Commutative Algebra Seminar

Date

Tuesday, 2 April, 2024, 4-5 pm

Venue

Room 215

Host

Tony J. Puthenpurakal

speaker

Om Prakash

Affiliation

IIT Bombay

Title

Numerical Semigroups and associated Semigroup Rings-I

Abstract

In this series of two lectures, we will study numerical semigroups and their associated semigroup rings. Initially, we will define numerical semigroups, state their fundamental properties, and introduce relevant invariants. Subsequently, we aim to prove the following fundamental results: (i) The Frobenius number of a numerical semigroup S equals the degree, viewed as a rational function, of the Hilbert series of the numerical semigroup ring k[S]. (ii) The Cohen-Macaulay type of the numerical semigroup ring $k[S]$ corresponds to the number of pseudo-Frobenius elements of $S$.  Consequently, we derive a well-known result concerning Gorenstein numerical semigroup rings (credited to Kunz) asserting that k[S] is Gorenstein if and only if S is symmetric.