Applications of DG Homological Algebra in Commutative Algebra

Absract:

Homological algebra comes from algebraic topology and provides valuable techniques in studying commutative ring theory. Auslander-Buchsbaum and Serre’s solution to the localization conjecture of regular local rings firmly established homological algebra as a powerful tool to solve problems in commutative algebra. Solving this conjecture by homological methods was a revolution that opened a new horizon to commutative algebraists and encouraged them to develop more homological techniques to study the properties of commutative rings. Another tool from algebraic topology (specifically, rational homotopy theory) that has sparked interest among commutative algebraists is differential graded (DG) homological algebra. The use of techniques from DG homological algebra was established by Avramov, Buchsbaum, Eisenbud, Foxby, Halperin, Kustin, Miller, and Weyman in commutative algebra, for instance, via DG algebra structures on Koszul complexes and free resolutions. It has been shown recently that these techniques can be applied to solve non-trivial problems in commutative algebra.

In this talk, we will discuss the following major problems:

Auslander-Reiten Conjecture (1975). If M is a finitely generated module over a local ring $R$ with ExtiR(M, M ⊕ R) = 0 for all i >> 0, then pdR(M) < ∞.

Vasconcelos’ Conjecture (1974). There are only finitely many semidualizing modules, up to isomorphism, over a local ring.

The Auslander-Reiten Conjecture originates from representation theory of Artin algebras. In a part of this talk, which is based on joint works with Luchezar Avramov, Srikanth Iyengar, and Sean Sather-Wagstaff, we present results about vanishing of homology over trivial extensions of DG algebras and use them to introduce new classes of commutative local rings that satisfy this conjecture. We then sketch a possible approach that involves techniques from differential equations which may result in a solution to this conjecture in full generality; this is based on an in-progress joint work with Maiko Ono and Yuji Yoshino. Vasconcelos’ Conjecture, is about an important class of finitely generated modules that are called semidualizing modules. Examples of these modules include the dualizing modules in the sense of Grothendieck. These modules were introduced by Foxby and rediscovered independently by several authors for different applications. Another part of the talk, which is based on a joint work with Sean Sather-Wagstaff, is devoted to sketch a complete solution to Vasconcelos’ Conjecture using geometric aspects of representation theory for DG algebras.

Thursday, January 9, 2020 - 12:00

427 Thackeray Hall

Speaker Information
Saeed Nasseh
Georgia Southern University

Research Area