Syzygies of semi-regular sequences

Abstract

A semi-regular sequence in $S=k[x_1, \ldots, x_n]$ is a sequence of polynomials $f_1, \ldots, f_r$ of degrees $d_1, \ldots, d_r$ which satisfy a certain generic condition. Suppose that $I\subset S$ is generated by such a semi-regular sequence and let $\rho$ be the Castelnuovo--Mumford regularity of $S/I$. We show that a minimal free resolution of $S/I$ is isomorphic to the Koszul complex on $f_1, \ldots, f_r$ in degrees $\le\rho-2$. If a common numerical condition is satisfied, then this isomorphism also holds in degree $\rho-1$. Therefore, the Betti diagram of $S/I$ and the Betti diagram of the Koszul complex always agree in rows $\le\rho-2$; we can sometimes determine that they also agree in row $\rho-1$. We also give a partial converse, that if the Betti diagram of $S/I$ agrees with the diagram of the Koszul complex except in possibly the last two rows, then~$I$ can be generated by a (not necessarily minimal) semi-regular sequence.