Combinatorial study of stable categories of graded Cohen–Macaulay modules over skew quadric hypersurfaces.

In this paper, we present a new connection between representation theory of noncommutative hypersurfaces and combinatorics. Let S be a graded ( ± 1 )-skew polynomial algebra in n variables of degree 1 and f = x 1 2 + ⋯ + x n 2 ∈ S . We prove that the stable category CM ̲ Z (S / (f)) of graded maximal Cohen–Macaulay module over S/(f) can be completely computed using the four graphical operations. As a consequence, CM ̲ Z (S / (f)) is equivalent to the derived category D b (mod k 2 r) , and this r is obtained as the nullity of a certain matrix over F 2 . Using the properties of Stanley–Reisner ideals, we also show that the number of irreducible components of the point scheme of S that are isomorphic to P 1 is less than or equal to r + 1 2 . [ABSTRACT FROM AUTHOR]