The What, Where, and Why of Almost Split Sequences

Transcription

1 Proceedings of the International Congress of Mathematicians Berkeley, California, USA, 1986 The What, Where, and Why of Almost Split Sequences MAURICE AUSLANDER Throughout this paper we assume that R is a noetherian, local Gorenstein ring. We also assume that all fi-algebras A are finitely generated E-modules. An Ä-module A is said to be complete if R is complete and is said to be Cohen- Macaulay if A is a (maximal) Cohen-Macaulay iï-module. Recall that if A is a complete iü-algebra, then mod A, the category of finitely generated A-modules, is a Krull-Schmidt category; i.e., an M in mod A is indecomposable if and only if EndA(M), the endomorphism ring of M, is a local ring and thus the decomposition of an X in mod A into a sum (direct) of indécomposables is unique up to isomorphism. When A is a Cohen-Macaulay i2-algebra, we denote by CM A the full subcategory of mod A consisting of the X is mod A which are (maximal) Cohen-Macaulay modules when viewed as iü-modules. Further, if A is a Cohen-Macaulay Ä-algebra we have the duality JD:CMA CMA op given by D(X) = B.oiriR(X,R) for all X in CM A. Moreover, this duality is exact in the sense that if 0 > A-^> B C 0 is exact in CM A, i.e., it is an exact sequence of A-modules with A, B, C in CM A, then 0 - D(C) - D(B) - D(A) - 0 is exact in CMA op. We say that A in CM A is CMA-injective if every exact sequence 0 y A > B y C 0 in CM A splits. From the exact duality D: CM A - CM A P it follows that A is CMA-injective if and only ifa = D(P) for some projective A op -module. We will be mainly interested in questions concerning almost split sequences in CM A when A is a complete Cohen-Macaulay i2-algebra. Recall that an almost split sequence in CM A is a nonsplit exact sequence 0 > A -+ B -^+ C > 0 m CM A having the following additional properties: (a) A and C are indecomposable; (b) if h:x y C is a nonisomorphism in CM A with X indecomposable, then h can be lifted to B\ i.e., there is a t:x y B such that gt = h\ (c) each nonisomorphism j:a Y in CM A with Y indecomposable can be extended to B; i.e., there is an s: B Y such that sf = j. It should be observed that (a) and (b) is equivalent to (a) and (c) and that two almost split sequences Written with partial support from National Science Foundation Grant MCS International Congress of Mathematicians

2 ALMOST SPLIT SEQUENCES > A - B - C -> 0 and 0 -+ A' -+ P' -+ C" -> 0 in CMA are isomorphic if and only if 4» 4' if and only if G C". Before stating our main existence theorem for almost split sequences in CMA, it is convenient to make the following definition. We say that a A-module X has a certain property outside the maximal ideal if the A p -module X p has the property for all nonmaximal ideals p of R. THEOREM 1 (AUSLANDER-REITEN [8]). Suppose A is a complete Cohen- Macaulay R-algebra. (a) An indecomposable nonprojective C in CMA has the property that it is projective outside the maximal ideal if and only if there is an almost split sequence 0-+ji->fi-*C->0in CMA. (b) An indecomposable A in CMA which is not CMA-injective is CMAinjective outside the maximal ideal if and only if there is an almost split sequence 0->,4->B->C->0m CMA. Let P m (CMA) and J m (CMA) denote the full subcategories of CMA consisting of the X which are projective (CMA-injective) outside the maximal ideal and have no projective (CMA-injective) indecomposable summands. Combining Theorem 1 with the fact that either end of an almost split sequence determines the sequence (up to isomorphism), we see that we obtain the following bijection between the isomorphism classes of indecomposable objects in P m (CMA) and J m (CMA). We send [C], the isomorphism class of an indecomposable C in P m (CMA), to the uniquely determined [A] with A indecomposable in J m (CMA) such that there is an almost split sequence 0 -^ A y B y C > 0. We now give a description of this bijection without reference to almost split sequences (Auslander [4]). f Let Pi y Po C y 0 be a minimal projective presentation of C in P m (CMA). As usual, define TrC, the transpose of C, by the exact sequence P 0 * C P* - TrC - 0 where X* = Hom A (X,A). Then define Tr L C = n d TrC where Q d is the dth syzygy in a miminal projective resolution of TrC and dimension R is d. Then Tr L C is in P m (CMA op ) and DTr L C is in J m (CMA). Moreover, if 0 y A y B y C 0 is an almost split sequence in CMA, then A = D Tr L C and C = Tr L DA. As a consequence of Theorem 1, we have the following characterization of when a complete Cohen-Macaulay P-algebra A is an isolated singularity; i.e., gl dim Ap = dimrp for all nonmaximal ideals p of R. THEOREM 2 (AUSLANDER [5, 4]). The following are equivalent for a complete, Cohen-Macaulay R-algebra A. (a) A is an isolated singularity. (b)pm(a) = CMA = / m (A). (c) For each indecomposable, nonprojective G in CMA there is an almost split sequence 0 +A-*B-^C^yOin CM A.

3 340 MAURICE AUSLANDER Recall that a complete, Cohen-Macaulay P-algebra is said to be of finite Cohen-Macaulay type if CM A has only a finite number of nonisomorphic indecomposable modules. Now it is not difficult to see that if A is a complete Cohen-Macaulay P-algebra of finite Cohen-Macaulay type, then every nonprojective G in CMA has an almost split sequence 0 y A > B >C 0 in CMA. Therefore we have the following result as an immediate consequence of Theorem 2. PROPOSITION 3 (AUSLANDER [3, 5]). A complete, Cohen-Macaulay R- algebra is an isolated singularity if it is of finite Cohen-Macaulay type. An interesting question is to classify the complete Cohen-Macaulay rings of the form A = fc[[xi,..., Xd\]/a which are of finite Cohen-Macaulay type, where o is an ideal in the formal power series ring fc[[xi,..., Xd]] over an algebraically closed field A:. This has been accomplished by Knörrer [24] and Buchweitz, Greuel, and Schreyer [17], when a is a principal ideal and characteristic k ^ 2. When chfc = 0, one obtains precisely the Ar'nold simple singularities. When fc = C, the complex numbers, and dimension A = 2, then Herzog [23] has shown that the quotient singularities with respect to finite subgroups of GL(2, C) are of finite Cohen-Macaulay type while Artin-Verdier (unpublished), Auslander [6], and Esnault [19] have shown the converse. Auslander-Reiten [12] have found the only known nonhypersurface singularities of finite Cohen-Macaulay type in dimensions greater than 2, both of which have dimension 3. They are the fixed ring C[[Xi, X 2, Xs]] z / 2Z where the generator a of Z/2Z operates by a(xi) = Xi for all i, which is the only quotient singularity of dimension greater than 2 of finite Cohen-Macaulay type, and the scroll C[[X Q,X 1,X 2,Yo,Y 1 ]]/(X 0 X 2 -XlX 0 X 1 -X 1 Y Q,X 1 Y 1 -X 2 Y Q ). In this connection, it should be noted that Artin [1] has classified an interesting class of noncommutative complete Cohen-Macaulay P-algebras A of finite Cohen-Macaulay type. Namely, when R = k[[xi,x2]], Chfc = 0 and A is a maximal order in a division ring of finite dimension over the field of quotients of R. Now when dimp = 0, then a complete, Cohen-Macaulay P-algebra A is nothing more than an P-algebra A and mod A = CMA = P m (CMA) = J m (CMA). For example, when R is a field, one is dealing with the theory of finite-dimensional modules over a finite-dimensional algebra. So under these circumstances Theorem 1 says that every indecomposable nonprojective A-module G has an almost split sequence 0 A -+ P y G 0 in mod A. While this now appears as a special case of Theorem 1, it was in the zero-dimensional case that almost split sequences were first introduced by Auslander-Reiten [13] and where they have been most extensively used, to very good effect. For instance, the first examples of tiltings [7] and coverings [26] depended in an essential way on almost split sequences. Since then tiltings and coverings as developed by Brenner-Butler [16], Happel-Ringel [22], and Bongartz-Gabriel [15] respectively have become

4 ALMOST SPLIT SEQUENCES 341 major tools in studying the representation theory of finite-dimensional algebras. The rest of this paper is devoted to showing how some ideas and results first developed in the zero-dimensional case have appeared in higher dimensions. From now on we assume that A is a complete Cohen-Macaulay P-algebra which is a commutative local ring. Since every commutative complete Cohen- Macaulay local ring S is a complete Cohen-Macaulay P-algebra for some P, we are really just talking about commutative complete Cohen-Macaulay rings S. To emphasize this point, we will denote A by S and often drop the P. In addition we assume that S is an isolated singularity which simply means that Sp is regular for all nonmaximal prime ideals of S. Therefore CMS = P m (CMS) = 7,n(CMS) so that there is an almost split sequence 0 y A-^y B y C > 0 in CM S for each indecomposable G ^ S and for each A ^ Hom^S, P). Let G be an indecomposable module in CMS. Then a morphism /:P > C in CM S is said to be a minimal right almost split morphism if it satisfies the following: (a) / is not a splittable surjection; (b) if t:x y G in CMS is not a splittable surjection, then it can be lifted to P; (c) if h:b y B is such that fh = f, then h is an automorphism. It is easily seen that if /: B C and /': B 1 y G are two minimal right almost split morphisms in CM S, then there is an isomorphism h:b yb / such that / = f ( h, so the minimal right almost split morphisms to G are uniquely determined (up to isomorphism) by G. If G ^ S and 0 y A -^y B y G 0 is an almost split sequence in CM S, then f:b > C is easily seen to be a minimal right almost split morphism. If C = S, then it can be shown that there is also a minimal right almost split morphism E > S in CM S. So for each indecomposable C in CM S there is a uniquely determined (up to isomorphism) minimal right almost split morphism f:b C. Also we have the dual notion of a minimal left almost split morphism g: A > B in CM S with A indecomposable and the dual result that given any indecomposable A in CMS, there is a uniquely determined (up to isomorphism) minimal left almost split morphism A -^> B which is the left-hand side of the almost split sequence 0->A-^P-^C->0in CMS when A Hom fi (S,P). A notion intimately connected to that of minimal right (left) almost split morphisms is that of an irreducible morphism first introduced in the case of Artin algebras by Auslander-Reiten [14]. A morphism f:x -+ Y between indecomposable modules in CM S is said to be irreducible in CM S if (a) / is not an isomorphism and (b) if / = hg in CMS, then g is a splittable injection or h is a splittable surjection. Let X and Y be indecomposable and suppose g:x B and h:b f yy are minimal left and right almost split morphisms respectively. Then for f:x y Y in CMS, the following are equivalent: (a) / is irreducible; (b) there exists splittable injection s:x >B f such that hs /; (c) there exists a splittable surjection t: B y Y such that tg = f. Thus for each indecomposable module X in CM S, there exist only a finite number of nonisomorphic indecomposable Y in CMS which have either an irreducible morphism Y X or an irreducible morphism X Y.

5 342 MAURICE AUSLANDER As our first example of how irreducible morphisms have been used in arbitrary dimensions we state the following criterion for determining when S is of finite Cohen-Macaulay type which is essentially the same as a criterion used extensively in the zero-dimensional case [4]. This criterion was used to show that the scroll k[[xo,x 1,X 2,Yo,Y 1 ]]/(X Q X 2 -XlX 0 X 1 -X 1 Y 0,X 1 Y 1 -X 2 Yo), cited earlier, is of finite Cohen-Macaulay type. PROPOSITION 4 (AUSLANDER-REITEN [12]). Suppose C is a subcategory of CM S closed under isomorphisms consisting of a finite number of nonisomorphic indecomposable modules including S. Further, assume that an indecomposable X in CM S is in C if there exists an irreducible G X or an irreducible X y G with G in G. Then S is of finite Cohen-Macaulay type and C consists of all the indecomposable modules in CMS. Assume now that S is a fc-algebra where fc is an algebraically closed field and S/m = fc where m is the maximal ideal of S. Then if U Y and X V are minimal right and left almost split morphisms in CMS with Y and X indecomposable, then the multiplicity that Y appears as a summand of V is the same as the multiplicity X appears as a summand of U. This common number is called the number of independent irreducible maps from X to Y. We are now in position to define the AR-quiver of S. This is a directed graph whose vertices are the isomorphism classes [Z] of indecomposable modules Z in CM S and where the number of arrows from [X] to [Y] is the number of independent irreducible morphisms from X to Y. When S is a Gorenstein ring, the AR-quiver of S minus the vertex [S] and the arrows to it and out of it, is called the stable AR-quiver of S. We then have the following striking result which is the consequence of work of Gonzalez-Springberg and Verdier [21], Artin-Verdier [2], Esnault-Knörrer [20], Auslander-Reiten [9]. THEOREM 5. Suppose S is a rational double point over an algebraically closed field of any characteristic. Then the desingularization graph of S is naturally isomorphic to the underlying graph of the stable AR-quiver of S. In connection with this theorem it is of interest to point out a different but related interpretation of the AR-quiver of two-dimensional quotient singularities in terms of finite group representations. This requires the notion of the McKayquiver of a representation of a finite group. Let fc be an algebraically closed field and G a finite group whose order is not divisible by Chfc. Let V\,..., V 9 be a complete list of nonisomorphic simple fcg-modules. Let V be an arbitrary (finite-dimensional) fc[g]-module. Then the McKay-quiver of V is defined as follows. The Vi are the vertices and one draws n arrows from Vi to Vj if Vi occurs in V k Vj with multiplicity n. Suppose now that V is a faithful two-dimensional fcg-module. Associated with this is a linear action of G as a group of fc-automorphisms of fc[[xi,x2]] whose fixed ring we denote by S. Then S is a local complete integrally closed two-dimensional domain and therefore an isolated Cohen-Macaulay singularity.

6 ALMOST SPLIT SEQUENCES 343 In addition, assume that the fcg-module V has the property that the height one primes of S are unramified in fc[[xi,.x2]] (if fc = C, the complex numbers, this means that G has no pseudo-reflections). Then, as noted before, S is of finite Cohen-Macaulay type and we have PROPOSITION 6 (AUSLANDER [6]), Under the above hypothesis there is a natural isomorphism between the McKay-quiver ofv and the AR-quiver of S. Since McKay [25] had already observed that for G a finite subgroup of SL(2,C), the McKay-quiver of V, the fcg-module given by the inclusion G C SL(2, C), is isomorphic to the disingularization graph of the associated fixed ring, Proposition 6 gave the first connection between almost split sequences and singularity theory. As our final remark concerning two-dimensional rings, we point out some special features of the structure of almost split sequences for arbitrary S of dimension 2 which played a significant role in the proofs of the above results. Suppose dim S = 2 and let u) = Hom^ (S,P), the dualizing module of S. Then there is a unique (up to isomorphism) exact sequence 0 u -^> E y S y S/m 0 with E in CMS called the fundamental sequence of S. It is not difficult to see that /: E y S is the minimal right almost split morphism to S. PROPOSITION 6 (AUSLäNDER [6] AND AUSLANDER-REITEN [10]). Suppose dim S = 2 and 0 y UJ -^ E y S y.s/m > 0 is the fundamental sequence and let M ^ S be an indecomposable module in CMS and let M* = Honig (M, S). Then the induced exact sequence 0 y Horns (M*, CJ) Homg (M*,E) y Horns (M*, S) M has the following properties. (a) 0 Horns(M*,u)) Horns(M*,E) y M y 0 is exact and is either split or almost split. (b) // ch S/m does not divide rank M, then is almost split Homs(M*,a;) -+ Homs(M*,E) -> M - 0 (c) If S/m is algebraically closed, then 0 - Homs(M*» -+ Homs(M*, ) -+ M -> 0 is almost split if and only if ch S/m does not divide rank M. We next turn our attention to computing Go(S), the Grothendieck group of S, which is the free abelian group with basis the isomorphism classes [X] of X in mods modulo the subgroup generated by [A] \B]-\~C whenever there is an exact sequence 0 y A P -» G 0 in mod S. Because S is a Cohen-Macaulay ring, it is easily seen that GQ(S) is also the free abelian group with basis [X] with X in CM S modulo the subgroup generated by [A] [B] + [C] whenever there is an exact sequence ino y A y B y G 0 in CMS. While the split and almost split sequences in CM S give some of the relations for Go (S) in general, it is when S is of finite Cohen-Macaulay type that we get a really definitive

7 344 MAURICE AUSLANDER connection between Grothendieck groups and almost split sequences which is a generalization of Butler's result [18] in dimensions at most 1. PROPOSITION 6 (AUSLANDER-REITEN [11]). Suppose S is of finite Cohen- Macaulay type. Then Go(S) is the free abelian group with basis the isomorphism classes of X in CMS modulo the subgroup generated by [A] - [B] + [C] whenever there is a split or almost split sequence Q-+A-+B >C yo. As an easy consequence of this result we have COROLLARY 7. Let S be of finite Cohen-Macaulay type. Then Go(S) is completely determined by the AR-quiver of S minus the vertex [S] and all arrows going into and coming out of it. We end this discussion of almost split sequences with the following global instance of almost split sequences. PROPOSITION 8 (AUSLANDER-REITEN [10]). LetX be a connected, Cohen- Macaulay projective curve over an infinite field fc. Then an indecomposable Cohen-Macaulay coherent sheaf M is locally free if and only if there is an almost split sequence 0» 7 > Q» # 0 in the category of Cohen-Macaulay coherent sheaves. An easy consequence of this result is the following which was first shown independently by Schofield [27] using sheaf theoretic methods and Auslander- Reiten [10] using graded modules. THEOREM 9. Let X be a connected nonsingular projective curve over an infinite field. Then the category of coherent sheaves over X has almost split sequences. BIBLIOGRAPHY 1. M. Artin, Maximal orders of global dimension and Krull dimension two. Invent. Math, (to appear). 2. M. Artin and J.-L. Verdier, Reflexive modules over rational double points, Math. Ann. 270 (1985), M. Auslander, Finite type implies isolated singularity (Proc. Oberwolfach, 1984), Lecture Notes in Math., Vol. 1142, Springer-Ver lag, Berlin and New York, 1985, pp , Functors and morphisms determined by objects. Applications of morphisms determined by objects (Proc. Conf. on Representation Theory, Philadelphia, Pa., 1976), Dekker, New York, 1978, pp , Isolated singularities and almost split sequences, Representation Theory. II, Groups and Orders (Ottawa, 1984), Lecture Notes in Math., Vol. 1178, Springer-Verlag, Berlin and New York, 1986, pp , Rational singularities and almost split sequences, Trans. Amer. Math. Soc. 293 (1986), no. 2, M. Auslander, M. I. Platzeck, and I. Reiten, Goxeter functors without diagrams, Trans. Amer. Math. Soc. 250 (1979), M. Auslander and I. Reiten, Almost split sequences for Cohen-Macaulay modules (to appear). 9., Almost split sequences for rational double points, Trans. Amer. Math. Soc. (to appear).

8 ALMOST SPLIT SEQUENCES , Almost split sequences in dimension 2, Adv. in Math, (to appear). 11., Grothendieck groups of algebras and orders, J. Pure Appi. Algebra 39 (1986), ,, The Cohen-Macaulay type of Cohen-Macaulay rings (to appear). 13., Representation theory of artin algebras. Ill, Almost split sequences, Comm. Algebra 3 (1975), , Representation theory of artin algebras. IV, Invariants given by almost split sequences, Comm. Algebra 5 (1977), K. Bongartz and P. Gabriel, Covering spaces in representation theory, Invent. Math. 65 (1982), S. Brenner and M. C. R. Butler, Generalizations of the Bernstein-G e Ifand- Ponomarev reflection functors, Representation Theory. II, Lecture Notes in Math., Vol. 832, Springer-Verlag, Berlin and New York, 1980, pp R.-O. Buchweitz, G.-M. Greuel, and F.-O. Schreyer, Cohen-Macaulay modules on hypersurfaces singularities. II, Preprint. 18. M. C. R. Butler, Grothendieck groups and almost split sequences (Proc. Oberwolfach Conf. on Integral Representations and Applications), Lecture Notes in Math., Vol. 882, Springer-Verlag, Berlin and New York, 1981, pp H. Esnault, Reflexive modules on quotient singularities, J. Reine Angew, Math, (to appear). 20. H. Esnault and H. Knörrer, Reflexive modules over rational double points, Math. Ann. 272 (1985), Gonzalez-Springberg and J.-L. Verdier, Construction geometric de la correspondence de McKay, Ann. Sci. École Norm. Sup. (4) 16 (1983), D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), J. Herzog, Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren Cohen-Macaulay-Moduln, Math. Ann. 233 (1978), H. Knörrer, Cohen-Macaulay modules on hypersurface singularities. I, Preprint. 25. J. McKay, Graphs, singularities and finite groups, Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc, Providence, R.I., 1980, pp C. Riedtmann, Algebren, Dor Stellungsköcher, Überlagerungen und Zurück, Comment. Math. Helv. 55 (1980), A. Schofield, Private communication. BRANDEIS UNIVERSITY, WALTHAM, MASSACHUSETTS 02254, USA