Abstract
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 2end{array}}right) .
Highlights
Triangulated categories play an increasingly important role in many areas of mathematics, including representation theory, algebraic geometry, algebraic topology, and mathematical physics
We denote by CMZ( Aε) the category of graded maximal Cohen–Macaulay Aε-modules with degree preserving Aε-module homomorphisms
The stable category of graded maximal Cohen–Macaulay modules, denoted by CMZ(Aε), has the same objects as CMZ(Aε) and the morphism space is given by HomCMZ(Aε)(M, N ) = HomCMZ(Aε)(M, N )/P(M, N ) where P(M, N ) consists of degree preserving Aε-module homomorphisms factoring through a graded projective module
Summary
Triangulated categories play an increasingly important role in many areas of mathematics, including representation theory, (commutative and noncommutative) algebraic geometry, algebraic topology, and mathematical physics. We can completely compute CMZ(Aε) by purely combinatorial methods Thanks to this theorem, we obtain the following two consequences. In [9, Conjecture 1.3], it was conjectured that the structure of CMZ(Aε) is determined by the number of irreducible components of the point scheme Eε of Sε that are isomorphic to P1. This is true if n ≤ 6 (see [6, Theorem 6.20]), but it is known to fail for n = 7. The results and discussion in this paper establish a novel connection between representation theory of noncommutative hypersurfaces and combinatorics
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