School of Mathematics Colloquium

Graded components of local cohomology modules supported on $\mathfrak{C}$-monomial ideals.

by Dr. Sudeshna Roy (TIFR, Mumbai)

Thursday, February 2, 2023 from to (Asia/Kolkata)
at AG-69
Abstract: The structure of local cohomology modules is quite mysterious owing to their non-finite generation. Over the last three decades, researchers have extensively investigated if they behave like finitely-generated modules. Let $A$ be a Dedekind domain of characteristic zero such that its localization at every maximal ideal has mixed characteristic with finite residue field. Let $R=A[X_1, \ldots, X_n]$ be a  polynomial ring equipped with the standard multigrading and let $I\subseteq R$ be a $\mathfrak{C}$-monomial ideal. We call an ideal in $R$ a \mathfrak{C}$-monomial ideal if it can be generated by elements of the form $aU$ where $a \in A$ (possibly nonunit) and $U$ is a monomial in $X_i$'s.
Local cohomology modules supported on usual monomial ideals of a polynomial ring over a field gain a great deal of interest due to their connections 
with combinatorics and toric varieties. The objective of this talk is to discuss a structure theorem for the multigraded components of the local 
cohomology modules $H^i_I(R)$ for $i \geq 0$. We will further show that if $A$ is a PID then each component can be written as a direct sum of its 
torsion part and torsion-free part. This result evinces the finiteness of their Bass numbers. This is joint work with Tony J. Puthenpurakal.