Spectra of tensor triangulated categories over category algebras

详细信息    查看全文

• 作者：Fei Xu
• 关键词：Primary 18E30 ; Secondary 20C20 ; 16S35 ; 16E35 ; Triangular spectrum ; Tensor triangulated category ; Category algebra
• 刊名：Archiv der Mathematik
• 出版年：2014
• 出版时间：September 2014
• 年：2014
• 卷：103
• 期：3
• 页码：235-253
• 全文大小：328 KB
• 参考文献：1. Auslander M., Reiten I.: Cohen-Macaulay and Gorenstein Artin algebras, in: Representation Theory of Finite Groups and Finite-dimensional Algebras. Progr. in Math. 95, 221-45 (1991)
  2. Balmer P.: The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math. 588, 149-68 (2005) CrossRef
  3. Balmer P.: Spectra, spectra, spectra-tensor triangular spectra versus Zariski spectra of endomorphism rings. Algebraic and Geometric Topology 10, 1521-563 (2010) CrossRef
  4. Benson D., Carlson J., Rickard J.: Thick subcategories of the stable module categories. Fund. Math. 153, 59-0 (1997)
  5. Benson D., Iyengar S., Krause H.: Local cohomology and support for triangulated categories. Ann. Sci. éc. Norm. Supér. 41, 573-19 (2008)
  6. Benson D., Iyengar S., Krause H.: Stratifying modular representations of finite groups. Ann. Math. 174, 1643-684 (2011) CrossRef
  7. R.-O. Buchweitz, Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings, unpublished notes (1986), 155 pages, available online.
  8. Happel D.: On Gorenstein algebras, in: Representation Theory of Finite Groups and Finite-dimensional Algebras. Progr. in Math. 95, 389-04 (1991)
  9. Krause H.: The stable derived category of a Noetherian scheme, Compositio Math 141, 1128-162 (2005) CrossRef
  10. Liu Y., Sierra S.: Recovering quivers from derived quiver representations. Comm. Algebra 41, 3013-031 (2013) CrossRef
  11. A. Neeman, Triangulated Categories, Ann. Math. Studies, Princeton University Press (2001).
  12. P. J. Webb, An introduction to the representations and cohomology of categories, in: Group Representation Theory, EPFL Press 2007, pp. 149-73.
  13. Xu F.: Hochschild and ordinary cohomology rings of small categories. Adv. Math. 219, 1872-893 (2008) CrossRef
  14. Xu F.: Tensor structure on \({k\mathcal{C}}\) -mod. Proc. Edinb. Math. Soc. 56, 349-70 (2013) CrossRef
  15. Xu F.: On local categories of finite groups. Math. Zeit. 272, 1023-036 (2012) CrossRef
  16. Xu F.: Support varieties for transporter category algebras. J. Pure Appl. Alg., 218, 583-01 (2014) CrossRef
  
• 作者单位：Fei Xu (1)
  
  1. Department of Mathematics, Shantou University, 243 University Road, Shantou, 515063, Guangdong, China
  
• ISSN：1420-8938

文摘

Let \({\mathcal{C}}\) be a finite EI category and k be a field. We consider the category algebra \({k\mathcal{C}}\) . Suppose \({\sf{K}(\mathcal{C})=\sf{D}^b(k \mathcal{C}-\sf{mod})}\) is the bounded derived category of finitely generated left modules. This is a tensor triangulated category, and we compute its spectrum in the sense of Balmer. When \({\mathcal{C}=G \propto \mathcal{P}}\) is a finite transporter category, the category algebra becomes Gorenstein, so we can define the stable module category \({\underline{\sf{CM}} k(G \propto \mathcal{P})}\) , of maximal Cohen–Macaulay modules, as a quotient category of \({{\sf{K}}(G \propto \mathcal{P})}\) . Since \({\underline{\sf{CM}} k(G\propto\mathcal{P})}\) is also tensor triangulated, we compute its spectrum as well. These spectra are used to classify tensor ideal thick subcategories of the corresponding tensor triangulated categories.