ABSTRACT
Contents 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.2 Quasipure Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.3 Low Codimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.3.1 Codimension Two, Cohen-Macaulay . . . . . . . . . . . . . . . . . . . . . . . 151 5.3.2 Codimension Two, Not Cohen-Macaulay . . . . . . . . . . . . . . . . . . . 153 5.3.3 Gorenstein Codimension Three . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.4 Monomial Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.4.1 Stable Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.4.2 Squarefree Strongly Stable Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.4.3 Componentwise Linear Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.4.4 Taylor Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.5 Resolutions for a Given Hilbert Function and Truncation . . . . . . . . . . . 169 5.5.1 Multiplicity and Resolutions for a Given Hilbert Function . . . 169 5.5.2 Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.6 Zero-Dimensional Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.1 Introduction Let S = k[x1, x2, . . ., xn] be the polynomial ring in n variables over a field
k and m = (x1, . . ., xn) the irrelevant maximal ideal. Let I be an ideal of S minimally generated by homogeneous polynomials f1, f2, . . ., ft in S. Then S/I has homogeneous or graded resolution F over S given by
0 → bn
⊕
j=1 S(−dnj ) δn→ · · · →
j=1 S(−di j ) δi→ · · · →
The numbers di j come from the degrees of the homogeneous polynomials in the maps in the resolution. Thus, the numbers d1 j , j = 1, . . ., n are simply the
degrees of the generators f j , j = 1, . . ., n of the ideal I . Much information about S/I can be recovered from the shifts in the resolution. For instance, the height of the ideal I is the smallest positive integer t such that ∑n
i=1 ∑
∑
∑
{
0 1 ≤ t ≤ h − 1 (−1)hh!e(S/I ) t = h.
