Hongmiao Yu: Square-free Gröbner Degenerations and N-fiber-full up to h Modules

Abstract.

In this talk, we introduce the notion of “N-fiber-full up to h modules” in order to investigate some extensions of the results obtained by Conca and Varbaro on square-free Gröbner degenerations. More concretely, given that \(R = K[X_1,\ldots,X_n]\) is the polynomial ring over a field \(K\) and \(I\) is a homogeneous ideal of \(R\), we aim to address the question: if the initial ideal \(in(I)\) is not square-free, under which assumptions does the equality in Conca and Varbaro’s theorem, namely \(\dim_K H_m^i (R/I)_j = \dim_K H_m^i (R/in(I))_j\), still hold for some \(i\)? Additionally, we will discuss when Gröbner degenerations preserve the componentwise linearity.