1. GORENSTEIN ARTIN ALGEBRAS AND DUALITY: INTERSECTION OF THE m1 AND THE LOEWY FILTRATIONS. 1A. Introduction. A Gorenstein Artin algebra A with maximal ideal m over a field k is self dual: there is an exact pairing -,• : AxA — - k, making A isomorphic as A-module to Homk(A,k) . The associated graded algebra A* = Grm(A) is in general no longer Gorenstein. We study below a stratification of A* by a descending sequence of ideals A* = C(0) ^ C(l) ^ . . . , whose successive quotients Q(a) = C(a)/C(a+1) are. reflexive A* modules. This reflexivity property imposes conditions on the Hilbert-Samuel function H(A), as well as on the deformations of A. If A has socle degree j (so mJ^O but mJ+1=0) then H (A) is the sum function of D (A) = (HA(0),HA(1),. . .,HA(j-2)) where the Hilbert function HA(a) = H(Q(a)) of Q(a) is symmetric around (j-a)/2 . The first subquotient Q(0) is always a graded Gorenstein algebra. An immediate consequence is that A* is itself Gorenstein iff H(A) is symmetric then A = A* = Q(0) (see Proposition 1.7). Our approach adds nothing new to what is already known about graded Gorenstein algebras: we generalize to the nongraded case. In two variables, each subquotient Q(a) is isomorphic to a graded complete intersection and the decomposition D(A) is determined by H(A). This structure underlies F.H.S. Macaulay's result determining the Hilbert functions possible for the intersection of two plane curves (see [Macl] and [16] §3, as well as §2 below). In three or more variables, the decomposition D (A) is not usually determined by H (A) even when A is a complete intersection, the subquotients Q(a) are in general not generated by a single element (Example 4.6). The decomposition D can be an obstruction to deforming: there is a partial order on the decompositions D of a given Hilbert function T, such that a GA quotient A having decomposition D, can have as deformation a GA Received by the editor June 30,1991, and in revised form October 1, 1992. 1

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