aa r X i v : . [ m a t h . A C ] F e b DEGENERATIONS OF GRADED COHEN-MACAULAYMODULES
NAOYA HIRAMATSU
Abstract.
We introduce a notion of degenerations of graded modules. Inrelation to it, we also introduce several partial orders as graded analogies ofthe hom order, the degeneration order and the extension order. We prove thatthese orders are identical on the graded Cohen-Macaulay modules if a gradedring is of graded finite representation type and representation directed. Introduction
The notion of degenerations of modules appears in geometric methods of rep-resentation theory of finite dimensional algebras. In [15] Yoshino gives a scheme-theoretical definition of degenerations, so that it can be considered for modulesover a Noetherian algebra which is not necessary finite dimensional. Now, a the-ory of degenerations is considered for derived module categories [10] or stablemodule categories [16]. The degeneration problem of modules has been studiedby many authors [6, 12, 14, 15, 18, 19]. For the study, several order relationsfor modules, such as the hom order, the degeneration order and the extensionorder, were introduced, and the connection among them has been studied. In theprevious paper [8], the author give the complete description of degenerations overa ring of even-dimensional simple hypersurface singularity of type ( A n ).In the present paper we consider degenerations of graded Cohen-Macaulaymodules over a graded Gorenstein ring with a graded isolated singularity. Firstwe consider an order relation on a category of graded Cohen-Macaulay moduleswhich is called the hom order (Definition 2.2). We shall show that it is actuallya partial order if a graded ring is Gorenstein with a graded isolated singularity.In section 3 we propose a definition of degenerations for graded modules andwe state several properties on it. We show that if the graded ring is of gradedfinite representation type and representation directed, then the hom order, thedegeneration order and the extension order are identical on the graded Cohen-Macaulay modules. We also consider stable analogue of degenerations of gradedCohen-Macaulay modules in section 4. Date : April 7, 2018.2000
Mathematics Subject Classification.
Primary 16W50 ; Secondary 13D10.
Key words and phrases. degeneration, graded Cohen-Macaulay module, finite representationtype. Hom order on graded modules
Throughout the paper R = ⊕ ∞ i =0 R i be a commutative Noetherian N -gradedCohen-Macaulay ring with R = k a field of characteristic zero. A graded ring issaid to be ∗ local if the set of graded proper ideals has a unique maximal element.Thus R is ∗ local since m = ⊕ i> R i is a unique maximal ideal of R . We denote bymod Z ( R ) the category of finitely generated Z -graded modules whose morphismsare homogenous morphisms that preserve degrees. For i ∈ Z , M ( i ) ∈ mod Z ( R )is defined by M ( i ) n = M n + i . Then Hom R ( M, N ( i )) consisting of homogenousmorphisms of degree i , and we set ∗ Hom R ( M, N ) = ⊕ i ∈ Z Hom R ( M, N ( i )) . For a graded prime ideal p of R , we denote by R ( p ) a homogenous localization of R by p . For M ∈ mod Z ( R ), take a graded free resolution · · · → F l d l −→ F l − → · · · → F d −→ F → M → . We define an l th syzygy module Ω l M of M by Im d l . We say that a graded R -module M is said to be a graded Cohen-Macaulay R -module if ∗ Ext iR ( R/ m , M ) = 0 for any i < d = dim R. In particular, this condition is equivalent to ∗ Ext iR ( M, ω R ) = 0 for any i > , where ω R is a ∗ canonical module of R . We denote by CM Z ( R ) the full sub-category of mod Z ( R ) consisting of graded Cohen-Macaulay R -modules. By ourassumption on R , mod Z ( R ) and CM Z ( R ) are Krull-Schmidt, namely each objectcan be decomposed into indecomposable objects up to isomorphism uniquely. For M ∈ mod Z ( R ) we denote by h ( M ) a sequence (dim k M n ) n ∈ Z of non-negative in-tegers. By the definition, it is easy to see that h ( M ) = h ( N ) if and only if theyhave the same Hilbert series. Moreover, we also have that h ( M ) = h ( N ) if andonly if h ( M ∗ ) = h ( N ∗ ) where ( − ) ∗ = ∗ Hom R ( − , ω R ). See [7, Theorem 4.4.5.]. Remark . If M and N give the same class in the Grothendieck group, i.e. [ M ] = [ N ] as an element of K (mod Z ( R )) then h ( M ) = h ( N ). However theconverse does not hold in general. Let R = k [ x, y ] / ( x − y ) with deg x = deg y = 1and set M = R/ ( x + y ) and N = R/ ( x − y ). Then h ( M ) = h ( N ) and [ M ] = [ N ]in K (mod Z ( R )). In fact, let p be an ideal ( x + y ) R . Then rank R ( p ) M ( p ) = 1and rank R ( p ) N ( p ) = 0. Note that [ M ] = [ N ] on K (mod Z ( R ( p ) )) yields that[ M ( p ) ] = [ N ( p ) ] on K (mod Z ( R ( p ) )). Thus [ M ] = [ N ] can never happen.Our motivation of the paper is to investigate the graded degenerations of gradedCohen-Macaulay modules in terms of some order relations. For the reason weconsider the following relation on CM Z ( R ) that is known as the hom order. EGENERATIONS OF GRADED COHEN-MACAULAY MODULES 3
Definition 2.2.
For M , N ∈ CM Z ( R ) we define M ≤ hom N if [ M, X ] ≤ [ N, X ]for each X ∈ CM Z ( R ). Here [ M, X ] is an abbreviation of dim k Hom R ( M, X ). Remark . For M , N ∈ mod Z ( R ), [ M, N ] is finite, and thus we can considerthe above relation. As a consequence in [5], if R is of dimension 0, 1 or 2, ≤ hom is a partial order on CM Z ( R ) since CM Z ( R ) is closed under kernels in such cases. Lemma 2.4.
Let R be a graded Gorenstein ring an let M and N be indecom-posable graded Cohen-Macaulay R -modules. Suppose that h ( M ) = h ( N ) . For Y ∈ mod Z ( R ) which is of finite projective dimension, we have [ M, Y ] = [
N, Y ] .Proof. For a graded R -module Y which is of finite projective dimension, Ω i Y isalso of finite projective dimension. Since R is Gorenstein, we have Ext R ( M, Ω i Y ) =0 for all graded Cohen-Macaulay R -modules M . Thus, take a graded free resolu-tion of Y and apply Hom R ( M, − ) to the resolution, we get an exact sequence0 → Hom R ( M, F l ) → Hom R ( M, F l − ) → · · · → Hom R ( M, F ) → Hom R ( M, Y ) → . Hence [
M, Y ] = l X i =0 ( − i [ M, F i ] . We also have [
N, Y ] = l X i =0 ( − i [ N, F i ] . Since R is Gorenstein, ω R = R ( l ) for some l ∈ Z . Then we have an equality[ M, F ] = [
N, F ] for each free module. Therefore we have[
M, Y ] = l X i =0 ( − i [ M, F i ] = l X i =0 ( − i [ N, F i ] = [ N, Y ] . (cid:3) In the paper we use the theory of Auslander-Reiten (abbr. AR) sequences ofgraded Cohen-Macaulay modules. For the detail, we recommend the reader tolook at [9, 3, 4] and [13, Chapter 15.]. We denote by CM Z ( R ) the full subcategoryof CM Z ( R ) consisting of M ∈ CM Z ( R ) such that M ( p ) is R ( p ) -free for any gradedprime ideal p = m . Theorem 2.5. [3, 4, 13, 9]
Let ( R, m ) be a Noetherian Z -graded Gorenstein ∗ localring. Then CM Z ( R ) admits AR sequences. Definition 2.6.
We say that ( R, m ) is a graded isolated singularity if each gradedlocalization R ( p ) is regular for each graded prime ideal p with p = m .It is easy to see that CM Z ( R ) = CM Z ( R ) if R is a graded isolated singularity,so that CM Z ( R ) admits AR sequences. We denote by µ ( M, Z ) the multiplicityof Z as a direct summand of M . NAOYA HIRAMATSU
Theorem 2.7.
Let R be a graded Gorenstein ring with R is an algebraicallyclosed field and let M and N graded Cohen-Macaulay R -modules. Assume that R is a graded isolated singularity. Then [ M, X ] = [
N, X ] for each X ∈ CM Z ( R ) if and only if M ∼ = N . Particularly, ≤ hom is a partial order on CM Z ( R ) .Proof. We decompose M as M = ⊕ M µ ( M,M i ) i where M i are indecomposablegraded Cohen-Macaulay R -modules. If M i is not free, we can take the AR se-quence ending in M i → τ M i → E i → M i → , where τ M i is an AR translation of M i . Apply Hom R ( M, − ) and Hom R ( N, − ) tothe sequence, since k is an algebraically closed field, we have0 → Hom R ( M, τ M i ) → Hom R ( M, E i ) → Hom R ( M, M i ) → k µ ( M,M i ) → → Hom R ( N, τ M i ) → Hom R ( N, E i ) → Hom R ( N, M i ) → k µ ( N,M i ) → . Counting the dimensions of terms, we conclude that µ ( M, M i ) = µ ( N, M i ).If M i is free, we may assume that M i = R . Let m be a ∗ maximal ideal of R .We consider the Cohen-Macaulay approximation (see [2]) of m → Y → X → m → . We note that X is a graded Cohen-Macaulay R -module and Y is of finite injectivedimension. We also note that Y is of finite projective dimension since R isGorenstein. Let f be a composition map of the approximation X → m and anatural inclusion m → R . Then we get the following commutative diagram.0 −−−→ K −−−→ X f −−−→ R x g x = x ⊆ −−−→ Y −−−→ X −−−→ m −−−→ . By a diagram chasing we see that g is surjective, so that K is of finite projectivedimension. Now we obtain an exact sequence0 → Hom R ( M, K ) → Hom R ( M, X ) → Hom R ( M, R ) → k µ ( M,R ) → . According to Lemma 2.4, [
M, K ] = [
N, K ]. Thus[
M, R ] + [
M, K ] − [ M, X ] = [
N, R ] + [
N, K ] − [ N, X ] . Hence µ ( M, R ) = µ ( N, R ). Consequently M ∼ = N . (cid:3) EGENERATIONS OF GRADED COHEN-MACAULAY MODULES 5 Graded degenerations of graded Cohen-Macaulay modules
We define a notion of degenerations for graded modules.
Definition 3.1.
Let R be a Noetherian N -graded ring where R = k is a fieldand let V = k [[ t ]] with a trivial gradation and K be the localization by t , namely K = V t = k ( t ). For finitely generated graded R -modules M and N , we saythat M gradually degenerates to N or N is a graded degeneration of M if thereis a finitely generated graded R ⊗ k V -module Q which satisfies the followingconditions:(1) Q is flat as a V -module.(2) Q ⊗ V V /tV ∼ = N as a graded R -module.(3) Q ⊗ V K ∼ = M ⊗ k K as a graded R ⊗ k K -module.In [15], Yoshino gives a necessary and sufficient condition for degenerations of(non-graded) modules. One can also show its graded version in a similar way.See also [12, 19]. Theorem 3.2. [15, Theorem 2.2]
The following conditions are equivalent forfinitely generated graded R -modules M and N . (1) M gradually degenerates to N . (2) There is a short exact sequence of finitely generated graded R -modules −−−→ Z −−−→ M ⊕ Z −−−→ N −−−→ . Remark . (1) As Yoshino has shown in [15], the endomorphism of Z inthe sequence of Theorem 3.2 is nilpotent. Note that we do not need thenilpotency assumption here. Actually, since End R ( Z ) is Artinian, by usingFitting theorem, we can describe the endomorphism as a direct sum of anisomorphism and a nilpotent morphism. See also [15, Remark 2.3.].(2) Assume that M and N are graded Cohen-Macaulay modules. Then wecan show that Z is also a graded Cohen-Macaulay R -module. See [15,Remark 4.3.].(3) Assume that there is an exact sequence of finitely generated graded R -modules 0 −−−→ L −−−→ M −−−→ N −−−→ . Then M gradually degenerates to L ⊕ N . See [15, Remark 2.5] for instance.(4) Let M and N be finitely generated graded R -modules and suppose that M gradually degenerates to N . Then the modules M and N give thesame class in the Grothendieck group.We can also prove in a similar way to the proof of [17, Theroem 2.1.] that,for L , M , N ∈ mod Z ( R ), if L gradually degenerates to M and if M graduallydegenerates to N then L gradually degenerates to N . And one can show that if L gradually degenerates to M then L ≤ hom M . Consideration into this fact wedefine partial orders as follows. NAOYA HIRAMATSU
Definition 3.4.
For finitely generated graded R -modules M , N , we define therelation M ≤ deg N , which is called the degeneration order, if M gradually degen-erates to N . We also define the relation M ≤ ext N if there are modules M i , N ′ i , N ′′ i and short exact sequences 0 → N ′ i → M i → N ′′ i → Z ( R ) so that M = M , M i +1 = N ′ i ⊕ N ′′ i , 1 ≤ i ≤ s and N = M s +1 for some s .For M ∈ CM Z ( R ), we take a first syzygy module of M ∗ → Ω M ∗ → F → M ∗ → . Applying ∗ Hom R ( − , ω R ) to the sequence, we have0 → M ∗∗ ∼ = M → F ∗ → (Ω M ∗ ) ∗ → . Then we denote (Ω M ∗ ) ∗ by Ω − M . Lemma 3.5.
Let R be a graded isolated singularity with ω R and let M and N be graded Cohen-Macaulay R -modules. Assume that h ( M ) = h ( N ) and M ≤ hom N . Then, for each graded Cohen-Macaulay R -module X , there exists an integer l X ≫ such that [ M, X ( ± l X )] = [ N, X ( ± l X )] .Proof. For each X ∈ CM Z ( R ), we can take an exact sequence as above0 → X → E → Ω − X → . Note that E is a direct sum of ω R ( n ) for some integers n . Applying ∗ Hom R ( M, − )to the sequence, we have0 → ∗ Hom R ( M, X ) → ∗ Hom R ( M, E ) → ∗ Hom R ( M, Ω − X ) → ∗ Ext R ( M, X ) → . Since R is a graded isolated singularity, dim ∗ Ext R ( M, X ) is finite. Thus ∗ Ext R ( M, X ) ± l =0 for sufficiently large l ≫
0. Similarly there also exists an integer l ≫ ∗ Ext R ( N, X ) ± l = 0.Set l = max { l , l } . Then we have0 → ∗ Hom R ( M, X ) ± l → ∗ Hom R ( M, E ) ± l → ∗ Hom R ( M, Ω − X ) ± l → . Therefore we obtain the equation[
M, X ( ± l )] = [ M, E ( ± l )] − [ M, Ω − X ( ± l )] . We also have [
N, X ( ± l )] = [ N, E ( ± l )] − [ N, Ω − X ( ± l )] . Suppose that [
M, X ( ± l )] < [ N, X ( ± l )]. Then the following inequality holds.[ M, E ( ± l )] − [ M, Ω − ( X )( ± l )] < [ N, E ( ± l )] − [ N, Ω − X ( ± l )] . Since h ( M ) = h ( N ), [ M, E ( ± l )] = [ N, E ( ± l )]. Hence we see that [ M, Ω − X ( ± l )] > [ N, Ω − X ( ± l )]. This is a contradiction since M ≤ hom N . Therefore we have someinteger l such that [ M, X ( ± l )] = [ N, X ( ± l )]. (cid:3) EGENERATIONS OF GRADED COHEN-MACAULAY MODULES 7
We say that the category CM Z ( R ) is of graded finite representation type if thereare only a finite number of isomorphism classes of indecomposable graded Cohen-Macaulay modules up to shift. We note that if CM Z ( R ) is of finite representationtype, then R is a graded isolated singularity. See [13, Chapter 15.] for the detail.As an immediate consequence of Lemma 3.5, we have the following. Corollary 3.6.
Let R be of finite representation type and let M and N be gradedCohen-Macaulay R -modules. Assume that h ( M ) = h ( N ) and M ≤ hom N . Thenthere are only finitely many indecomposable graded Cohen-Macaulay R -modules X such that [ N, X ] − [ M, X ] > . For graded Cohen-Macaulay R -modules M and N , we consider the followingset of all the isomorphism classes of indecomposable graded Cohen-Macaulaymodules F M,N = { X | [ N, X ] − [ M, X ] > } / ∼ = . Note from Corollary 3.6 that F is a finite set if h ( M ) = h ( N ) and M ≤ hom N .We also note that ω R
6∈ F in the case.
Proposition 3.7. [12]
Let R be a graded Gorenstein ring which is of gradedfinite representation type and let M and N be graded Cohen-Macaulay R -modules.Assume that h ( M ) = h ( N ) and M ≤ hom N . Then there exists some gradedCohen-Macaulay R -module L such that M ⊕ L degenerates to N ⊕ L .Proof. Although a proof of the proposition is given in [12], we refer the argumentof the proof in the present paper. For this reason we briefly recall the proof ofthe proposition.Since R is a graded isolated singularity, CM Z ( R ) admits AR sequences. Foreach X ∈ F M,N , we can take an AR sequence starting from X .Σ X : 0 → X → E X → τ − X → . Now we consider a sequence which is a direct sum of [
N.X ] − [ M, X ] copies ofΣ X where X runs through all modules in F M,N . Namely M X ∈F M,N Σ [ N.X ] − [ M,X ] X = 0 → U → V → W → . For any indecomposable Z ∈ CM Z ( R ), we obtain0 → Hom R ( W, Z ) → Hom R ( V, Z ) → Hom R ( U, Z ) → k [ N,Z ] − [ M,Z ] → . This implies that the equality[
U, Z ] + [
W, Z ] − [ V, Z ] = [
N, Z ] − [ M, Z ] , thus [ U, Z ] + [
W, Z ] + [
M, Z ] = [
N, Z ] + [
V, Z ]holds for all Z ∈ CM Z ( R ). Hence this yields that M ⊕ U ⊕ W ∼ = N ⊕ V. NAOYA HIRAMATSU
Since V degenerates to U ⊕ W , therefore M ⊕ V degenerates to M ⊕ U ⊕ W ∼ = N ⊕ V . (cid:3) Now we focus on the case that a graded Gorenstein ring is of graded finiterepresentation type and representation directed. We say that a graded Cohen-Macaulay ring R is representation directed if the AR quiver of CM Z ( R ) has nooriented cyclic paths. Bongartz [6] has studied such a case over finite dimensional k -algebras. In our graded settings, the similar results hold. Actually we shallprove the following. Theorem 3.8.
Let R be a graded Gorenstein ring which is of graded finite rep-resentation type and representation directed. Then the following conditions areequivalent for M and N ∈ CM Z ( R ) . (1) h ( M ) = h ( N ) and M ≤ hom N . (2) M ≤ deg N . (3) M ≤ ext N . To prove the theorem, we modify the arguments in [6].
Lemma 3.9 (Cancellation property) . Let M , N and X be finitely generatedgraded R -modules. (1) Assume that [ X, M ] = [
X, N ] . If M ⊕ X gradually degenerates to N ⊕ X , M gradually degenerates to N . (2) Assume that R is Gorenstein and M and N graded Cohen-Macaulay R -modules. If M gradually degenerates to N ⊕ F for some graded free R -module F then M/F gradually degenerates to N .Proof. (1) Since M ⊕ X gradually degenerates to N ⊕ X , there exist an exactsequence 0 → W → M ⊕ X ⊕ W → N ⊕ X → . We construct a pushout diagram. 0 0 x x
X X x x −−−→ W −−−→ M ⊕ X ⊕ W −−−→ N ⊕ X −−−→ (cid:13)(cid:13)(cid:13) x x −−−→ W −−−→ E −−−→ N −−−→ x x . EGENERATIONS OF GRADED COHEN-MACAULAY MODULES 9
For the middle column sequence,[
X, X ] + [
X, E ] − [ X, M ⊕ X ⊕ W ] = [ X, E ] − [ X, M ] − [ X, W ] ≥ . On the other hand, for the bottom row sequence, since [
X, M ] = [
X, N ],[
X, N ] + [
X, W ] − [ X, E ] = [
X, M ] + [
X, W ] − [ X, E ] ≥ . Thus we have [
X, X ] + [
X, E ] − [ X, M ⊕ X ⊕ W ] = 0 . This implies that the middle column sequence splits, so that X ⊕ E ∼ = M ⊕ X ⊕ W .Therefore E ∼ = M ⊕ W and we get0 → W → M ⊕ W → N → . Namely M gradually degenerates to N .(2) Since M gradually degenerates to N ⊕ F , we have an exact sequence0 → Z → M ⊕ Z → N ⊕ F → . Suppose that Z contains a graded free module G as a direct summand. Weconstruct a pushout diagram0 0 x x −−−→ Z/G −−−→ E −−−→ N ⊕ F −−−→ x x (cid:13)(cid:13)(cid:13) −−−→ Z −−−→ M ⊕ Z −−−→ N ⊕ F −−−→ x x G G x x . The left column sequence is a split sequence induced by the decomposition Z ∼ = Z/G ⊕ G . Note that E is also a graded Cohen-Macaulay module. Since R isGorenstein, the middle column sequence is also split. Hence E ⊕ G ∼ = M ⊕ Z ∼ = M ⊕ Z/G ⊕ G. This yields that E ∼ = M ⊕ Z/G . Hence we may assume that Z has no graded freemodules as direct summands.Consider a composition of the surjection M ⊕ Z → N ⊕ F and the projection N ⊕ F → F . Then the composition mapping is split, so that M contains F as adirect summand. Hence M ∼ = M/F ⊕ F gradually degenerates to N ⊕ F . Since[ F, M/F ⊕ F ] = [ F, M ] = [
F, N ⊕ F ], by (1), we conclude that M/F graduallydegenerates to N . (cid:3) For indecomposable graded Cohen-Macaulay modules M and N , we write X (cid:22) Y if X ∼ = Y or if there exists a finite path from X to Y in the AR quiver ofCM Z ( R ). Lemma 3.10.
Let R be a graded Gorenstein ring which is of graded finite repre-sentation type and representation directed and let M and N ∈ CM Z ( R ) . Assumethat h ( M ) = h ( N ) , M ≤ hom N and M and N have no common direct sum-mands. Let X ∈ CM Z ( R ) be an indecomposable module such that (cid:22) -minimalwith the property [ N, X ] − [ M, X ] > and E be a middle term of an AR sequencestarting from X . Then [ E, N ] = [
E, M ] .Proof. As in the proof of Proposition 3.7, we can construct the sequence0 → U → V → W → Z ( R ) such that U ⊕ W ⊕ M ∼ = V ⊕ N via taking a direct sum of AR sequencesstarting from modules in F M,N . This isomorphism implies that [
Z, U ] + [
Z, W ] − [ Z, V ] = [
Z, N ] − [ Z, M ] for each Z ∈ CM Z ( R ). Thus it is enough to show thatthe equality [ E, U ] + [
E, W ] − [ E, V ] = 0 holds. If there is a Y ∈ F M,N such that,for the AR sequence 0 → Y → G → τ − Y → E, Y ] + [
E, τ − Y ] − [ E, G ] > . If Y ∼ = X , then one can show that τ − X is a direct summand of E , so that thereis a cyclic path. This is a contradiction since R is represented directed. Thus Y is not isomorphic to X . The inequality also show that τ − Y is a direct summandof E . Thus there is an irreducible map from X to τ − Y . Hence X is a directsummand of G , so that Y (cid:22) X . This is a contradiction. Consequently, for each Y ∈ F M,N , [
E, Y ] + [
E, τ − Y ] − [ E, G ] = 0, therefore [
E, N ] = [
E, M ]. (cid:3) Proof of Theorem 3.8.
The implication (3) ⇒ (2) ⇒ (1) is trivial. Note thatExt R ( X, X ) = 0 for an indecomposable X ∈ CM Z ( R ) since R is representationdirected. Thus the implication (2) ⇒ (3) can be shown by the same argument in[18, 3.5.]. Now we shall show (1) ⇒ (2). Set V = ⊕ W ∈F M,N W . Since [ N, X ] − [ M, X ] = 0 for any X
6∈ F
M,N , we can show the implication by induction on d = [ N, V ] − [ M, V ]. If d = 0, [ N, Z ] = [
M, Z ] for each X ∈ CM Z ( R ). Thus M ∼ = N . Hence assume that d > M and N have no summand in commonin the inductive step. We take X ∈ CM Z ( R ) in Lemma 3.10 and let E be amiddle term of the AR sequence starting from X . By virtue of Lemma 3.10and Lemma 3.9 (1) it is enough to show that E ⊕ M ≤ deg E ⊕ N . Now wehave E ⊕ M ≤ ext X ⊕ τ − X ⊕ M . By the property of AR sequence, for eachindecomposable Y , [ X, Y ] + [ τ − X, Y ] = [
E, Y ] + δ X,Y where δ X,Y = 1 if X ∼ = Y and otherwise 0. Thus X ⊕ τ − X ⊕ M ≤ hom E ⊕ N .We should remark that F X ⊕ τ − X ⊕ M,E ⊕ N is contained in F M,N since X ∈ F M,N . EGENERATIONS OF GRADED COHEN-MACAULAY MODULES 11
Then [ E ⊕ N, V ] − [ X ⊕ τ − X ⊕ M, V ]= [
N, V ] − [ M, V ] + [
E, V ] − [ X ⊕ τ − X, V ]= d − E ⊕ M ≤ ext X ⊕ τ − X ⊕ M ≤ deg E ⊕ N so that E ⊕ M ≤ deg E ⊕ N . (cid:3) Remark . The implication M ≤ ext N ⇒ M ≤ deg N does not hold in general.Let R = k [ x, y ] / ( x ) with deg x = deg y = 1. Then R ( −
1) gradually degeneratesto ( x, y ) R . In fact we have an exact sequence0 −−−→ R/ ( x )( − (cid:18) x (cid:19) −−−→ R ( − ⊕ R/ ( x )( −
2) ( x y ) −−−−−→ ( x, y ) R −−−→ . Since ( x, y ) R is an indecomposable graded Cohen-Macaulay module which isnot isomorphic to R ( − R ( − ≤ ext ( x, y ) R can never happen. See also [8,Remark 2.5.]. Proposition 3.12.
Let R be a graded Gorenstein ring which is of graded finiterepresentation type and let M and N be graded Cohen-Macaulay R -modules. As-sume that h ( M ) = h ( N ) and M ≤ hom N . Then for each indecomposable non-freegraded Cohen-Macaulay R -module X we have the following equality. [ N, X ] − [ M, X ] = [ τ − X, N ] − [ τ − X, M ] . Proof.
Under the assumption, as in the proof of Proposition 3.7, we can constructan exact sequence 0 → U → V → W → Z ( R ) such that [ U, X ] + [
W, X ] − [ V, X ] = [
N, X ] − [ M, X ]. It is enough toshow that the equality [ U ⊕ W, X ] − [ V, X ] = [ τ − X, U ⊕ W ] − [ τ − X, V ] holds foreach indecomposable X . Moreover the sequence is a direct sum of AR sequences,we may assume that 0 → U → V → W → U .Let X ∈ CM Z ( R ) be indecomposable and non projective. By the property of anAR sequence, we have[ U ⊕ W, X ] − [ V, X ] = δ U,X [ τ − X, U ⊕ W ] − [ τ − X, V ] = δ W,τ − X . Since W ∼ = τ − U , we can get the equality. (cid:3) Remarks on stable degenerations of graded Cohen-Macaulaymodules
In the rest of the paper we consider the stable analogue of degenerations ofgraded Cohen-Macaulay modules.Let R be a graded Gorenstein ring where R = k is a field and let V = k [[ t ]]with a trivial gradation and K = k ( t ). Note that R ⊗ k V and R ⊗ k K are graded Gorenstein rings as well. Then CM Z ( R ⊗ k V ) and CM Z ( R ⊗ k K ) aretriangulated categories. We denote by L : CM Z ( R ⊗ k V ) → CM Z ( R ⊗ k K ) (resp. R : CM Z ( R ⊗ k V ) → CM Z ( R )) the triangle functor defined by the localizationby t (resp. taking − ⊗ V V /tV ). See also [16, Definition 4.1].
Definition 4.1.
Let
M , N ∈ CM Z ( R ). We say that M stably degenerates to N if there exists a graded Cohen-Macaulay module Q ∈ CM Z ( R ⊗ k V ) such that L( Q ) ∼ = M ⊗ k K in CM Z ( R ⊗ k K ) and R ( Q ) ∼ = N in CM Z ( R ).One can show the following characterization of stable degenerations similarlyto the proof of [16, Theorem 5.1]. Theorem 4.2.
Let R be a graded Gorenstein ring where R = k is a field. Thefollowing conditions are equivalent for graded Cohen-Macaulay R -modules M and N . (1) F ⊕ M degenerates to N for some graded free R -module F . (2) There is a triangle in CM Z ( R ) Z −−−→ M ⊕ Z −−−→ N −−−→ Z [1] . (3) M stably degenerates to N .Proof. We should note that the implication (3) ⇒ (1). In our setting, R ⊗ k V and R ⊗ k K are ∗ local. Then graded projective R ⊗ k V (resp. R ⊗ k K ) -modules aregraded free R ⊗ k V (resp. R ⊗ k K ) -modules. Hence we can show the implicationas in the Artinian case of the proof of [16, Theorem 5.1]. (cid:3) Remark . A theory of degenerations for derived categories has been studiedin [10]. They have shown that, for complexes M , N in the bounded derivedcategory of a finite dimensional algebra, M degenerates to N if and only if thereexists a triangle of the form which appears in the above theorem. Let R be agraded Gorenstein ring with R = k is an algebraically closed field. As shownin [1, 11, 9], suppose that R has a simple singularity then there exists a Dynkinquiver Q such that we have a triangle equivalenceCM Z ( R ) ∼ = D b ( kQ )where D b ( kQ ) is a bounded derived category of the category of finitely generatedleft modules over a path algebra kQ . By virtue of Theorem 4.2, we can describethe degenerations for D b ( kQ ) in terms of the graded degenerations for CM Z ( R ).Since the graded ring R is of graded finite representation type and representationdirected, we have already seen them in Theorem 3.8. Acknowledgments
The author express his deepest gratitude to Tokuji Araya and Yuji Yoshino forvaluable discussions and helpful comments.
EGENERATIONS OF GRADED COHEN-MACAULAY MODULES 13
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Department of general education, Kure National College of Technology,2-2-11, Agaminami, Kure Hiroshima, 737-8506 Japan
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