IPDF Talk
Speaker: Dr. Kanoy Kumar Das
Host: Ananthnarayan Hariharan
Title: Square-free powers of Cohen--Macaulay simplicial forests
Time, day and date: 4:00:00 PM – 5:00:00 PM, Monday, March 23
Venue: Online (https://meet.google.com/kof-mymh-vim)
Abstract: In this talk, we study square-free powers of facet ideals of simplicial complexes, a notion that lies at the intersection of combinatorial commutative algebra and matching theory. A fundamental contrast motivates our work: while ordinary powers of a graded radical ideal are rarely Cohen--Macaulay (except in the complete intersection case), square-free powers exhibit significantly different behavior.

Given a simplicial complex $\Delta$, let $I(\Delta)^{[k]}$ denote the $k$-th square-free power of its facet ideal in a polynomial ring $R$. In this talk, we show that if $\Delta$ is a Cohen--Macaulay simplicial forest, then the quotient $R/I(\Delta)^{[k]}$ is Cohen--Macaulay for all $k \geq 1$.

Our approach is primarily combinatorial, complemented by tools from homological algebra. For a Cohen--Macaulay simplicial complex $\Delta$, we derive explicit formulas for $\mathrm{depth}(R/I(\Delta)^{[k]})$ and $\dim(R/I(\Delta)^{[k]})$ for all $k \geq 1$. We introduce a new structural notion, called a \emph{special leaf}, which enables us to control the behavior of the depth of square-free powers. As a consequence, we show that the normalized depth function associated to a Cohen--Macaulay simplicial forest is non-increasing.

This talk is based on a recent work with A. Roy and K. Saha.