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
 Description 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.  Material: