Algebra & Number Theory

The radius of a subcategory of modules

Hailong Dao and Ryo Takahashi

Full-text: Open access

Abstract

We introduce a new invariant for subcategories X of finitely generated modules over a local ring R which we call the radius of X. We show that if R is a complete intersection and X is resolving, then finiteness of the radius forces X to contain only maximal Cohen–Macaulay modules. We also show that the category of maximal Cohen–Macaulay modules has finite radius when R is a Cohen–Macaulay complete local ring with perfect coefficient field. We link the radius to many well-studied notions such as the dimension of the stable category of maximal Cohen–Macaulay modules, finite/countable Cohen–Macaulay representation type and the uniform Auslander condition.

Article information

Source
Algebra Number Theory, Volume 8, Number 1 (2014), 141-172.

Dates
Received: 14 July 2012
Revised: 10 August 2013
Accepted: 14 September 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730137

Digital Object Identifier
doi:10.2140/ant.2014.8.141

Mathematical Reviews number (MathSciNet)
MR3207581

Zentralblatt MATH identifier
1308.13015

Subjects
Primary: 13C60: Module categories
Secondary: 13C14: Cohen-Macaulay modules [See also 13H10] 16G60: Representation type (finite, tame, wild, etc.) 18E30: Derived categories, triangulated categories

Keywords
radius of subcategory resolving subcategory thick subcategory Cohen–Macaulay module complete intersection dimension of triangulated category Cohen–Macaulay representation type

Citation

Dao, Hailong; Takahashi, Ryo. The radius of a subcategory of modules. Algebra Number Theory 8 (2014), no. 1, 141--172. doi:10.2140/ant.2014.8.141. https://projecteuclid.org/euclid.ant/1513730137


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References

  • T. Aihara and R. Takahashi, “Generators and dimensions of derived categories”, preprint, 2011.
  • T. Araya, K.-i. Iima, and R. Takahashi, “On the structure of Cohen–Macaulay modules over hypersurfaces of countable Cohen–Macaulay representation type”, J. Algebra 361 (2012), 213–224.
  • M. Auslander, Anneaux de Gorenstein, et torsion en algèbre commutative, Secrétariat mathématique, Paris, 1967.
  • M. Auslander and M. Bridger, Stable module theory, Memoirs of the American Mathematical Society 94, Amer. Math. Soc., Providence, R.I., 1969.
  • M. Auslander and R.-O. Buchweitz, “The homological theory of maximal Cohen–Macaulay approximations”, pp. 5–37 Mém. Soc. Math. France $($N.S.$)$ 38, 1989.
  • L. L. Avramov and R.-O. Buchweitz, “Support varieties and cohomology over complete intersections”, Invent. Math. 142:2 (2000), 285–318.
  • L. L. Avramov, V. N. Gasharov, and I. V. Peeva, “Complete intersection dimension”, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 67–114.
  • L. L. Avramov, R.-O. Buchweitz, S. B. Iyengar, and C. Miller, “Homology of perfect complexes”, Adv. Math. 223:5 (2010), 1731–1781.
  • P. A. Bergh, “Modules with reducible complexity”, J. Algebra 310:1 (2007), 132–147.
  • R. O. Buchweitz, “Maximal Cohen–Macaulay modules and Tate–cohomology over Gorenstein rings”, preprint, 1986, http://hdl.handle.net/1807/16682.
  • L. W. Christensen, Gorenstein dimensions, Lecture Notes in Mathematics 1747, Springer, Berlin, 2000.
  • L. W. Christensen and H. Holm, “Algebras that satisfy Auslander's condition on vanishing of cohomology”, Math. Z. 265:1 (2010), 21–40.
  • H. Dao and R. Takahashi, “The dimension of a subcategory of modules”, preprint, 2012.
  • H. Dao and R. Takahashi, “Classification of resolving subcategories and grade consistent functions”, 2013. To appear in Int. Math. Res. Not.
  • D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series 119, Cambridge University Press, 1988.
  • C. Huneke and D. A. Jorgensen, “Symmetry in the vanishing of Ext over Gorenstein rings”, Math. Scand. 93:2 (2003), 161–184.
  • S. B. Iyengar, “Stratifying derived categories associated to finite groups and commutative rings, Kyoto RIMS Workshop on Algebraic Triangulated Categories and Related Topics”, Lecture notes, 2009, http://www.math.unl.edu/~siyengar2/Papers/RIMS0709.pdf.
  • D. A. Jorgensen and L. M. Şega, “Nonvanishing cohomology and classes of Gorenstein rings”, Adv. Math. 188:2 (2004), 470–490.
  • H. Minamoto, “A note on dimension of triangulated categories”, Proc. Amer. Math. Soc. 141:12 (2013), 4209–4214.
  • R. Rouquier, “Dimensions of triangulated categories”, J. K-Theory 1:2 (2008), 193–256.
  • L. M. Şega, “Vanishing of cohomology over Gorenstein rings of small codimension”, Proc. Amer. Math. Soc. 131:8 (2003), 2313–2323.
  • G. Stevenson, “Duality for bounded derived categories of complete intersections”, 2013. To appear in Bull. London Math. Soc.
  • G. Stevenson, “Subcategories of singularity categories via tensor actions”, 2013. To appear in Compos. Math.
  • R. Takahashi, “Modules in resolving subcategories which are free on the punctured spectrum”, Pacific J. Math. 241:2 (2009), 347–367.
  • R. Takahashi, “Classifying thick subcategories of the stable category of Cohen–Macaulay modules”, Adv. Math. 225:4 (2010), 2076–2116.
  • R. Takahashi, “Classifying resolving subcategories over a Cohen–Macaulay local ring”, Math. Z. 273:1-2 (2013), 569–587.
  • H.-J. Wang, “On the Fitting ideals in free resolutions”, Michigan Math. J. 41:3 (1994), 587–608.
  • Y. Yoshino, “A functorial approach to modules of G-dimension zero”, Illinois J. Math. 49:2 (2005), 345–367.
  • M. Yoshiwaki, “On self-injective algebras of stable dimension zero”, Nagoya Math. J. 203 (2011), 101–108.