Ph.D thesis defense
Speaker: Samarendra Sahoo
Date: 13 October 2025
Day: Monday
Venue: Ramanujam Hall
Time: 4-5 pm
Title: Minimal free resolutions, $I$-stable filtrations and some lower
bounds of Hilbert coefficients.
Abstract: Let $(A, \mathfrak{m})$ be a Cohen-Macaulay local ring and $M$ a
Cohen-Macaulay $A$-module. We study certain lower bounds for the Hilbert
coefficients of $M$ and investigate the conditions under which the
associated graded module $G_{\mathfrak{m}}(M)$ is Cohen--Macaulay. Let
$M_i$ denote the $i$-th syzygy of $M$, and suppose that $(A,
\mathfrak{m})$ is a complete intersection ring. We examine the asymptotic
behavior of $e_1(M_i)$, the first Hilbert coefficient of $M_i$, and
$\operatorname{reg}(G_{\mathfrak{m}}(M_i))$, the Castelnuovo--Mumford
regularity of the associated graded module, for sufficiently large $i$.
Furthermore, for an $\mathfrak{m}$-primary ideal $I$ and an $I$-stable
filtration $F = \{I_n\}_{n \ge 0}$, we analyze the situation when $\dim
A(F)/A(I) = \dim A$, where $A(F)$ denotes the Rees algebra with respect to
the filtration $F$.