**Date & Time:** Tuesday, July 19, 2016, 15:30-17:00.

**Venue:** Room No. 216

**Title:** Koszul Algebras

**Speaker: ** Dr. Neeraj Kumar, Indian Statistical Institute, Bangalore

**Abstract:** In this talk, we shall first see some examples of a minimal
graded free resolution of a finitely generated graded module $M$ over a
commutative ring $R$.
Given a field $K$, a positively graded $K$-algebras $R$ with $R_0=K$ is
called "Koszul" if the field $K$ has an $R$-linear free resolution when
viewed as an $R$-module via the identification $K=R/R_{+}$.
We shall review the classical invariant Castelnouvo-Mumford
regularity of a module and define Koszul algebras in terms of
regularity. We shall also discuss several other characterizations
of Koszul algebras. Then I will present some results on Koszul
property of diagonal subalgebras of bigraded algebras; in
particular, Koszul property of diagonal subalgebras of Rees
algebras for a complete intersection ideal generated by
homogeneous forms of equal degrees. At the end, I will present
several problems concerning Koszul algebras.