrMod(A) with Nprojective. Considered as an exact sequence in Mod(A), it
yields that Kis Gorenstein projective in Mod(A) since the class of Gorenstein projective modules
is closed under taking kernel of epimorphisms.
Since Miis Gorenstein projective in Mod(A), it follows from Lemma 2.2 that Miis also
Gorenstein projective as an R-module. Then the sequence
0→Y
i∈Z
HomR(Mi, P i)→Y
i∈Z
HomR(Ni, P i)→Y
i∈Z
HomR(Ki, P i)→Y
i∈Z
Ext1
R(Mi, P i) = 0
is exact. This yields the desired exact sequence
0−→ HomGr(M, P )−→ HomGr(N, P )−→ HomGr(K, P )−→ 0.

Proof of Theorem 3.2. (1)⇒(2) is precisely the result of Lemma 3.5. (2)⇒(3) follows from
Lemma 2.2 since A=R[x]/(x2) is a Frobenius extension of R.
(2)⇒(1). Let M∈GrMod(A), and Mis Gorenstein projective in Mod(A). By Lemma 3.7,
there is an exact sequence 0 →K1→P1→M→0 in GrMod(A), where P1is projective
and K1is Gorenstein projective in Mod(A), which is also HomGr(−, P )-exact for any projective
module P∈GrMod(A). Repeat this procedure, we get a HomGr(−, P )-exact exact sequence
· · · → P2→P1→M→0 in GrMod(A) with Piprojective. Similarly, by applying Lemma 3.6,
12

we have a HomGr(−, P )-exact exact sequence 0 →M→P0→P−1→ · · · in GrMod(A) with Pi
projective. Splice this two sequences together, and then we obtain a totally acyclic complex of
projectives in GrMod(A), such that Mis Gorenstein projective in GrMod(A).
(3)⇒(2). By Lemma 2.3(1), it suffices to construct the right part of the totally acyclic complex
of projective A-modules. Since Mis a Gorenstein projective R-module, the argument in Lemma
3.6 works, that is, there is an exact sequence 0 →M→P0→L1→0 in GrMod(A), where P0is
projective and L1is Gorenstein projective in Mod(R). Moreover, the sequence is HomR(−, D)-
exact for any projective R-module D. Let Pbe any projective A-module. Thus, the above
sequence is HomA(−, A⊗RP)-exact, and furthermore, HomA(−, P )-exact. Successively, we build
a HomA(−, P )-exact exact sequence 0 →M→P0→P−1→ · · · with Pibeing projective A-
modules. This completes the proof. 
Finally, let us mention recent works on R[x]/(x2)-modules. Note that A=R[x]/(x2) is the
ring 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] introduce
projective, free and flat classes for differential modules and give some inequalities. These results
specialize 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 hereditary
Artin algebra R, a bijective correspondence between the stable category of finitely generated
Gorenstein projective differential R-modules and the category of finitely generated R-modules is
given. Wei [24] shows that for any ring, a differential module is Gorenstein projective if and only
if 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 this
topic. This research was completed when author was a postdoctor at Fudan University supervised
by Professor Wu Quan-Shui. The author thanks the referee for helpful comments and suggestions.
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