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Periodicity of Betti numbers of monomial curves. (English) Zbl 1317.13037

Let \(\underline{a}=(a_1,\ldots,a_n),\;a_1<\cdots <a_n\) be a sequence of positive integers and \(C(\underline{a})\) the affine monomial curve in \(\mathbb{A}^n\) parametrised by \(a_i\mapsto \underline{t}^{a_i}\), \(1\leq i\leq n\). Let \(I(\underline{a})\) be the defining ideal of \(C(\underline{a})\) in the polynomial ring \(S=k[x_1,\ldots,x_n]\), where \(k\) is a field, and \(\underline{a}+j=(a_1+j,\ldots,a_n+j)\), for \(j\) a positive integer. Herzog and Srinivasan conjectured that the Betti numbers of the ideal \(I(\underline{a}+j)\) are eventually periodic in \(j\) with the period \(a_n-a_1\). The author proves this conjecture by using the squarefree divisor simplicial complexes.

MSC:

13D02 Syzygies, resolutions, complexes and commutative rings
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
05E40 Combinatorial aspects of commutative algebra

Software:

Macaulay2
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References:

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