2019/10/23 by Akihiro Higashitani, Higashitani, Akihiro, Kenta Ueyama +1
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1910.10612
openalex publication_date 2019/10/23 · openalex created_date 2019/11/01 · openalex updated_date 2026/07/28
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 =x12 + ⋯ +xn2 ∈ S. We prove that the stable category \mathsf\underlineCM\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\underlineCM\mathbb Z(S/(f)) is equivalent to the derived category Db(mod k2r), and this r is obtained as the nullity of a certain matrix over \mathbb 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 \mathbb P1 is less than or equal to \binomr+12.