On the structure of Ext groups of strongly stable ideals
Let \(k\) be a field and \(A = k[X_1, \dots, X_n]\) a polynomial ring. A monomial ideal \(I \subset A\) is said to be strongly stable if, for any monomial \(u \in I\), \(X_j u /X_i \in I\) whenever \(X_i|u\) and \(j < i\). For example, a lexicographic ideal is strongly stable. In the present paper, the author computes the Ext module \(\text{Ext}_A^i (A/I, A)\) for a strongly stable ideal \(I\). He applies it to a certain upper bound theorem. That is, if a polynomial \(P(z)\) is given, then the length of the cohomology module \(H^i(\mathbb P_k^{n-1}, \widetilde I(z))\) is bounded where \(I\) runs through homogeneous ideals such that the Hilbert polynomial of \(A/I\) is \(P(z)\). Furthermore, the length reaches the maximum when \(I\) is a lexicographic ideal.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00042].
scientific article; zbMATH DE number 1604765