Consecutive cancellations in filtered free resolutions
Bull. Korean Math. Soc.
Published online July 9, 2019
Leila Sharifan
Department of mathematics, Hakim Sabzevari university
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 [RSh, Theorem 3.1] 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 σ, Betti numbers of I are
obtained from those of LT_σ(I) by so-called zero σ-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, σ-Gr¨obner filtration.
MSC numbers : Primary 13H05, Secondary 13D02.
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