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 author, a family of complexes that contains important information 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 $I/{\mathfrak {a}}.$