Minimal resolutions of some monomial ideals
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 applications of this theorem, when k is a field, one gets a formula for the Betti numbers \(\beta_ q(I)\) and the Poincaré series \(P(I,t)\) of I: \[ \beta_ q(I)=\dim_ k Tor_ q^ R(I,k) = \text{rank}(L_ q(I))=\sum_{u\in G(I)} {{\max(u)-1}\choose q}. \] \[ P(I,t)=\sum^{\infty}_{q=0}(-1)^ q\sum_{u\in G(I)} {{\max(u)- 1}\choose q} \frac{t^{\deg (u)}}{(1-t)^ n}=\sum_{u\in G(I)} \frac{t^{\deg (u)}}{(1-t)^{n-\max (u)+1}}. \]
scientific article; zbMATH DE number 4148248