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Erschienen in:
2017 | OriginalPaper | Buchkapitel
Betti Sequences over Standard Graded Commutative Algebras with Two Relations
verfasst von : Luchezar L. Avramov, Zheng Yang
Erschienen in: Homological and Computational Methods in Commutative Algebra
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Abstract
It is well known that if Q is a standard graded polynomial ring in d variables over a field, and R is a quotient of Q with \(\mathop{\mathrm{proj\,dim}}\nolimits _{Q}R \leq 1\), then the Betti sequence of every R-module N is constant after d steps. We prove that if \(\mathop{\mathrm{proj\,dim}}\nolimits _{Q}R = 2\), then the Betti sequence of N is determined explicitly by some initial segment of known length, which can be read off a presentation of N. There are three cases, depending on the degrees of the defining relations of R and on whether these have a common divisor.