Afshin Goodarzi: On sequential Cohen-Macaulayness

In this talk, I introduce a bivariate polynomial, BW-polynomial, associated with any homogenous ideal J in a polynomial ring R. The BW-polynomial is a finer invariant than the Hilbert series, in the sense that the Hilbert series can be computed from the BW-polynomial. It can be shown that the BW-polynomial is an algebraic counterpart to the combinatorially defined h-triangle of simplicial complexes introduced by Björner and Wachs.

It will be shown that for a homogenous ideal J the quotient ring R/J is sequentially Cohen-Macaulay (SCM) if and only if R/J has a stable BW-polynomial under passing to the reverse lexicographic generic initial ideal. If S/J is SCM, then the Hilbert series of the local cohomology modules (supported on the maximal graded ideal of R) can be computed from the BW-polynomial. As a consequence, in the SCM case, the extremal Betti numbers (and in particular, the depth and the regularity) can be read from the BW-polynomial. Some numerical results concerning BW-polynomials of SCM algebras and Betti diagram of componentwise linear ideals will be also discussed.