Quasi-Gorenstein extended Rees algebras associated with filtrations
Naoki Endo
TL;DR
The paper addresses when the quasi-Gorenstein property deforms from the associated graded ring to the extended Rees algebra for Hilbert filtrations over Noetherian local rings. It develops a deformation criterion based on the Cohen–Macaulayness of the Matlis dual of a top local cohomology module and a precise length equality between $H^{d-1}_{\mathfrak m}(R)$ and $H^{d-1}_{\mathfrak M}(G)$, under suitable depth and finiteness hypotheses. The main result shows that, when $G(\mathcal{F})$ is quasi-Gorenstein and $\mathrm{depth}\,\mathcal{R}'(\mathcal{F})\ge d$ with finite $H^{d-1}_{\mathfrak M}(G(\mathcal{F}))$, the extended Rees algebra $\mathcal{R}'(\mathcal{F})$ is quasi-Gorenstein if and only if the stated length equality holds; this yields concrete, filtration-based criteria and special cases in small dimensions. The work extends prior deformation results by linking canonical modules, local duality, and filtration blow-ups, and provides practical criteria for dimension $3$ and $4$ with FLC, broadening the understanding of quasi-Gorenstein properties in blow-up algebras.
Abstract
This paper investigates the quasi-Gorenstein property of extended Rees algebras associated with the Hilbert filtrations on a Noetherian local ring. We provide necessary and sufficient conditions for the deformation of the quasi-Gorenstein property, characterized by the Cohen-Macaulayness of the Matlis dual of local cohomology modules. As a consequence, we offer a characterization of the quasi-Gorenstein property of extended Rees algebras in terms of conditions on the length of local cohomology.