Clifford deformations of Koszul Frobenius algebras and noncommutative quadrics

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 Knörrer Periodicity Theorem for quadric hypersurfaces. As an application, we recover Knörrer Periodicity Theorem without using of matrix factorizations.

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References in corpus (1)

• Noncommutative Knörrer's periodicity theorem and noncommutative quadric hypersurfaces

Cited by in corpus (4)

• Noncommutative Knörrer's periodicity theorem and noncommutative quadric hypersurfaces
• Generalized Knörrer's Periodicity Theorem
• Noncommutative conics in Calabi-Yau quantum projective planes
• Pre-resolutions of noncommutative isolated singularities