Let $R$ be a polynomial ring and $I$ a monomial ideal in $R$. One important invariant associated to the ideal $I$ is the projective dimension, ${\rm{projdim}}\; R/I^t$, namely the length a minimal free resolution of $R/I^t$, for $t\in \mathbb{N}$. In general, since $R$ is a polynomial ring, we know that  ${\rm{projdim}}\; R/I^t \le n$, where $n=\dim R$. One can then ask what are effective upper bounds for  ${\rm{projdim}}\; R/I^t$, in the case $I$ is a monomial ideal. 

Equivalently, since  ${\rm{depth}}\; R/I^t=n-{\rm{projdim}}\; R/I^t$, one can ask for effective lower bounds for ${\rm{depth}}\; R/I^t$.

The asymptotic depth for such ideals is well understood by classical results of Burch and Broadmann. We construct certain types of (initially) regular sequences on $R/I$ that give effective bounds on the depth of $R/I$. Moreover, we will  discuss when these sequences remain (initially) regular sequences on $R/I^t$ and give lower bounds on $\depth R/I^t$ for $t\ge 2$. This is joint work with  T\`ai  Huy H\`a and Susan Morey.

Here is a zoom link for the seminar:  

https://unm.zoom.us/j/95467082856

Meeting ID: 954 6708 2856
Passcode: algebra