Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2019; 56(4): 1077-1097

Online first article July 9, 2019      Printed July 31, 2019

https://doi.org/10.4134/BKMS.b180907

Copyright © The Korean Mathematical Society.

Consecutive cancellations in filtered free resolutions

Leila Sharifan

Institute for research in Fundamental Sciences (IPM)

Abstract

Let $M$ be a finitely generated module over a regular local ring $(R,\n)$. We will fix an $\n$-stable filtration for $M$ and show that the minimal free resolution of $M$ can be obtained from any filtered free resolution of $M$ by zero and negative consecutive cancellations. This result is analogous to \cite[Theorem 3.1]{RSh} in the more general context of filtered free resolutions. Taking advantage of this generality, we will study resolutions obtained by the mapping cone technique and find a sufficient condition for the minimality of such resolutions. Next, we give another application in the graded setting. We show that for a monomial order $\sigma$, Betti numbers of $I$ are obtained from those of $\LT_\sigma(I)$ by so-called zero $\sigma$-consecutive cancellations. This provides a stronger version of the well-known cancellation ``cancellation principle" between the resolution of a graded ideal and that of its leading term ideal, in terms of filtrations defined by monomial orders.

Keywords: minimal free resolution, filtered module, associated graded module, filtered free resolution, consecutive cancellation, mapping cone, leading term ideal, $\sigma$-Gr\"{o}bner filtration

MSC numbers: Primary 13H05; Secondary 13D02

Supported by: This research was in part supported by a grant from IPM (No. 95130058).