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Minimal free resolution of the associated graded ring of monomial curves of generalized arithmetic sequences. (English) Zbl 1167.13001

Let \(A=K[[\tau^{a_0},\ldots,\tau^{a_n}]]=K[[x_0,\ldots,x_n]]/I\) be the coordinate ring of a monomial curve \(\mathcal C\subset\mathbb A^n\). Suppose the semigroup \(S=\langle a_0,\ldots,a_n\rangle\) is minimally generated by \(a_0<\cdots<a_n\) and let \(\underline{m}\) be the maximal ideal of \(A\). Let \(G=\text{gr}_{\underline{m}}(A)= \text{gr}_{\underline{m}}(S)=\bigoplus_{i\geq0}(\underline{m}^i/\underline{m}^{i+1})\) be the associated graded ring of \(A\). It is known that \(G\simeq P/I^{*}\), where \(P=K[x_0,\ldots,x_n]\) and \(I^{*}=\langle in_{\underline{m}}(I)\rangle\). Both \(A\) and \(G\) have been studied a lot before, but in general their good algebraic properties cannot be carried from one to another (see Section 1 for a nice exposition of the results).
The article studies the numerical invariants of \(G\) (see Theorem 4.1) in the case when \(a_0,\ldots,a_n\) is a generalized arithmetic sequence (i.e., \(a_i=ha_0+id,1\leq i\leq n\) for some integers \(h,d\) with \(\gcd(a_0,d)=1\)). In the same setting, Corollary 4.11 compares the properties of \(A\) and \(G:I\) and \(I^{*}\) have the same minimum number of generators, \(A\) and \(G\) have the same Cohen-Macaulay type or \(A\) is a complete intersection iff \(G\) is a complete intersection.
Theorem 4.1 (the main result) is proven by induction on \(h\geq 1\). For the base case (\(h=1\), and so \(a_0,\ldots,a_n\) is an “arithmetic sequence”) Eagon-Northcott resolution and mapping cone are used (Theorem 4.4 and Proposition 4.6). When \(h>1\), the results from Section 3 are very useful (Corollary 3.5): a generalized arithmetic sequence can be obtained in a “natural” way from an arithmetic sequence such that the corresponding associated algebras have the same properties (e.g. they have the same Hilbert function; see Theorem 3.4).
Besides the relevance of its results, the article is a beautiful detailed sample of two very powerful techniques in commutative algebra: Eagon-Northcott complex and mapping cone.

MSC:

13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13D02 Syzygies, resolutions, complexes and commutative rings
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series

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References:

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