Conjectures on Cohen–Macaulay residual intersections and their canonical modules

Let $R$ be a Cohen–Macaulay local or $^*\!$local ring, and let $I$ be a strongly Cohen–Macaulay ideal, or more generally an ideal satisfying sliding depth. Let $J=(\mathfrak{a}:_R I)$ be an $s$-residual intersection of $I$. When $R$ is Gorenstein, write $g=\operatorname{ht}(I)\leq s$. Residual-intersection conjectures. The following assertions should hold: (1) $R/J$ is Cohen–Macaulay; (2) if $R$ is Gorenstein, the canonical module of $R/J$ is the $(s-g+1)$-th symmetric power of $I/\mathfrak{a}$; (3) $\mathfrak{a}$ is minimally generated by $s$ elements; (4) $J$ is unmixed; and (5) when $R$ is positively graded over a field, the Hilbert series of $R/J$ depends only on $I$ and the degrees of the generators of $\mathfrak{a}$. These assertions extend classical results for residual intersections under stronger hypotheses, but the general Cohen–Macaulayness, canonical-module, minimal-generation, unmixedness, and Hilbert-series claims were presented as long-standing conjectures.