Stanley filtrations and strongly stable ideals (Q2473593)

The author gives a new proof of the fact that the Castelnuovo-Mumford regularity of a strongly stable ideal is the highest degree of a minimal monomial generator. More precisely, the author constructs a Stanley filtration for strongly stable ideals which provides a bound for the Castelnuovo-Mumford regularity. Let \(S = k[x_0,x_1,\ldots ,x_n]\) be the polynomial ring in \(n+1\) variables over a field \(k\), and let \(I\) be a strongly stable ideal of \(S\). Let \(\prec\) be the pure lexicographic order on \(S\) with \(x_0 \succ x_1 \succ \ldots \succ x_n\). For a monomial \(x^a \in S\) and a subset \(\sigma \subset \{0,1, \ldots ,n\}\), we put \((a, \sigma) = \{ x^a x^v \mid \;\text{supp}(v) \subset \sigma \}\) where \(\text{supp}(v) = \{ i \mid \;v_i > 0 \}\). Let \(u\) be the highest degree of a minimal monomial generator for \(I\). For a monomial \(x^a \in S \setminus I\) of degree at most \(u-1\), the set \(g(a)\) is defined as follows. If \(\deg(x^a) = u-1\) then \(g(a) = (a, \{ i, i+1, \ldots , n \})\) where \(i\) is the least integer such that \(i \geq \max \{ i \mid \;a_i > 0 \}\) and \(x^a x_i \in S \setminus I\), provided that such an integer exists. If either \(x^a x_n \in I\) or \(\deg(x^a) < u-1\) then \(g(a) = (a, \emptyset)\). Let \(\mathfrak {G}^{u-1} = \{ g(a) \mid \;x^a \in S \setminus I, \;\deg(x^a) \leq u-1 \}\). The main result of the paper under review shows for a strongly stable monomial ideal \(I\) of \(S\) the following assertions hold: (1) The set \(\mathfrak {G}^{u-1}\) is a Stanley decomposition for \(S/I\). (2) Consider the order where \(g(a)\) precedes \(g(b)\) if \(\deg(x^a) < \deg(x^b)\) or \(\deg(x^a) = \deg(x^b)\) and \(x^a \prec x^b\). The Stanley decomposition \(\mathfrak {G}^{u-1}\) with this order is a Stanley filtration for \(S/I\). (3) As a conclusion, the Castelnuovo-Mumford regularity of \(I\) is equal to \(u\).