INDECOMPOSABLE GORENSTEIN MODULES OF ODD RANK Christel Rotthaus, Dana Weston and Roger Wiegand July 17, 1998 Abstract. Let (R,m) be a local Cohen-Macaulay ring with m-adic completion R. A Gorenstein R-module is a non-zero finitely generated R-module whose m-adic completion is isomorphic to a direct sum of copies of the canonical module ω R. The rank of the Gorenstein module G is the positive integer r such that Ĝ = r ω R (the direct sum of r copies of ω R). In this note we show that for any given positive integer r there is a Cohen-Macaulay ring R with an indecomposable Gorenstein module G of rank r. Throughout this paper (R, m) denotes a Cohen-Macaulay local Noetherian ring with m- adic completion R, and ω R denotes the canonical module of R. A Gorenstein R-module is a non-zero finitely generated R-module G such that Ĝ = r ω R (the direct sum of r copies of ω R) for some integer r (called the rank of G). Equivalently, [Sh3, (2.7)], G is a maximal Cohen-Macaulay R-module with finite injective dimension. If R has a Gorenstein module, then R has, up to isomorphism, a unique indecomposable Gorenstein module G, and every Gorenstein module is isomorphic to a direct sum of copies of G. (See [FFGR, (4.6)].) Gorenstein modules were introduced by Rodney Sharp [Sh1], [Sh2] and studied extensively in [FFGR]. For many years it was unknown whether a Cohen-Macaulay ring having a Gorenstein module must also have a canonical module (i.e., a Gorenstein module of rank 1). This was settled in a 1988 paper [We1] by Weston, who produced a Cohen-Macaulay unique factorization domain having an indecomposable Gorenstein module of rank 2. In this note we will show that there are Cohen-Macaulay unique factorization domains having indecomposable Gorenstein modules of arbitrary rank. (The first example of a Cohen-Macaulay, non-gorenstein, unique factorization domain was given by Ogoma [O].) One of the interesting features of [FFGR] is the use of the Brauer group of R. It turns out that Λ := End R (G) is an Azumaya R-algebra if G is a Gorenstein R-module. The order of Λ in the Brauer group is always a divisor of the rank r, [KO, Ch. IV, (6.1)], and if Λ is trivial in Br(R) then R has a canonical module. It follows that if G is an indecomposable 1991 Mathematics Subject Classification. Primary 13C14, 13C11; Secondary 13A20, 13B22, 13B35. Key words and phrases. Gorenstein module, canonical module, unique factorization domain, Brauer group. Rotthaus and Wiegand thank the National Science Foundation for partial research support. 1 Typeset by AMS-TEX
2 CHRISTEL ROTTHAUS, DANA WESTON AND ROGER WIEGAND Gorenstein module of prime rank p, then the endomorphism ring Λ must have order p in Br(Λ). Our results contradict Theorem 4.9 of [FFGR], which says that the order of Λ in Br(R) must be either 1 or 2, and that, consequently, an indecomposable Gorenstein module of rank > 1 must have even rank. (The error in the proof of [FFGR, (4.9)] seems to be in the second displayed isomorphism on page 210 of [FFGR]: One needs instead Λ op A End A (G) = End Λ (Λ A G) in order for the obvious module homomorphsm to be a ring homomorphism.) The examples. Let k be an algebraically closed field of characteristic 0, and let B = k[[x,y ]], the power series ring in two variables. Fix an integer r 2 and put A = A r := k[[x r,x r 1 Y,...,XY r 1,Y r ]]. Then A is the fixed ring of the k-automorphism of B that multiplies each variable by ζ r = primitive r th root of unity. We see that A is a complete local normal domain of dimension 2. By Heitmann s theorem [Hei] there is a local unique factorization domain (R,m) = (R r,m) such that R = A. We will show that R r has an indecomposable Gorenstein module of rank r if r is odd and of rank r/2 if r is even. The divisor class group of A r. We need to compute the divisor class group of A r and identify the canonical module. Computations of this sort occur often in the literature, e.g., [F], [Her], [Wa1], [Wa2], [We2]. Here, following [Her], we will give a direct argument, which has the feature of identifying all of the indecomposable reflexive modules. The trick is to decompose B as an A-module. We have B = A J 1 J r 1, where J i is the A-submodule of B generated by the monomials of degree i in X and Y. Let I = (X r,x r 1 Y ), the ideal of A generated by X r and X r 1 Y. Then J i = I i for i = 1,...,r 1. Thus we have the decomposition ( ) B = A I I 2 I r 1. Since depth(b) = 2 each I i is a divisorial ideal of A and therefore an indecomposable reflexive (= maximal Cohen-Macaulay) A-module. Following [Her], we will show that these divisorial ideals form a complete set of representatives for the indecomposable reflexive A- modules. Write V W to indicate that the A-module V is isomorphic to a direct summand of W. Suppose M is an indecomposable finitely generated reflexive A-module. Since A B by ( ) we see that M B A M. Therefore M (B A M) (duals over A). Now M = M ; and (B A M) is a B-module of depth 2 and is therefore B-free. By the Krull-Schmidt theorem (for finitely generated modules over the complete ring A), M must be isomorphic to one of the I i. Since µ A (I i ) = i + 1 for 0 i r 1 (where µ = number of generators), the divisorial ideals I i are pairwise non-isomorphic. Thus we have the following result: Proposition 1. The divisor class group Cl(A r ) is cyclic of order r with generator [I] and with [I] i = [I i ] for 0 i r 1. ( The ideal I r is not divisorial since it is isomorphic to m, so [I] r = [A].) Next we identify the canonical module of A r. Proposition 2. The canonical module ω r of the complete local ring A r is the ideal I r 2. In particular, µ Ar (ω r ) = r 1. Moreover, the order of [ω r ] in Cl(A r ) is r if r is odd and r/2 if r is even.
GORENSTEIN MODULES 3 Proof. We have already seen that [I] has order r and that [I r 2 ] = [I] r 2. It follows that [I r 2 ] has the order indicated. Since ω r must be isomorphic to one of the ideals I i,0 i r 1, it will suffice to show that µ Ar (ω r ) = r 1. Let C = A r /(X r,y r ), and let n be the maximal ideal of C. We have µ Ar (ω r ) = µ C (ω r /(X r,y r )ω r ), which is the k-dimension of the socle of C. (See [BH, 3.3].) But n 2 = 0, so n is the socle, and we have µ Ar (ω r ) = dim k (n) = r 1. Recall [B, Chap.VII, 4.7] that if S is a normal Noetherian domain one can associate to each finitely generated S-module M a divisor class [M] Cl(S) in such a way that (1) if 0 M M M 0 is an exact sequence of finitely generated S-modules, then [M] = [M ] + [M ], and (2) if J is a fractional ideal of S, then [J] = [J ], the isomorphism class of J. The following result is a slight reformulation of [We1, (1.5)]: Proposition 3. Let (S,m S ) be a local Noetherian ring of dimension two whose m S -adic completion T is a normal domain, and let N be a finitely generated torsion-free T-module. Then N is extended from S (that is, there is an S-module M necessarily finitely generated and torsion-free such that N = T S M) if and only if [N] is in the image of the natural map Φ : Cl(S) Cl(T). Proof. Suppose N is extended, say, N = T S M. By faithful flatness, M is finitely generated and torsion-free. By Bourbaki s theorem [B, chap. VII, $4.9, Thm.6] there is an exact sequence (1.1) 0 F M J 0, in which F is a free S-module and J is an ideal of S. Tensoring (1.1) with T and using properties (1) and (2), we see that [N] = [T S J] = [(T S J) ] in Cl(T). But (T S J) = T S J where the latter duals are computed over S. Thus [N] = Φ([J ]) Im(Φ). (In fact, not surprisingly, [N] = Φ([M]).) For the converse, choose a Bourbaki sequence (1.2) 0 G N L 0, where now G is a free T-module and L is an ideal of T. Then [L] = [N], and since [N] is in the image of Φ there is a divisorial ideal D of S such that T S D = L. Put V := L /L. This is a T-module of finite length and is therefore a finitely generated S-module. Now Hom T (L,V ) = T S Hom S (D,V ) = Hom S (D,V ), since Hom S (L,V ) has finite length. In other words, the quotient map L V is extended from an S-homomorphism f : D V. Putting J := Ker(f), we see that T S J = L. Let F be a free S-module such that T S F = G. The exact sequence in (1.2) represents an element of Ext 1 T (L,G) = T S Ext 1 S (J,F) = Ext1 S (J,F), since Ext1 S (J,F) has finite length. In other words, the exact sequence in (1.2) is actually extended from an exact sequence of S-modules as in (1.1). It follows that N = T S M.
4 CHRISTEL ROTTHAUS, DANA WESTON AND ROGER WIEGAND The hypothesis that M be torsion-free cannot be omitted. For example, in our ring T := A 2 the ideal I is not extended from S = R 2. Therefore M := I A/I cannot be extended either, even though [M] = 0 in Cl(T). Returning to our examples, we fix an arbitrary positive integer s. Put r = s if s is odd; and if s is even put r = 2s. Let A = A r, and let ω = ω r, the canonical module of A. We know that [ω] has order s in Cl(A). Since R = R r is a unique factorization domain (whence Cl(R) = 0), Proposition 3 implies that the direct sum c ω of c copies of ω is extended from R if and only if c is a multiple of s. Therefore R has an indecomposable Gorenstein module of rank s. Suppose now that s is prime. The order of Λ := End(G) divides s by [KO, Chap. IV, (6.1)]; and by [FFGR,(4.1)] Λ must be non-trivial in Br(R). Therefore Λ has order s in Br(R). We conclude on a positive note. Proposition 4. Let (R,m,k) be a local ring with a Gorenstein module G of rank 2. Suppose char(k) 2 and the group R of units of R is 2-divisible. Then R has a canonical module. Proof. We consider the splitting ring S, constructed in [AG, (6.3)], for the Azumaya algebra Λ := End R (G). This splitting ring is free of rank 2 as an R-module and is constructed as follows: Since [Λ] is in the kernel of the canonical map Br(R) Br(k), we can identify Λ/mΛ with the ring of 2 2 matrices over k. Then S = R[β], where β Λ is a preimage of α := [ 1 0 0 1 ] under the natural map π : Λ Λ/mΛ. By the proof of [AG, (6.3)], β 2 = a+bβ ], we see that a R and b m. for suitable a,b R. Since π(β 2 ) = α 2 = [ 1 0 0 1 Put γ = b 2 + β. Then π(γ) = α and γ2 = c, where c = a + b2 4 R. It follows that S = R[γ]. Moreover, since S is a free R-module (with basis 1 and γ) the division algorithm shows that S = R[X]/(X 2 c). Since c is a square in R (as R is 2-divisible), S = R R. It follows that Λ already splits over R, and by [FFGR, Theorem 4.1 (4)] R has a canonical module. Remark. One can obtain the canonical module explicitly as the fixed module of the ordertwo automorphism of G : x 1 c γ(x). References [AG] M. Auslander and O. Goldman, The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367 409. [B] N. Bourbaki, Commutative Algebra, Springer Verlag, New York, 1983. [BH] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Math., vol. 39, Cambridge University Press, Cambridge, 1993. [F] R. Fossum, The Divisor Class Group of a Krull Domain, Springer-Verlag, New York, 1973. [FFGR] R. Fossum, H.-B. Foxby, P. Griffith and I.Reiten, Minimal injective resolutions with applications to dualizing modules and Gorenstein modules, IHES Publ. Math. 45 (1976), 193 215. [Hei] R. Heitmann, Characterizations of completions of unique factorization domains, Trans. Amer. Math. Soc. 337 (1993), 379 387.
GORENSTEIN MODULES 5 [Her] J. Herzog, Ringe mit nur endlich vielen Isomorphieklassen von maximalen unzerlegbaren Cohen- Macaulay Moduln, Math. Ann. 233 (1978), 21 34. [Hi] V. Hinich, Rings with approximation property admit a dualizing complex, Math. Nachr. 163 (1993), 289 296. [KO] M.-A. Knus and M. Ojanguren, Théorie de la Descente et Algèbres d Azumaya, Lecture Notes in Math., vol. 389, Springer, Berlin. [M] D. Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math. Inst. Hautes Etud. Sci. 9 (1961), 5 22. [O] T. Ogoma, Cohen Macaulay factorial domain is not necessarily Gorenstein,, Mem. Fac. Sci. Kochi University (Math.) 3 (1982), 65 74. [Sh1] R. Sharp, The Cousin complex for a module over a commutative Noetherian ring, Math. Z. 112 (1969), 340 356. [Sh2], Gorenstein modules, Math. Z. 115 (1970), 117 139. [Sh3], On Gorenstein modules over a complete Cohen-Macaulay local ring, Quart. J. Math. 22 (1971), 425 434. [St] R. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc.(N.S.) 1 (1979), 475 511. [Wa1] K. Watanabe, Certain invariant subrings are Gorenstein I, Osaka J. Math. 11 (1974), 1 8. [Wa2], Certain invariant subrings are Gorenstein II, Osaka J. Math. 11 (1974), 379-388. [Wa3]. [We1] D. Weston, On descent in dimension two and non-split Gorenstein modules, J. Algebra 118 (1988), 263 275. [We2] D. Weston, Divisorial properties of the canonical module for invariant subrings, Comm. Algebra 19 (1991), 2741 6666. Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027 E-mail address: rotthaus@math.msu.edu Department of Mathematics, University of Missouri, Columbia, MO 65211 E-mail address: weston@math.missouri.edu Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0323 E-mail address: rwiegand@unl.edu