Syzygies of semi-regular sequences
Let \(S = k[x_1,\dots,x_n]\) be the polynomial ring in \(n\) variables over a field \(k\) and let \(f_1,\dots,f_r\) be a sequence of homogeneous polynomials of degrees \(d_1,\dots,d_r\), respectively. If \(r > n\) then this cannot be a regular sequence; a \textit{semi-regular sequence} is the next best thing: we require only that for each \(i\) (\(2 \leq i \leq r\)) and \(a\), the induced homomorphism \( \times f_i : [S/(f_1,\dots,f_{i-1})]_{a-d_i} \rightarrow [S/(f_1,\dots,f_{i-1})]_a \) be either injective or surjective. (To be regular would mean that this map is injective for all \(i\) and \(a\).) Notice that the cokernel of this map is \([S/(f_1,\dots,f_i)]_a\). Fröberg conjectured that ``most'' sequences of fixed degrees \(d_1,\dots,d_r\) are semi-regular. In particular, this means that the Hilbert functions for the ideals generated by ``most'' such sequences are well-understood. Some cases of this conjecture are known, but in general it remains open. This paper considers the more difficult question of determining the graded Betti numbers of an ideal generated by a semi-regular sequence. The main results show, in the language of Macaulay 2, that for an ideal generated by a semi-regular sequence, the Betti diagram looks like a Koszul diagram, except for the last two rows (or, if a certain numerical condition holds, then the exception is only the last row). In some cases they also provide a converse. The first quoted result was also shown by the reviewer and \textit{R. M. Miró-Roig} [J.\ Pure Appl.\ Algebra 182, No. 1, 79--107 (2003; Zbl 1041.13011)], but with an entirely different proof.
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