Frontiers in Mathematical Sciences
7th Conference University of Isfahan  January 13, 2020 
Title:
Vanishing of (CO)Homology over Local Rings
Speaker:
Arash Sadeghi, IPM
Date, Time, and Venue: Thursday, January 2  09:4510:30  Hall 2
Abstract:
In this talk, we will discuss about the vanishing of (co)homology over commutative
Noetherian local rings. A remarkable consequence of the vanishing of homology
is the depth formula, $\text{depth}_{R}(M)+\text{depth}_{R}(N) = \text{depth}(R)+\text{depth}_{R}(M\otimes_{R}N),$ established
by Auslander when $R$ is regular. In the first part of this talk, we will discuss about
the depth formula over Gorenstein rings. In the second part, we will talk about Auslander–Reiten Conjecture. This is one of the most celebrated conjectures in the representation
theory of algebras. We will present various criteria for freeness of modules over local
rings in terms of vanishing of cohomology, which recover a lot of known results on the
Auslander–Reiten Conjecture.
