Clifford deformations of Koszul Frobenius algebras and noncommutative quadrics

TL;DR

This paper explores Clifford deformations of Koszul Frobenius algebras, revealing their connection to noncommutative quadrics, derived categories, and singularities, and provides a new perspective on Kn"{o}rrer Periodicity.

Contribution

It introduces a novel link between Clifford deformations and noncommutative quadrics, and relates derived categories to Cohen-Macaulay modules, offering a new proof of Kn"{o}rrer Periodicity.

Findings

Derived category equivalence with Cohen-Macaulay modules

Characterization of noncommutative isolated singularities

Recovery of Kn"{o}rrer Periodicity without matrix factorizations

Abstract

Let $E$ be a Koszul Frobenius algebra. A Clifford deformation of $E$ is a finite dimensional $\mathbb Z_2$-graded algebra $E(\theta)$, which corresponds to a noncommutative quadric hypersurface $E^!/(z)$, for some central regular element $z\in E^!_2$. It turns out that the bounded derived category $D^b(\text{gr}_{\mathbb Z_2}E(\theta))$ is equivalent to the stable category of the maximal Cohen-Macaulay modules over $E^!/(z)$ provided that $E^!$ is noetherian. As a consequence, $E^!/(z)$ is a noncommutative isolated singularity if and only if the corresponding Clifford deformation $E(\theta)$ is a semisimple $\mathbb Z_2$-graded algebra. The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to the Kn\"{o}rrer Periodicity Theorem for quadric hypersurfaces. As an application, we recover Kn\"{o}rrer Periodicity Theorem without using of matrix factorizations.