Gorenstein projective and injective dimensions over Frobenius extensions
aa r X i v : . [ m a t h . K T ] J u l Gorenstein pro jective and injective dimensions overFrobenius extensions
Wei Ren
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China
Abstract
Let R ⊂ A be a Frobenius extension of rings. We prove that: (1) for any left A -module M , A M is Gorenstein projective (injective) if and only if the underlyingleft R -module R M is Gorenstein projective (injective). (2) if G-proj . dim A M < ∞ ,then G-proj . dim A M = G-proj . dim R M ; the dual for Gorenstein injective dimensionalso holds. (3) if the extension is split, then G-gldim( A ) = G-gldim( R ). Key Words:
Frobenius extension, Gorenstein projective module, Gorenstein homo-logical dimension.
Introduction
The study of Gorenstein homological algebra stems from finitely generated modules of G-dimension zero over any noetherian rings, introduced by Auslander [1] as a generalization of finitegenerated projective modules. In order to complete the analogy, Enochs and Jenda introduced theGorenstein projective modules (not necessarily finitely generated) over any associative rings; anddually, Gorenstein injective modules are defined [11]. Then, Gorenstein homological dimensionsare defined in the standard way, by using resolutions of Gorenstein modules. As shown in[9, Theorem 4.2.6], the Gorenstein projective dimension of a finitely generated module over acommutative noetherian ring agrees with its G-dimension. For finitely generated Gorensteinprojective modules, there are several different terminologies in the literature, such as maximalCohen-Macaulay modules, totally reflexive modules and modules of G-dimension zero.In this paper, we intend to study the properties of Gorenstein projective (injective) modules andGorenstein homological dimensions along Frobenius extension of rings. The theory of Frobeniusextensions was developed by Kasch [19] as a generalization of Frobenius algebras, and was furtherstudied by Nakayama-Tsuzuku [21] and Morita [20] et. al. For example, the integral group ringextension Z ⊂ Z G for a finite group G , and the ring extension of dual numbers of an algebra Supported by the National Nature Science Foundation of China (11401476, 11561039) and a project ofChongqing Normal University (18XLB001).E-mail address: [email protected]. ⊂ R [ x ] / ( x ), are Frobenius extensions. There are other examples include Hopf subalgebras,finite extensions of enveloping algebras of Lie super-algebras, enveloping algebras of Lie coalgebrasetc. We refer to a lecture due to Kadison [17] for more details.We are motivated by a question raised by Chen [8]. He introduced a generalization of Frobeniusextension, called the totally reflexive extension of rings, and proved that totally reflexive modulestransfer along such extension. He asked if this is true for not necessarily finitely generatedGorenstein projective modules, and claimed that a new method for argument is needed. Our firstmain result gives an affirmative answer to this problem for Frobenius extensions. In Theorem 2.2,we show that: for any Frobenius extension R ⊂ A and any left A -module M , A M is Gorensteinprojective in Mod( A ) if and only if the underlying left R -module R M is Gorenstein projective;and a similar result for Gorenstein injective modules also holds.In the study of Gorenstein homological algebra, an interesting assertion that “every result inclassical homological algebra has a counterpart in Gorenstein homological algebra” (see Holm’sthesis [16]), can be confirmed by many works, see for example [9, 12, 15]. As a contribution tothis, we give a Gorenstein counterpart of the result due to Nakayama-Tsuzuku ([21, Theorem 8,Theorem 8 ′ ]): Let R ⊂ A be a Frobenius extension, M be any left A -module. If G-proj . dim A M < ∞ , then G-proj . dim A M = G-proj . dim R M ; the dual for Gorenstein injective dimension also holds;see Proposition 3.1.For any ring Λ, Bennis and Mahdou proved an equality ([5, Theorem 1.1]):sup { G-proj . dim Λ M | M ∈ Mod(Λ) } = sup { G-inj . dim Λ M | M ∈ Mod(Λ) } . As a Gorenstein counterpart of the global dimension, they named the common value of thisequality the Gorenstein global dimension of Λ, and denoted it by G-gldim(Λ). Following [4], aring Λ is left-Gorenstein provided that the category Mod(Λ) of left Λ-modules is a Gorensteincategory. This is equivalent to the condition that the Gorenstein global dimension of Λ is finite.According to a classical result established by Auslander, Buchsbaum and Serre, a commutativenoetherian local ring is regular if and only if the projective dimension of its residue field is finite;moreover, in this case the ring has finite global dimension. So left-Gorenstein rings may be called(left) Gorenstein regular rings, meaning a Gorenstein counterpart of regular rings.Let R ⊂ A be any Frobenius extension. If the extension is split (i.e. R is a direct sum-mand of A as an R -bimodule), then we prove in Theorem 3.3 that A is Gorenstein regular ifand only if R is Gorenstein regular; and moreover, we show that G-gldim( A ) = G-gldim( R ),that is, the Gorenstein global dimensions are invariant along Frobenius extensions; see Theorem3.4. Consequently, it follows immediately from [10, Theorem 4.1] that for a Gorenstein regu-lar Frobenius extension R ⊂ A (i.e. either R or A is Gorenstein regular), there are equalities:spli( A ) = silp( A ) = fin . dim( A ) = spli( R ) = silp( R ) = fin . dim( R ); see Corollary 3.5. Here, thesupremum of the projective lengths of injective left R -modules spli( R ), and the supremum of theinjective lengths of projective left R -modules silp( R ), are two invariants introduced by Gedrichand Gruenberg [13] in connection with the existence of complete cohomological functors in the ategory of left R -modules. The left finitistic dimension fin . dim( R ) of R is defined as the supre-mum of the projective dimensions of those left R -modules that have finite projective dimension.The paper is organized as follows. In Section 2, we prove the first main result on transfer ofGorenstein projective and injective modules along Frobenius extensions, see Theorem 2.2 and 2.3.The result in Theorem 2.2 gives an affirmative answer to Chen’s question. In Section 3, we studyGorenstein homological dimensions along Frobenius extensions. A Gorenstein counterpart of [21,Theorem 8 and 8 ′ ] is given in Proposition 3.1, which shows that under the finiteness condition,the Gorenstein projective and injective dimensions of modules are invariant under Frobeniusextensions. Then, we show that the Gorenstein regular property of rings (i.e. finiteness ofGorenstein global dimension), and furthermore, the Gorenstein global dimension, are invariantalong split Frobenius extensions; see Theorem 3.3 and 3.4. Consequently, some equalities follows;see Corollary 3.5.2. Gorenstein projective and injective modules over Frobenius extensions
Let R be a ring. Recall that an R -module M is said to be Gorenstein projective if M is asyzygy of a totally acyclic complex of projective modules, i.e. if there exists an acyclic complexof projective R -modules P := · · · → P → P → P − → · · · which remains acyclic when applyingthe functor Hom R ( − , P ) for any projective R -module P , such that M = Ker( P → P − ). Dually,Gorenstein injective modules are defined [12]. The study of Gorenstein homological algebra hasfound interesting applications in some areas such as representation theory, Tate cohomology andthe theory of singularity categories, see for example [2, 7, 14, 24]. Moreover, it may prove tobe useful in studying certain group-theoretical problems, such as characterizing algebraically thegroups that admit a finite dimensional model for the classifying space for proper actions ([3]).Throughout, all rings are associative with a unit. Homomorphisms of rings are required to sendthe unit to the unit. 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 betweenthem. A right R -module M is sometimes written as M R . We identify right R -modules withleft R op -modules, where R op is the opposite ring of R . For two right R -modules M and N , theabelian 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 anotherring. 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 is givenby rar ′ := ι ( r ) · a · ι ( r ′ ). Similarly, we consider R A and R A A etc.The theory of Frobenius extensions was developed by Kasch [19] as a generalization of Frobe-nius algebras. Since then, Nakayama-Tsuzuku [21] and Morita [20] et. al. defined naturalgeneralizations of Frobenius extensions of different kinds. The definition of Frobenius extension e chose is a condition in [20]. We refer to [17, Definition 1.1, Theorem 1.2] for the followingdefinition of Frobenius extensions.A functor between abelian categories is generally called “Frobenius” if it has left and rightadjoints which are naturally equivalent. For a ring extension R ⊂ A , there is a restricted functor Res : Mod( A ) → Mod( R ) sends A M to R M . In the opposite direction, there are functors T = A ⊗ R − : Mod( R ) → Mod( A ) and H = Hom R ( A, − ) : Mod( R ) → Mod( A ). It is clear that( T, Res ) and (
Res, H ) are adjoint pairs.
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 . By [21, Proposition 1], if we choose the automorphism of R to be the identity of R , then theabove definition coincides with the 2. Frobenius extension (or a Frobenius extension of 2nd kind)introduced by Nakayama-Tsuzuku.There is an observation due to Buchweitz: for a finite group G , a Z G -module, or equivalentlyan 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, the underlying Z -module is free, see [6,Section 8.2]. Note that the classical example Z ⊂ Z G is a Frobenius extension. In [8], Chenintroduced a generalization of Frobenius extension, called the totally reflexive extension of rings,and proved that totally reflexive modules transfer along such extension. He asked if this is truefor not necessarily finitely generated Gorenstein projective modules, and claimed that a newmethod is needed for the question. We have the following, which gives an affirmative answer ofChen’s problem in the case of Frobenius extensions. Moreover, it generalizes [22, Theorem 2.5and 2.11]. Theorem 2.2.
Let R ⊂ A be a Frobenius extension of rings, M be any left A -module. Thefollowing are equivalent: (1) A M is Gorenstein projective in Mod( A ) . (2) The underlying left R -module R M is Gorenstein projective. (3) A ⊗ R M and Hom R ( A, M ) are Gorenstein projective left A -modules.Proof. (1)= ⇒ (2). It follows from [22, Lemma 2.2]. Indeed, for the Gorenstein projective left A -module M , there exists a totally acyclic complex of projective A -modules P := · · · → P → P → P − → · · · such that M = Ker( P → P − ). By restricting P one gets an acyclic complexof projective R -modules. For any projective left R -module Q , Hom R ( A, Q ) ∼ = A ⊗ R Q is a rojective A -module, and it follows from isomorphisms Hom R ( P , Q ) ∼ = Hom R ( A ⊗ A P , Q ) ∼ =Hom A ( P , Hom R ( A, Q )) that Hom R ( P , Q ) is acyclic. The assertion follows.(2)= ⇒ (3). Let P := · · · → P → P → P − → · · · be a totally acyclic complex of projective R -modules such that R M = Ker( P → P − ). It is direct to check that A ⊗ R P is a totally acycliccomplex of projective A -modules, and A ⊗ R M = Ker( A ⊗ R P → A ⊗ R P − ). Hence, A ⊗ R M and Hom R ( A, M ) are Gorenstein projective left A -modules.(3)= ⇒ (2). Note that for the ring extension R ⊂ A and any A -module M , the module R M is adirect summand of the R -module A ⊗ R M . If A ⊗ R M is a Gorenstein projective left A -module,then since (1) implies (2), we get that A ⊗ R M is Gorenstein projective in Mod( R ), and hence R M is Gorenstein projective.(3)= ⇒ (1). Let P be any projective A -module. Since (3) implies (2), we have, from A ⊗ R M being a Gorenstein projective left A -module, that the module R M is Gorenstein projective.Then it follows from [22, Lemma 2.3] that Ext iA ( M, P ) = 0. Indeed, note that the module R P isprojective, and then we have 0 = Ext iR ( M, P ) ∼ = Ext iR ( A ⊗ A M, P ) ∼ = Ext iA ( M, Hom R ( A, P )) ∼ =Ext iA ( M, A ⊗ A P ). Moreover, since A P is a direct summand of A ⊗ R P , and then Ext iA ( M, P ) = 0.It only remains to construct the right part of the totally acyclic complex of A M .Since Hom R ( A, M ) is a Gorenstein projective A -module, there is an exact sequence 0 → Hom R ( A, M ) f → P → L → A -modules, where P is projective and L is Gorenstein projective.There is a map ϕ : M → Hom R ( A, M ) given by ϕ ( m )( a ) = am , which is an A -monomorphism,and is split when we restrict it as an R -homomorphism. So we have an R -homomorphism ϕ ′ : Hom R ( A, M ) → M such that ϕ ′ ϕ = id M . Let Q be any projective R -module, and g : M → Q be any R -homomorphism. Since L is also Gorenstein projective as an R -module, for the R -homomorphism gϕ ′ : Hom R ( A, M ) → Q , there is an R -homomorphism h : P → Q , such that gϕ ′ = hf . That is, we have the following commutative diagram: Q / / Hom R ( A, M ) gϕ ′ O O f / / P ∃ h e e ▲▲▲▲▲▲▲▲▲▲▲▲▲ / / L / / A -monomorphism f ϕ : M → P . Consider the exact sequence 0 → M fϕ → P → L → A -modules, where P is projective, and L = Coker( f ϕ ). Restricting the sequence, wenote that it is Hom R ( − , Q )-exact for any projective R -module Q , since for any R -homomorphism g : M → Q , there exists an R -homomorphism h : P → Q such that g = ( gϕ ′ ) ϕ = h ( f ϕ ). Then,it follows from the exact sequence Hom R ( P , Q ) → Hom R ( M, Q ) → Ext R ( L , Q ) → R ( L , Q ) = 0. Moreover, R M is Gorenstein projective by (3) ⇒ (2) and R P is projective, itfollows from [15, Corollary 2.11] that L is a Gorenstein projective R -module. et P be any projective A -module. There is a split epimorphism ψ : A ⊗ R P → P of A -modules given by ψ ( a ⊗ R x ) = ax for any a ∈ A and x ∈ P , and then there exists an A -homomorphism ψ ′ : P → A ⊗ R P such that ψψ ′ = id P . Note that P is also projective as an R -module. Then, it follows from Ext A ( L , A ⊗ R P ) ∼ = Ext R ( L , P ) = 0 that the exact sequence0 → M fϕ → P → L → A ( − , A ⊗ R P ).For any A -homomorphism α : M → P , we consider the following diagram P ψ ′ / / A ⊗ R P / / M α O O fϕ / / P / / ∃ β O O c c ❍ ❍ ❍ ❍ ❍ L / / ψ ′ α : M → A ⊗ R P , there exists an A -map β : P → A ⊗ R P such that ψ ′ α = β ( f ϕ ). Andthen, we have ψβ : P → P , such that α = ( ψψ ′ ) α = ( ψβ )( f ϕ ). This implies that the sequence0 → M fϕ → P → L → A ( − , P )-exact.Note that L is a Gorenstein projective R -module, and then Hom R ( A, L ) is a Gorensteinprojective A -module. Repeating the process we followed with M , we inductively construct anexact sequence 0 → M → P → P → P → · · · in Mod( A ), with each P i projective and whichis also exact after applying Hom A ( − , P ) for any projective A -module P . This completes theproof. (cid:3) Dually, we have the following.
Theorem 2.3.
Let R ⊂ A be a Frobenius extension of rings, M be any left A -module. Thefollowing are equivalent: (1) A M is Gorenstein injective in Mod( A ) . (2) The underlying left R -module R M is Gorenstein injective. (3) A ⊗ R M and Hom R ( A, M ) are Gorenstein injective left A -modules. Gorenstein projective and injective dimensions over Frobenius extensions
Unless otherwise mentioned we will be working with left modules. The Gorenstein projectiveand injective dimensions of modules are defined in the standard way by using resolutions ofGorenstein modules. That is, the Gorenstein projective dimension of a Λ-module M , denotedby G-proj.dim Λ M , is defined by declaring that G-proj.dim Λ M ≤ n ( n ∈ N ) if, and only if, M has a Gorenstein projective resolution 0 → G n → · · · → G → G → M → n .We set G-proj.dim Λ M = ∞ if there is no such a resolution. Similarly, the Gorenstein injectivedimension is defined; see for example [12, 15].It follows from [21, Theorem 8, Theorem 8 ′ ] that: for a Frobenius extension R ⊂ A and anyleft A -module M , if the A -projective dimension ( A -injective dimension, respectively) of M isfinite, then one has proj.dim A M = proj.dim R M (inj.dim A M = inj.dim R M , respectively). We an extend this result to corresponding Gorenstein homological dimensions. This serves as anexample to support the metatheorem (see Holm’s thesis [16]) “every result in classical homologicalalgebra has a counterpart in Gorenstein homological algebra”. Proposition 3.1.
Let R ⊂ A be a Frobenius extension of rings. For any left A -module M , if G - proj . dim A M < ∞ , then G - proj . dim A M = G - proj . dim R M . Dually, if G - inj . dim A M < ∞ ,then G - inj . dim A M = G - inj . dim R M .Proof. By Theorem 2.2, any Gorenstein projective A -module is also Gorenstein projective in R -Mod. It is easy to see that G-proj . dim R M ≤ G-proj . dim A M . For the converse, we canassume that G-proj . dim R M = n < ∞ . Let P be any projective A -module. Then, P is alsoprojective as an R -module. By [15, Theorem 2.20], for any i > n + iR ( M, P ) = 0.Moreover, since Ext n + iA ( M, A ⊗ R P ) ∼ = Ext n + iR ( M, P ) and P is a direct summand of A ⊗ R P as A -modules, we have Ext n + iA ( M, P ) = 0. This implies that G-proj . dim A M ≤ n . Then, the equalityG-proj . dim A M = G-proj . dim R M holds. Analogously, we can prove the assertion for Gorensteininjective dimension. (cid:3) Proposition 3.2.
Let R ⊂ A be a Frobenius extension of rings, and M be any left A -module.Then G - proj . dim R M = G - proj . dim A ( A ⊗ R M ) = G - proj . dim R ( A ⊗ R M ) . Similarly, we have G - inj . dim R M = G - inj . dim A ( A ⊗ R M ) = G - inj . dim R ( A ⊗ R M ) .Proof. It follows from Theorem 2.2 that G-proj . dim R ( A ⊗ R M ) ≤ G-proj . dim A ( A ⊗ R M ). Forany Gorenstein projective R -module G , it follows from Theorem 2.2 that A ⊗ R G is a Gorensteinprojective A -module. Then G-proj . dim A ( A ⊗ R M ) ≤ G-proj . dim R M is easy to see. As R -modules, M is a direct summand of A ⊗ R M . It follows immediately from [15, Proposition 2.19]that G-proj . dim R M ≤ G-proj . dim R ( A ⊗ R M ). Hence, we get the desired equality. (cid:3) By [4, Definition VII2.1, VII2.5], a ring Λ is left-Gorenstein provided the category Mod(Λ) ofleft Λ-modules is a Gorenstein category, that is, if both spli(Λ) and silp(Λ) are finite. Here, spli(Λ)is the supremum of the projective lengths of injective left Λ-modules, and silp(Λ) is the supremumof the injective lengths of projective left Λ-modules. These two invariants are introduced byGedrich and Gruenberg [13], in connection with the existence of complete cohomological functorsin the category of left Λ-modules.According to a classical result established by Auslander, Buchsbaum and Serre, a commutativenoetherian local ring is regular if and only if the projective dimension of its residue field is finite;moreover, in this case the ring has finite global dimension. It is known that Λ is left-Gorensteinif and only if the left Gorenstein global dimension of Λ is finite, see for example [10]. We preferto call left-Gorenstein ring as (left) Gorenstein regular ring, meant a Gorenstein counterpart ofregular ring. By [12, Theorem 10.2.14], each Iwanaga-Gorenstein ring (i.e. two-sided noetherianring with finite left and right self-injective dimension) is Gorenstein regular. ecall that an extension R ⊂ A of rings is split if A = R ⊕ S as R -bimodules for somesubbimodule S in A ; see for example [18]. In this case, it is clear to see that for any R -module M , M = R ⊗ R M is a direct summand of A ⊗ R M . Theorem 3.3.
Let R ⊂ A be a split Frobenius extension of rings. Then A is Gorenstein regularif and only if R is Gorenstein regular.Proof. Let R ⊂ A be a Frobenius extension. We claim that every projective R -module has finite R -injective dimension if, and only if every projective A -module has finite A -injective dimension.For the “only if” part, let P be a projective left A -module. Consider P as an R -module, thenby the assumption inj.dim R P is finite. Assume that inj.dim R P = n and 0 → P → I → I →· · · → I n → R -injective resolution of R P . For any injective R -module I , it follows fromthe isomorphism A ⊗ R I ∼ = Hom R ( A, I ) that A ⊗ R I is an injective left A -module. Hence, theexact sequence 0 → A ⊗ R P → A ⊗ R I → A ⊗ R I → · · · → A ⊗ R I n → A -injectiveresolution of A ⊗ R P . Moreover, A P is a direct summand of A ⊗ R P , so P is of finite A -injectivedimension.Conversely, for the “if” part, let Q be a projective left R -module. By the assumption, theprojective left A -module A ⊗ R Q has finite A -injective dimension. It follows from [21, Theorem8 ′ ] that inj.dim R ( A ⊗ R Q ) = inj.dim A ( A ⊗ R Q ) < ∞ . Moreover, Q is a direct summand of A ⊗ R Q ,and then Q has finite R -injective dimension.Similarly, we can prove that every injective R -module has finite R -projective dimension if andonly if every injective A -module has finite A -projective dimension. This will imply the desiredassertion that A is Gorenstein regular if and only if R is Gorenstein regular. (cid:3) Moreover, we have the following. It shows that not only the finiteness of Gorenstein globaldimension, but also Gorenstein global dimension itself, is invariant under Frobenius extensions.
Theorem 3.4.
Let R ⊂ A be a split Frobenius extension of rings. Then G - gldim( A ) = G - gldim( R ) .Proof. We deduce from Theorem 3.3 that G-gldim( A ) = ∞ if and only if G-gldim( R ) = ∞ . Nowwe assume that both G-gldim( A ) and G-gldim( R ) are finite.By Proposition 3.1, there is an equality G-proj . dim A M = G-proj . dim R M for any A -module M .Hence, G-gldim( A ) ≤ G-gldim( R ). Let N be any R -module. By Proposition 3.1, G-proj . dim R N ≤ G-proj . dim R ( A ⊗ R N ) = G-proj . dim A ( A ⊗ R N ). This implies that G-gldim( R ) ≤ G-gldim( A ).Then, the desired equality follows. (cid:3) For a ring Λ of finite Gorenstein global dimension (i.e. Gorenstein regular ring, or left-Gorenstein ring), Emmanouil got the following equalities: G-gldim(Λ) = spli(Λ) = silp(Λ) =fin . dim(Λ), by comparing Gorenstein projective and injective dimensions with some invariantsof rings; see [10, Theorem 4.1]. Here, the left finitistic dimension fin . dim(Λ) of Λ is defined asthe supremum of the projective dimensions of those left Λ-modules that have finite projectivedimension. The following is immediate. orollary 3.5. Let R ⊂ A be a split Frobenius extension of rings. If either A or R is Gorensteinregular, then there are equalities: spli( A ) = silp( A ) = fin . dim( A ) = spli( R ) = silp( R ) = fin . dim( R ) . Remark 3.6. (1)
For any finite group G , the integer group ring extension Z ⊂ Z G is a splitFrobenius extension. For any ring R , R ⊂ R [ x ] / ( x ) is a split Frobenius extension, where x is avariable which is supposed to commutate with all the elements of R . (2) Every excellent extension (see e.g. [23] for the collection of definition and examples) is asplit Frobenius extension. (3)
Recall that R ⊂ A is a Frobenius extension of rings if and only if there exists an R - R -homomorphism τ : A → R and elements x i , y i in A , such that for any a ∈ A , one has P i x i τ ( y i a ) = a and P i τ ( ax i ) y i = a ; the triple ( τ, x i , y i ) is called a Frobenius system, τ aFrobenius homomorphism. It follows from [18, Corollary 4.2] that if τ (1) = 1 then the Frobeniusextension R ⊂ A is split. Indeed, for the R -homomorphism ϕ : A ⊗ R R → R given by ϕ ( a ⊗ r ) = τ ( a ) r for any a ∈ A and r ∈ R , we have ψ : R → A ⊗ R R , r → re with e = P i x i ⊗ τ ( y i ) , suchthat the composition ϕψ is the identity map of R . Moreover, the relation between split Frobeniusextensions and separable Frobenius extensions is studied by Kadison [18, Proposition 4.1]. ACKNOWLEDGEMENTS.
The author is grateful to the referee for several comments thatimproved the paper, and he thanks Professor Xiao-Wu Chen for helpful suggestions.
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