Curved DG Modules and Matrix Factorizations from Noncommutative Quadric Hypersurfaces

Abstract

The category of noncommutative quadratic quadric hypersurfaces, $\mathtt{Quad}\text{-}\mathtt{QHS}$, consists of pairs $(A,f)$, where $A$ is a quadratic algebra and $f\in A$ is a nonzero degree $2$ element. We associate to such $(A,f)$ a pair $({\bar{A}}^{!},f^{!})$, and show that this association makes $\mathtt{Quad}\text{-}\mathtt{QHS}$ into a category with duality. We construct a faithful functor from the category of graded modules over ${\bar{A}}^{!}$ to the homotopy category of curved DG modules over a canonical curved DG algebra $(A\otimes {\bar{A}}^{!},d,f\otimes f^{!})$. If $A$ satisfies the left strong rank condition and $f\in A$ is not a right zero divisor, we show that the restriction of our functor to a natural full subcategory of the category of graded modules over ${\bar{A}}^{!}$ is valued in a stable category of noncommutative matrix factorizations of $f$. When $A$ is Koszul of finite global dimension and $f\in A$ is normal and regular, we prove that the even Clifford algebra, ${\bar{A}}^{!}[(f^{!}{)}^{-1}{]}_0$, is isomorphic to a canonical PBW-deformation of a Zhang twist of the $2$-Veronese subalgebra of the Koszul dual $A^{!}$. Finally, we study several classes of Artin-Schelter regular algebras to illustrate our results.

30 pages, submitted version, comments welcome