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 (\pm 1)-skew polynomial algebra in n variables of degree 1 and f =x_1^2 + \cdots +x_n^2 \in S. We prove that the stable category \mathsf {\underline{CM}}^{\mathbb Z}(S/(f)) of graded maximal Cohen–Macaulay module over S/(f) can be completely computed using the four graphical operations. As a consequence, \mathsf {\underline{CM}}^{\mathbb Z}(S/(f)) is equivalent to the derived category \mathsf {D}^{\mathsf {b}}({\mathsf {mod}}\,k^{2^r}), and this r is obtained as the nullity of a certain matrix over {\mathbb 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 {\mathbb P}^1 is less than or equal to \left( {\begin{array}{c}r+1\\ 2\end{array}}\right).