Minimal resolutions of some monomial ideals
type: Item
instance of: scholarly article
title: Minimal resolutions of some monomial ideals
zbMATH Open document ID: 0701.13006
DOI: 10.1016/0021-8693(90)90237-I
author: Shalom Eliahou
author: Michel Kervaire
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: Let \(R=k[x_ 1,...,x_ n]\) be a (graded) polynomial ring in n variables over a commutative ring, e.g., a field, k, where \(\deg (x_ i)=1\). A monomial ideal I has a uniquely determined minimal set of monomial generators G(I), which is called the canonical generator system of I. If w is a monomial in R, let max(w) denote the largest index of the variables dividing w. A monomial ideal I is said to be stable if for every monomial \(w\in I\) and \(i<m=\max (w),\) \(x_ iw/x_ m\in I\). Remark that, if I is a stable monomial ideal with canonical generator set G(I), every monomial \(w\in I\) has a unique decomposition, called the canonical decomposition, \(w=u\cdot y\) with \(u\in G(I)\) and max(u)\(\leq \min (y)\) (lemma 1.1). Now define a symbol \(e(i_ 1,...,i_ q;u)\) to be admissible if the following three conditions are satisfied: (1) \(u\in G(I);\) (2) \(i_ 1,...,i_ q\) are integers such that \(1\leq i_ 1<i_ 2<...<i_ q\leq n\); (3) \(i_ q<\max (u).\) Let \(L_ q=L_ q(I)\) be the free R-module on the set of all admissible symbols \(e(i_ 1,...,i_ q;u)\) for fixed \(q\geq 0\). Let \(\alpha:L_ 0\to I\) be the map of R-modules with \(\alpha (e(u))=u\). And, for \(q\geq 1\), define the R-module map \(d:L_ q\to L_{q-1}\) by \[ d(e(\sigma;u))=\sum^{q}_{r=1}(-1)^ rx_{i_ r}e(\sigma_ r;u)\quad -\sum_{r\in A(\sigma;u)}(-1)^ ry_ re(\sigma_ r;u_ r) \] where \(\sigma =(i_ 1,...,i_ q)\), \(\sigma_ r=(i_ 1,...,\hat i_ r,...,i_ q)\), and \(x_{i_ r}u=u_ r\cdot y_ r\), is the canonical decomposition of \(x_{i_ r}u\), with the set \(A(\sigma;u)\subset \{1,...,q\}\) of values of r such that \(\max \{i_ 1,...,\hat i_ r,...,i_ q\}<\max (u_ r).\)- With these notation, the main result of the paper is: Theorem 2.1. Let \(R=k[x_ 1,...,x_ n]\) be the polynomial ring in n variables over the unitary ring k. Let I be a stable monomial ideal in R. Then, \((L_*(I),d)\) as described above is a free graded resolution of I over R. If k is a field, then \((L_*(I),d)\) is the minimal resolution. As applicati…
reviewed by: Jun Nishimura
Mathematics Subject Classification ID: 13D25
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 4148248
zbMATH Keywords: resolution of stable monomial ideal
zbMATH Keywords: canonical generator system
zbMATH Keywords: admissible symbols
zbMATH Keywords: polynomial ring
zbMATH Keywords: Betti numbers
zbMATH Keywords: Poincaré series
MaRDI profile type: Publication
Wikidata QID: Q61632105
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description: scientific article; zbMATH DE number 4148248