Mixed Hilbert coefficients of homogeneous \(d\)-sequences and quadratic sequences
Let \(R\) be a Noetherian ring, \({\mathfrak m}\) a maximal ideal of height \(n\), \(K=R/{\mathfrak m}\), and \(I\subset{\mathfrak m}\) an ideal. The Rees algebra \(R[It]\) has a natural \(\mathbb{N}^2\)-filtration \({\mathcal F}\) by the ideals \({\mathfrak m}^rI^s\), and the homogeneous component of bidegree \((r,s)\) of the corresponding associated graded ring \(\text{gr}_{\mathcal F}R[It]\) is \(Q_{rs}={\mathfrak m}^rI^s/{\mathfrak m}^{r+1}I^s\). Therefore \(h_{rs}=\dim_KQ_{rs}\) is a polynomial in \(r\) and \(s\) for \(r,s\gg 0\), \[ h_{rs}=\sum_{i+j\leq n-1}a_{ij}({\mathfrak m}\mid I){r+i\choose i}{s+j\choose j}. \] The coefficients of the degree \(n-1\) part of this polynomial are the mixed multiplicities of the pair \(({\mathfrak m},I)\). This paper is devoted to the computation of the mixed multiplicities and the bigraded Hilbert series \(\sum h_{ij}\lambda^i_1\lambda^j_2\) for certain cases of ideals \(I\) (and natural choices of \({\mathfrak m})\) that arise from so-called quadratic homogeneous sequences: homogeneous \(d\)-sequences (in particular regular sequences), straightening closed ideals in algebras with straightening law, maximal minors of a generic matrix of indeterminates (a special case of the latter), defining ideals of projective monomial space curves on the quadric \(xy-zw=0\), and Huckaba-Huneke ideals of analytic deviation 1 and 2. In view of the formidable complexity of the problem one can of course not expect explicit formulas in all cases. For example, in the case of maximal minors, the formulas involve certain natural invariants related to the poset formed by the minors.
scientific article; zbMATH DE number 1076272
scientific article