Abstract
We propose the notion of partial resolution of a ring, which is by definition the endomorphism ring of a certain generator of the given ring. We prove that the singularity category of the partial resolution is a quotient of the singularity category of the given ring. Consequences and examples are given.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Auslander, M., Representation Dimension of Artin Algebras, Queen Mary College Math. Notes, Queen Mary College, London, 1971.
Bondal, A. I. and Kapranov, M. M., Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 1183–1205 (Russian). English transl.: Math. USSR-Izv. 35 (1990), 519–541.
Buchweitz, R. O., Maximal Cohen–Macaulay Modules and Tate Cohomology over Gorenstein Rings, Unpublished Manuscript, Hannover, 1987.
Chen, X. W. and Yang, D., Homotopy categories, Leavitt path algebras and Gorenstein projective modules, Algebr. Represent. Theory 17 (2014), 713–733.
Chen, X. W. and Ye, Y., Retractions and Gorenstein homological properties, to appear in Algebr. Represent. Theory. doi:10.1007/s10468-013-9415-1.
Gabriel, P. and Zisman, M., Calculus of Fractions and Homotopy Theory, Springer, New York, 1967.
Happel, D., Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, London Mathematical Society Lecture Note Ser. 119, Cambridge Univ. Press, Cambridge, 1988.
Happel, D., On Gorenstein algebras, in Representation Theory of Finite Groups and Finite-Dimensional Algebras (Bielefeld, 1991), Progress in Math. 95, pp. 389–404, Birkhäuser, Basel, 1991.
Kalck, M. and Yang, D., Relative singularity categories I: Auslander resolutions, Preprint, 2012. arXiv:1205.1008v2.
Keller, B., Deriving DG categories, Ann. Sci. Éc. Norm. Supér. 27 (1994), 63–102.
Kong, F. and Zhang, P., From CM-finite to CM-free, Preprint, 2012. arXiv:1212.6184.
Krause, H., The Spectrum of a Module Category, Mem. Amer. Math. Soc. 707, 2001.
Miyachi, J. I., Localization of triangulated categories and derived categories, J. Algebra 141 (1991), 463–483.
Neeman, A., The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. Éc. Norm. Supér. 25 (1992), 547–566.
Orlov, D., Triangulated categories of singularities and D-branes in Landau–Ginzburg models, Tr. Mat. Inst. Steklova 246 (2004), 240–262 (Russian). English transl.: Proc. Steklov. Inst. Math. 2004:3(246), 227–248.
Ringel, C. M., The Gorenstein projective modules for the Nakayama algebras, I, J. Algebra 385 (2013), 241–261.
Ringel, C. M. and Zhang, P., Representations of quivers over the algebra of dual numbers, Preprint, 2011. arXiv:1112.1924.
Schofield, A. H., Representation of Rings over Skew Fields, Cambridge Univ. Press, Cambridge, 1985.
Schofield, A. H., Semi-invariants of quivers, J. Lond. Math. Soc. 43 (1991), 385–395.
Stenstrom, B., Rings of Quotients: An Introduction to Methods of Ring Theory, Springer, Berlin, 1975.
Van den Bergh, M., Non-commutative crepant resolutions, in The Legacy of Niels Henrik Abel, pp. 749–770, Springer, Berlin, 2004.
Verdier, J. L., Catégories Dérivées, in Cohomologie étale (SGA \(4\frac{1}{2}\) ), Lecture Notes in Math. 569, pp. 262–311, Springer, Berlin, 1977.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, XW. A note on the singularity category of an endomorphism ring. Ark Mat 53, 237–248 (2015). https://doi.org/10.1007/s11512-014-0200-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11512-014-0200-0