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

Akihiro Higashitani, Kenta Ueyama

• 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=x21+⋯+x2n∈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 Db(modk2r), and this r is obtained as the nullity of a certain matrix over F2. 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 P1 is less than or equal to (r+12).