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

Higashitani, Akihiro ^[1] ; Ueyama, Kenta ^[2]
1. [1] Osaka University
  
  Osaka University
  
  Kita Ku, Japón
2. [2] Hirosaki University
  
  Hirosaki University
  
  Japón
Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 73, Fasc. 1, 2022, págs. 43-54
Idioma: inglés
DOI: 10.1007/s13348-020-00306-1
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Resumen
- 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).