CHAPTER 1 Introduction Let R = k[xi,..., xn] = (&ioRi, k an algebraically closed field of characteristic 0, and let I be a homogeneous ideal of i?, A = R/I. The Hilbert function of A, HA : N N, (or sometimes H(A, —)) defined by: HA(t) = dim/e Rt - dimfe It has been much studied. In case / is the ideal of a subscheme, X of P n _ 1 , (in which case the Hilbert function of A = R/I is sometimes denoted Hx(—) or H(X, —)) then this function contains a great deal of information about the geometry of this subscheme. What possible functions arise in this context? This question was successfully considered by Macaulay in [51]. That solution was not, however, the end of the story. Many other, related, questions have also been considered: i) What can H ^ be if A is a domain? (see [65]) ii) What can H ^ be if / = ix is the ideal of a reduced set of points, X, in P71"1? (see [25]) in) What can H ^ be if / = Ix is the ideal of a set of points, X, which is the generic hyperplane section of a curve in P n ? (see [26], [34], [52]) iv) What can HA be if A is a Gorenstein ring? (see [3], [11], [16], [17], [28], [30], [32], [33], [37], [46], [55], [64], [71]). We can rephrase iv) above as follows: let R be as above and let / be a homo- geneous ideal for which \fl (#i,... , xn). If s -f 1 is the least integer such that (#i,... , xn)s+1 C / then i 4 = fc0i10...®is where As ^ 0. The socle of A, denoted soc(A), is defined by soc(A) : ann^(m) where m = 0| = 1 Ai. Since m is a homogeneous ideal of A, soc(A) is also a homogeneous ideal of A. Clearly, As C soc(A). Write soc(A) = 2li © 0 2ls (noting that 2ls = As) and let ai = dim^(2li). The integer vector s = s(A) = (ai,...,a s ) is called the socle vector of A. Notice that as = dim/- 2ls ^ 0. We also call s the socle degree of A. It is well-known that A is a Gorenstein ring if and only if s(A) = (0,..., 0,1). 3
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