0-Borel fixed ideals
Let \(k\) be a field and \(S = k[x_1,\ldots,x_r]\) the polynomial ring in \(r\) variables. A monomial ideal \(I\) in \(S\) is \textit{\(0\)-Borel fixed } if for any minimal monomial generator \(a\) of \(I\), if \(x_i\) divides \(a\), then \(x_j(a/x_i) \in I\) for all \(1 \leq j < i.\) By a result of Bayer, Galligo and Stillman, the generic initial ideal of any homogeneous ideal in \(S\) is \(0\)-Borel fixed when char(\(k) = 0.\) The author shows that, for any \(0\)-Borel fixed ideal \(I\), the minimal free resolution of \(S/I\) has a commutative associative differential graded algebra structure. This is a useful tool for investigating the properties of the resolution. As an immediate consequence, for any \(0\)-Borel fixed ideal \(I\) contained in the square of \((x_1,\ldots,x_r)\), \(S/I\) is a Golod ring [\textit{E. Golod}, Sov. Math., Dokl. 3, 745-749 (1962); translation from Dokl. Akad. Nauk SSSR 144, 479-482 (1962; Zbl 0122.04201)]. Consequently, one gets information about the Betti numbers of any finitely generated module over \(S/I.\) In particular, for char(\(k) = 0\), given any homogeneous ideal \(Q\), the graded Betti numbers of \(k\) over \(S/Q\) are not greater than the graded Betti numbers of \(k\) over \(S/L\), where \(L\) denotes the lexicographic segment ideal with the same Hilbert function.
scientific article; zbMATH DE number 941275