Clifford Deformations of Koszul Frobenius Algebras and Noncommutative Quadrics

A Clifford deformation of a Koszul Frobenius algebra [Formula: see text] is a finite dimensional [Formula: see text]-graded algebra [Formula: see text], which corresponds to a noncommutative quadric hypersurface [Formula: see text] for some central regular element [Formula: see text]. It turns out that the bounded derived category [Formula: see text] is equivalent to the stable category of the maximal Cohen-Macaulay modules over [Formula: see text] provided that [Formula: see text] is noetherian. As a consequence, [Formula: see text] is a noncommutative isolated singularity if and only if the corresponding Clifford deformation [Formula: see text] is a semisimple [Formula: see text]-graded algebra. The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to Knörrer's periodicity theorem for quadric hypersurfaces. As an application, we recover Knörrer's periodicity theorem without using matrix factorizations.