Cohen-Macaulayness and canonical module of residual intersections

Abstract

We show the Cohen-Macaulayness and describe the canonical module of residual intersections $J=\mathfrak{a}\colon_R I$ in a Cohen-Macaulay local ring $R$, under sliding depth type hypotheses. For this purpose, we construct and study, using a recent article of Hassanzadeh and the second named author, a family of complexes that contains important informations on a residual intersection and its canonical module. We also determine several invariants of residual intersections as the graded canonical module, the Hilbert series, the Castelnuovo-Mumford regularity and the type. Finally, whenever $I$ is strongly Cohen-Macaulay, we show duality results for residual intersections that are closely connected to results by Eisenbud and Ulrich. It establishes some tight relations between the Hilbert series of some symmetric powers of $I/\mathfrak{a}$. We also provide closed formulas for the types and for the Bass numbers of some symmetric powers of $I/\mathfrak{a}.$