Structure Theory for a Class of Grade Four Gorenstein Ideals

An ideal / in a commutative noetherian ring R is a Gorenstein ideal of grade g if pdR(R/I) = grade I = g and the canonical module HxtsR(R/I, R) is cyclic. Serre showed that if g = 2 then / is a complete intersection, and Buchsbaum and Eisenbud proved a structure theorem for the case g = 3. We present generic resolutions for a class of Gorenstein ideals of grade 4, and we illustrate the structure of the resolution with various specializations. Among these examples there are Gorenstein ideals of grade 4 in k][x,y, z, v]] that are A»-generated for any odd integer n > 7. We construct other examples from almost complete intersections of grade 3 and their canonical modules. In the generic case the ideals are shown to be normal primes. Finally, we conclude by giving an explicit associative algebra structure for the resolutions. It is this algebra structure that we use to classify the different Gorenstein ideals of grade 4, and which may be the key to a complete structure theorem. Introduction. One of the most useful techniques in the study of ideals of finite projective dimension has been to realize such ideals as specializations of generic ideals whose structure and resolutions are well understood. Of these the various determinantal ideals play a preeminent role, and in many cases explicit resolutions have been constructed. Moreover, for perfect ideals the structure of the resolution is preserved under specialization so long as the grade of the ideal is maintained. In §2 we present generic resolutions for a class (defined below) of Gorenstein ideals of grade four. As a corollary we show that there are Gorenstein ideals of grade four in local rings that are minimally generated by any number n elements, where n > 6. We exploit the structure of the resolution to prove a regularity result, which shows that the generic ideals are normal primes. In §3 we present a construction that produces examples in our class. Finding the generic resolution represents the first step toward a description and classification of all Gorenstein ideals of grade four. The next step is to show that any ideal in the class is indeed a specialization of the generic case. A similar program has been carried out successfully for Gorenstein local rings R/I of codimension three by Buchsbaum and Eisenbud [4]. To accomplish this they exploited the algebra structure on a minimal free resolution of R/I. They Received by the editors July 31, 1980 and, in revised form, February 17, 1981. Some of the contents of this paper were presented to the special session on commutative algebra at the American Mathematical Society Meeting in Philadelphia on April 18, 1980. 1980 Mathematics Subject Classification. Primary 13C05; Secondary: 13D25, 13H10, 14M12, 16A03. ' Research supported by grant #3093-XO-0038 of the University of Kansas General Research Allocation. 2 Research supported by a grant from the University of Tennessee. © American Mathematical Society 0002-9947/81/0000-1035/$06.25 287 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 288 ANDREW KUSTIN AND MATTHEW MILLER found that there is essentially just one kind of resolution: the ideal / is presented by an alternating matrix and is generated by the maximal pfaffians of this matrix. The situation is more complicated in codimension four. In [17] we proved that if (R, m, k) is a local Gorenstein ring in which 2 is a unit, then a minimal resolution of R/I admits an associative algebra structure. In §4 we give an explicit algebra structure on the resolutions discussed in §2, and we are able to drop the restrictive hypotheses on R. The induced algebra structure on A.= Torf (R/I, k) agrees with the usual multiplication on the homology algebra. In codimension three, with the exception of the complete intersections, one always has A2 = 0, but in cc>,, and yu = 0) with entries in a commutative ring R. If r < ai we define Pf,_ ¡(Y) to be the pfaffian of the alternating submatrix of Y formed by deleting rows and columns ix, . . . , ir. (The determinant of such a matrix is a perfect square in R; the pfaffian is a uniquely determined square root. See Lang [19, p. 372] or Artin [1, p. 140].) Let (i) denote the multi-index i,i2 . . . ir. Define a(i) to be 0 if (i) has a repeated index and otherwise to be the sign of the permutation that rearranges ix, .. ., ir in ascending order. Let |/| = 2^_, /, and let Y(i) = (-lf+xa(i)Pf0)(Y). If r=n let Y(i) = (l)|,| + 1o(/), and if r > ai let y(l) = 0. Finally, let y = [Yx, . . . , Yn]be the vector of pfaffians of Y of order ai 1, signed appropriately according to the conventions described above. There is a "Laplace expansion" for developing pfaffians in terms of ones of lower order. Lemma 1.1. Let Y be an n X n alternating matrix andj a fixed integer, 1 < j < n. Then (a)Pf(Y) = -ZUxyijYij,and (b) yr = 0. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use GRADE FOUR GORENSTEIN IDEALS 289 Proof. The first assertion is an exercise in Artin [1, p. 142]; the second follows by applying the first to an augmented matrix formed by repeating a row and corresponding column of Y. The next result also follows from (a). Lemma 1.2. Let Y be an n X n alternating matrix. Suppose that a, b, c, d, and I are in {1.ri) with a, b, c, d distinct. Then 2 yklYkab = ~8laYb + SIbYa k=l and Zj yklYkabcd-^laYbcd + $IbYacd "icYabd + °ldYabcfc-1 Lemma 1.3. Let B and Z be 3 X 3 alternating matrices. Then bZ 4xB = 0. Proof. A direct computation establishes the result. The following example illustrates the expansion formula, and will be used in the sequel. Example 1.4. Let t > 3 be an odd integer and let X = (t 4l)/2. Let Y be the T X T alternating matrix with entries x if / is odd and/ = / + 1, y a = y if ' is even and/ = / + 1, z if y = t — / + 1 for i 7TT_1pflT_lT(7)