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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.
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참고문헌 (15)
참고문헌 신청
A. Capani / CoCoA: a system for doing Computations in Commutative Algebra
J. Elias / 2008 / Structure theorems for certain Gorenstein ideals / Michigan Math. J 57:269~292
M. L. Green / 2010 / Six lectures on commutative algebra, Mod. Birkhauser Class / Birkhauser Verlag :119~186
G.-M. Greuel / 2008 / A Singular Introduction to Commutative Algebra / Springer
M. Kreuzer / 2005 / Computational Commutative Algebra. 2 / Springer-Verlag
J. Elias / 2008 / Structure theorems for certain Gorenstein ideals / Michigan Math. J 57:269~292
M. L. Green / 2010 / Six lectures on commutative algebra, Mod. Birkhauser Class / Birkhauser Verlag :119~186
G.-M. Greuel / 2008 / A Singular Introduction to Commutative Algebra / Springer
M. Kreuzer / 2005 / Computational Commutative Algebra. 2 / Springer-Verlag
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