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.
Leila Sharifan
Institute for research in Fundamental Sciences (IPM)
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).
2017; 54(1): 99-124
2016; 53(6): 1707-1714
2015; 52(3): 977-986
2012; 49(4): 715-736
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