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).

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