Araya, Tokuji

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A Characterization of One Dimensional N-Graded Gorenstein Rings of Finite Cohen-Macaulay Representation Type

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Mathematical Journal of OkayamaUniversityVolume 42, Issue 1 2000 Article 1JANUARY 2000A Characterization of One DimensionalN-Graded Gorenstein Rings of FiniteCohen-Macaulay Representation TypeTokuji Araya∗∗Okayama UniversityCopyright c©2000 by the authors. Mathematical Journal of Okayama University is produced byThe Berkeley Electronic Press (bepress). http://escholarship.lib.okayama-u.ac.jp/mjouMath. J. Okayama Univ. 42 (2000), 61-66A CHARACTERIZATION OF ONE DIMENSIONALN-GRADED GORENSTEIN RINGS OF FINITECOHEN-MACAULAY REPRESENTATION TYPETOKUJI ARAYA1. IntroductionLet R =LRn be an N-graded Cohen-Macaulay ring where R0 = kis a ¯eld. We denote by modR the category of ¯nitely generated gradedR-modules whose morphisms are graded R-homomorphisms that preservedegrees. We also denote by CMR the full subcategory of modR consistingof all graded maximal Cohen-Macaulay modules. In the paper [1] , we haveshown that if k is an algebraically closed ¯eld of characteristic 0 and if Ris a one dimensional N-graded Gorenstein ring of ¯nite Cohen-Macaulayrepresentation type, then there exists an MCM generating exceptional se-quence. In that work, we had to compute the dimension of ExtnR(X;Y ) ask-vector space, for all indecomposable graded maximal Cohen-Macaulaymodules X and Y and for all n 2 N. Through this computation, we no-ticed the importance of the invariants d(R) and dn(R) of R that are de¯nedas follows:d(R) := supfXn¸0dimk ExtnR(X;Y ) j X;Y 2 CMR are indecomposableg;dn(R) := supfdimk ExtnR(X;Y ) j X;Y 2 CMR are indecomposableg:In the present paper, we will give a characterization of one dimen-sional N-graded Gorenstein rings of ¯nite Cohen-Macaulay representationtype utilizing d(R) and d0(R). More precisely, let R be a positively dimen-sional N-graded Gorenstein ring with isolated singularity where R0 = k isan algebraically closed ¯eld of characteristic 0. Then the invariant d(R)can take only 7 values in f1; 2; 3; 4; 6; 9;1g. Moreover, if d(R) <1, thendimR = 1 and R is isomorphic to one of the rings in the list (1) below andin each case we are able to compute d(R) and dn(R).611Araya: A Characterization of One Dimensional N-Graded GorensteinProduced by The Berkeley Electronic Press, 200062 TOKUJI ARAYAA2m+1 A2m D2m D2m+1 E6 E7 E8d(R) = d0(R) 1 2 3 4 4 6 9dn(R) (n ¸ 1) 1 2 1 2 3 4 6Acknowledgment : The author thanks to Professor Yuji Yoshinofor his great help in writing this paper.2. PreliminariesIn this section, we assume R =LRn is a one dimensional N-gradedGorenstein ring of ¯nite Cohen-Macaulay representation type and assumethat R0 = k is an algebraically closed ¯eld of characteristic 0. In this case,it is known that R is isomorphic to one of the following rings (c.f.[4] ).(1)(An) R = k[x; y]=(y2 ¡ xn) (n ¸ 2)(Dn) R = k[x; y]=(xy2 ¡ xn) (n ¸ 3)(E6) R = k[x; y]=(x3 + y4)(E7) R = k[x; y]=(x3 + xy3)(E8) R = k[x; y]=(x3 + y5)Moreover the Auslander-Reiten quiver of CMR for each type can bedescribed as they are shown in [1, Figures (1)¡ (7)] . We denote by ¡ theAuslander-Reiten quiver of CMR.For indecomposable graded maximal Cohen-Macaulay modules Xand Y , we write X ¹ Y if X »= Y or if there exists a ¯nite path fromX to Y in ¡.Lemma 2.1. [1, Lemma 3.3.] The following hold for indecomposablegraded maximal Cohen-Macaulay modules X and Y .( i ) There are no cyclic paths in ¡.( ii ) If Hom(X;Y ) 6= 0, then X ¹ Y .( iii ) If Ext1R(X;Y ) 6= 0, then Y ¹ ¿X. Here, ¿X denotes theAuslander-Reiten translation of X.It follows from lemma 2.1.(iii) that, for a ¯xed X, ¿X is the rightbound of the set fY 2 CMR j indecomposable; Ext1R(X;Y ) 6= 0g in ¡.Now, we are giving the left bound of this set.Lemma 2.2. For indecomposable graded maximal Cohen-Macaulay mod-ules X and Y , if Ext1R(X;Y ) 6= 0 then we have ­X ¹ Y ¹ ¿X. Here, ­Xdenotes the ¯rst syzygy module of X.2Mathematical Journal of Okayama University, Vol. 42 [2000], Iss. 1, Art. 1http://escholarship.lib.okayama-u.ac.jp/mjou/vol42/iss1/1A CHARACTERIZATION OF ONE DIMENSIONAL GORENSTEIN RINGS 63Proof. Let 0 ! Y ! Z ¼! X ! 0 be a non-split exact sequence. Takingthe ¯rst syzygy of X; 0 ! ­X ! F ! X ! 0 where F is free, we havethe commutative diagram:0 ¡¡¡¡! ­X ¡¡¡¡! F ¡¡¡¡! X ¡¡¡¡! 0f??y g??y °°°0 ¡¡¡¡! Y ¡¡¡¡! Z ¼¡¡¡¡! X ¡¡¡¡! 0Suppose f = 0 in this diagram. Then the morphism g will induce a mor-phism X ! Z which contradicts the fact that ¼ is not a split epimorphism.Therefore f 6= 0, and we get ­X ¹ Y . 2Lemma 2.3. For any indecomposable graded maximal Cohen-Macaulaymodules X and Y , we have ]fn 2 N j ExtnR(X;Y ) 6= 0g · 1 .Proof. If X is free, then the lemma is obviously true. Thus we may assumethat X is non-free, and hence ­iX (i > 0) are also non-free and ¿­iX(i ¸ 0) are well-de¯ned. Now assume that ExtnR(X;Y ) 6= 0 for some n > 0.Since Ext1R(­n¡1X;Y ) »= ExtnR(X;Y ) 6= 0, we have ­nX ¹ Y ¹ ¿­n¡1Xby lemma 2.2. On the other hand, since there exists a sequence ¢ ¢ ¢ Á­i+1X ¹ ¿­iX Á ­iX ¹ ¿­i¡1X Á ­i¡1X ¹ ¿­i¡2X Á ¢ ¢ ¢ ¹ ¿­X Á­X ¹ ¿X Á X and since there is no cyclic path in ¡, one sees thatY 6¹ ¿­mX for all m ¸ n and ­mX 6¹ Y for all 0 · m < n. Therefore wehave ExtmR (X;Y ) = Ext1R(­m¡1X;Y ) = 0 for all m 6= n by lemma 2.2. 23. Main theoremIn this section, we de¯ne the invariants d(R) and d0(R) by which wewill give a characterization of one dimensional N-graded Gorenstein ringsof ¯nite Cohen-Macaulay representation type.De¯nition 3.1. For anN-graded Cohen-Macaulay ring R (not necessarilyof dimension one) with R0 = k being a ¯eld, we de¯ne d(R) and dn(R) asfollow:d(R) := supfXn¸0dimk ExtnR(X;Y ) j X;Y 2 CMR are indecomposableg;dn(R) := supfdimk ExtnR(X;Y ) j X;Y 2 CMR are indecomposableg:Now we are ready to state our main theorem of this paper.Theorem 3.2. Let k be an algebraically closed ¯eld of characteristic 0 andlet R be a positively dimensional N-graded Gorenstein ring with isolatedsingularity where R0 = k. Then the following conditions are equivalent.3Araya: A Characterization of One Dimensional N-Graded GorensteinProduced by The Berkeley Electronic Press, 200064 TOKUJI ARAYA( i ) R is a one dimensional N-graded Gorenstein ring of ¯nite Cohen-Macaulay representation type.( ii ) d(R) <1( ii') d0(R) <1( iii ) d(R) · 9( iii') d0(R) · 9To show this theorem, we need the graded version of Brauer-Thrall 1theorem for graded maximal Cohen-Macaulay modules, due to [4], [3] and[2].Theorem 3.3 (graded version of Brauer-Thrall 1 theorem). Let R be anN-graded Cohen-Macaulay ring with isolated singularity where R0 = k isa perfect ¯eld. If supfe(X) j X 2 CMR is indecomposableg < 1, thenR is of ¯nite Cohen-Macaulay representation type. Here e(X) denotes themultiplicity of the irrelevant maximal ideal along X.Proof of 3.2. The implications (iii) ) (ii) ) (ii') and (iii) ) (iii') ) (ii')are trivial. First, we show (ii') ) (i). Since d0(R) < 1, we see thatdimk Rn = dimk Hom(R;R(n)) · d0(R) < 1 for all n. Therefore theHilbert polynomial of R is constant. Hence dimR = 1. For any in-decomposable graded maximal Cohen-Macaulay module X, dimkXn =dimk Hom(R;X(n)) · d0(R) for all n. Therefore the multiplicity e(X) ofX is bounded by d0(R). Hence R is of ¯nite Cohen-Macaulay representa-tion type by theorem 3.3.To prove (i) ) (iii'), it is enough to compute supfdimk Hom(R; Y );dimk Hom(Y;R); dimk Hom(Xi; Y ); dimk Hom(Yi; Y ) j Y 2 CMR is inde-composable g where Xi and Yi are in [1, Figures (1) ¡ (7)] . For an inde-composable graded maximal Cohen-Macaulay module X, we denote by X+(resp. X¡) the smallest additive full subcategory of CMR containing allindecomposable graded maximal Cohen-Macaulay modules Y with X ¹ Y(resp. Y ¹ X). Then, by induction on the length of the path from X to Y(resp. from Y to X), one can easily check that dimk Hom(X;Y ) = 1 (resp.dimk Hom(Y;X) = 1) for all indecomposable Y 2 X+ (resp. Y 2 X¡) withY is not free and ¿Y =2 X+ (resp. ¿¡Y =2 X¡). Since R is a one dimen-sional N-graded Gorenstein ring of ¯nite Cohen-Macaulay representationtype, we may assume that R is one of the rings given in (1) . Thus we areable to compute dimk Hom(R;R(n)) = dimk Hom(R(¡n); R) for all n byHilbert function. Since the functor Hom(R;¡) (resp. Hom(¡; R)) is anexact functor on R+ (resp. R¡), it is possible to compute dimk Hom(R; Y )(resp. dimk Hom(Y;R)) for all Y 2 R+ (resp. Y 2 R¡) by using Auslander-Reiten quiver. Since Hom(R; Y ) = 0 (resp. Hom(Y;R) = 0) for all Y =2 R+(resp. Y =2 R¡) by lemma 2.1, it is possible to compute dimk Hom(R; Y )4Mathematical Journal of Okayama University, Vol. 42 [2000], Iss. 1, Art. 1http://escholarship.lib.okayama-u.ac.jp/mjou/vol42/iss1/1A CHARACTERIZATION OF ONE DIMENSIONAL GORENSTEIN RINGS 65and dimk Hom(Y;R) for all Y 2 CMR. For any X 2 fXi; Yigi, since wehave already computed dimk Hom(X;R(n)) = dimk Hom(X(¡n); R) andsince Hom(X;¡) is an exact functor on X+, it is also possible to computedimk Hom(X;Y ) for all Y 2 X+ by using Auslander-Reiten quiver. Inthis way we can accomplish the computation of dimk Hom(X;Y ) for anyindecomposable X;Y 2 CMR and get the invariant d0(R). The result isshown in table 1. Looking at this table we have d0(R) · 9.Table 1type A2m+1 A2m D2m D2m+1 E6 E7 E8d0(R) 1 2 3 4 4 6 9Finally, we prove (iii') ) (iii). Because we have already proved (iii')) (i), we may assume that R is one given in (1). Since ExtnR(X;Y ) »=Ext1R(­n¡1X;Y ) for all n > 0 and by lemma 2.3, it is enough to showd0(R) ¸ d1(R). For any indecomposable graded maximal Cohen-MacaulaymoduleX, the ¯rst syzygy ­X ofX is also an indecomposable graded max-imal Cohen-Macaulay module. Since there exists a natural epimorphismHom(­X;Y )³ Ext1R(X;Y ), one can see d0(R) ¸ d1(R) and get d(R) · 9.2Remark 3.4. Let R be a one dimensional N-graded Gorenstein ring of ¯-nite Cohen-Macaulay representation type with R0 = k being algebraicallyclosed ¯eld of characteristic 0 (i.e. R is isomorphic to one of the ringsgiven in (1)). In the above proof, we showed how to compute the in-variant d0(R). Remark that we can also compute the invariant dn(R)(n ¸ 1) by using Auslander-Reiten quiver in a similar way to this. SinceExtnR(X;Y ) »= Ext1R(­n¡1X;Y ), we have dn(R) = d1(R) for n ¸ 1. Wewill show how to compute d1(R). For an indecomposable graded maximalCohen-Macaulay module X, we denote by X(1) the smallest additive fullsubcategory of CMR containing all indecomposable graded non-free max-imal Cohen-Macaulay modules Y with ­X ¹ Y ¹ ¿X. We also denoteby X(1)0the smallest additive full subcategory of CMR containing all in-decomposable graded non-free maximal Cohen-Macaulay modules Y with¿X Á Y and X  Y . It turns out from lemma 2.1 and lemma 2.2 thatExt1R(X;Y ) = 0 for all Y =2 X(1) and ExtnR(X;Y ) = 0 for all Y 2 X(1)0and for all n. And it follows from lemma 2.3 that Ext1R(X;¡) is an exactfunctor on X(1) [X(1)0 . Hence it is possible to compute d1(R) (and there-fore dn(R) for all n ¸ 1) by using Auslander-Reiten quiver. The resultsare given in following table.5Araya: A Characterization of One Dimensional N-Graded GorensteinProduced by The Berkeley Electronic Press, 200066 TOKUJI ARAYATable 2A2m+1 A2m D2m D2m+1 E6 E7 E8d(R) = d0(R) 1 2 3 4 4 6 9dn(R) (n ¸ 1) 1 2 1 2 3 4 6References[1] T. Araya: Exceptional sequences over graded Cohen-Macaulay rings, Math. J.Okayama Univ. (to appear).[2] E. Dieterich: Reduction of isolated singularities, Comment. Math. Helv. 62 (1987),654-676.[3] Y. Yoshino: Brauer-Thrall type theorem for maximal Cohen-Macaulay modules, J.Math. Soc. Japan 39 (1987), 719-739.[4] Y. Yoshino: Cohen-Macaulay Modules over Cohen-Macaulay Rings, London Math.Soc., Lecture Note Series vol. 146, Cambridge U.P.(1990).Tokuji ArayaGraduate school of Natural Science and TechnologyOkayama University,700-8531 JAPAN(Received November 24, 2000)6Mathematical Journal of Okayama University, Vol. 42 [2000], Iss. 1, Art. 1http://escholarship.lib.okayama-u.ac.jp/mjou/vol42/iss1/1

A Characterization of One Dimensional N-Graded Gorenstein Rings of Finite Cohen-Macaulay Representation Type

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