Consecutive cancellations in filtered free resolutions
Bull. Korean Math. Soc. 2019 Vol. 56, No. 4, 1077-1097
https://doi.org/10.4134/BKMS.b180907
Published online July 31, 2019
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
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