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Noncommutative Kn\"orrer's periodicity theorem and noncommutative quadric hypersurfaces

Research paper by Izuru Mori, Kenta Ueyama

Indexed on: 29 Dec '19Published on: 29 May '19Published in: arXiv - Mathematics - Rings and Algebras



Abstract

Noncommutative hypersurfaces, in particular, noncommutative quadric hypersurfaces are major objects of study in noncommutative algebraic geometry. In the commutative case, Kn\"orrer's periodicity theorem is a powerful tool to study Cohen-Macaulay representation theory for hypersurfaces, so, in this paper, we show a noncommutative version of graded Kn\"orrer periodicity theorem. Moreover, under high rank property defined in this paper, we show that computing the stable category of graded maximal Cohen-Macaulay modules over a noncommutative smooth quadric hypersurface up to 6 variables can be reduced to one or two variables cases. We also give a complete classification of the stable category of graded maximal Cohen-Macaulay modules over a smooth quadric hypersurface in a skew $\mathbb P^{n-1}$ up to 6 variables without high rank property using graphical methods.