Almost complete intersections (Q952561)

Let \(R\) be a polynomial ring in \(n\) variables over a field \(k\) with standard grading and let \(I\subseteq R\) be a homogeneous ideal of codimension \(h\). Consider the graded minimal free \(R\)-resolution of \(R/I\): \[ 0\to \bigoplus R(-j)^{\beta_{p,j}}\to \dots \to \bigoplus R(-j)^{\beta_{1,j}}\to R\to R/I\to 0. \] The multiplicity conjecture of Herzog-Huneke-Srinivasan asserts that if \(R/I\) is a Cohen-Macaulay ring, then the multiplicity of \(R/I\) satisfies the inequalities \[ \frac{\prod_{i=1}^{h}m_i}{h!}\leq e(R/I)\leq \frac{\prod_{i=1}^{h}M_i}{h!}, \] where \[ m_i=\min\{j\mid b_{i,j}\neq 0\}\quad\text{and}\quad M_i=\max\{j\mid b_{i,j}\neq 0\}. \] Herzog, Huneke, and Srinivasan also conjecture that the inequality on the right holds even if \(R/I\) is not Cohen-Macaulay. One motivation for this conjecture is the theorem of \textit{C. Huneke} and \textit{M. Miller} [Can. J. Math. 37, 1149--1162 (1985; Zbl 0579.13012)] which proves the conjecture when the resolution is pure; that is, when \(M_i=m_i\) for all \(i\). After the present paper was written, but before it appeared, Boij and Söderberg [arXiv:0803.1645] proved the monomial conjecture using results of \textit{D. Eisenbud} and \textit{F.-O. Schreyer} [Betti Numbers of Graded Modules and Cohomology of Vector Bundles, \url{arXiv:0712.1843}] and \textit{D. Eisenbud, G. Fløystad}, and \textit{J. Weyman} [The Existence of Pure Free Resolutions, \url{arXiv:0709.1529}]. The present paper focuses on codimension three almost complete intersections. Such ideals are one link from a codimension three Gorenstein ideal; and therefore, it is possible to use the results of \textit{D. A. Buchsbaum} and \textit{D. Eisenbud} [Am. J. Math. 99, 447--485 (1977; Zbl 0373.13006)] in order to record their minimal resolutions, and thereby verify the monomial conjecture for these ideals. The present paper also studies almost complete intersections which are one link from a complete intersection, such ideals are sometimes called Northcott ideals.