Consecutive cancellations in Tor modules over local rings

TL;DR

This paper extends a theorem by Rossi and Sharifan, showing that the bigraded Hilbert series of Tor modules over local rings can be derived from associated graded modules through negative consecutive cancellations.

Contribution

It generalizes the understanding of Hilbert series of Tor modules over local rings by linking them to associated graded modules via consecutive cancellations.

Findings

Hilbert series of Tor modules can be obtained from associated graded modules.

Negative consecutive cancellations describe the relationship between these Hilbert series.

Extension of Rossi and Sharifan's theorem to a broader context.

Abstract

Let $M, N$ be finite modules over a Noetherian local ring $R$, and let $G$ be the associated graded ring of $R$. We show that the bigraded Hilbert series of $gr(Tor^R(M,N))$ is obtained from that of $Tor^G(gr(M),gr(N))$ by negative consecutive cancellations, thus extending a theorem of Rossi and Sharifan.