+91-674-249-4082
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 $K$-algebra $R=\bigoplus_{i \in \mathbb{N}} R_i$
with $R_0=K$ is \emph{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
recent progress on the Charney-Davis-Stanley conjecture and on several problems concerning Koszul algebras.
School of Mathematical Sciences
NISER, PO- Bhimpur-Padanpur, Via- Jatni, District- Khurda, Odisha, India, PIN- 752050
Tel: +91-674-249-4081