Classifying Serre subcategories of finitely presented modules
Authors:
Grigory Garkusha and Mike Prest
Journal:
Proc. Amer. Math. Soc. 136 (2008), 761-770
MSC (2000):
Primary 13C05, 13C11, 18E30, 18G35
DOI:
https://doi.org/10.1090/S0002-9939-07-08844-2
Published electronically:
November 30, 2007
MathSciNet review:
2361847
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Given a commutative coherent ring $R$, a bijective correspondence between the thick subcategories of perfect complexes $\mathcal D_{\operatorname {per}}(R)$ and the Serre subcategories of finitely presented modules is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of isomorphism classes of indecomposable injective modules are used in an essential way.
- M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
- Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821
- Ivo Herzog, The Ziegler spectrum of a locally coherent Grothendieck category, Proc. London Math. Soc. (3) 74 (1997), no. 3, 503–558. MR 1434441, DOI https://doi.org/10.1112/S002461159700018X
- M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43–60. MR 251026, DOI https://doi.org/10.1090/S0002-9947-1969-0251026-X
- Michael J. Hopkins, Global methods in homotopy theory, Homotopy theory (Durham, 1985) London Math. Soc. Lecture Note Ser., vol. 117, Cambridge Univ. Press, Cambridge, 1987, pp. 73–96. MR 932260
- Mark Hovey, Classifying subcategories of modules, Trans. Amer. Math. Soc. 353 (2001), no. 8, 3181–3191. MR 1828603, DOI https://doi.org/10.1090/S0002-9947-01-02747-7
- Henning Krause, The spectrum of a locally coherent category, J. Pure Appl. Algebra 114 (1997), no. 3, 259–271. MR 1426488, DOI https://doi.org/10.1016/S0022-4049%2895%2900172-7
- Amnon Neeman, The chromatic tower for $D(R)$, Topology 31 (1992), no. 3, 519–532. With an appendix by Marcel Bökstedt. MR 1174255, DOI https://doi.org/10.1016/0040-9383%2892%2990047-L
- M. Prest, The Zariski spectrum of the category of finitely presented modules, preprint (available at maths.man.ac.uk/~mprest).
- Jan Erik Roos, Locally Noetherian categories and generalized strictly linearly compact rings. Applications, Category Theory, Homology Theory and their Applications, II (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Two), Springer, Berlin, 1969, pp. 197–277. MR 0407092
- Bo Stenström, Rings of quotients, Springer-Verlag, New York-Heidelberg, 1975. Die Grundlehren der Mathematischen Wissenschaften, Band 217; An introduction to methods of ring theory. MR 0389953
- R. W. Thomason, The classification of triangulated subcategories, Compositio Math. 105 (1997), no. 1, 1–27. MR 1436741, DOI https://doi.org/10.1023/A%3A1017932514274
- Martin Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984), no. 2, 149–213. MR 739577, DOI https://doi.org/10.1016/0168-0072%2884%2990014-9
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13C05, 13C11, 18E30, 18G35
Retrieve articles in all journals with MSC (2000): 13C05, 13C11, 18E30, 18G35
Additional Information
Grigory Garkusha
Affiliation:
Department of Mathematics, Swansea University, SA2 8PP Swansea, United Kingdom
MR Author ID:
660286
ORCID:
0000-0001-9836-0714
Email:
G.Garkusha@swansea.ac.uk
Mike Prest
Affiliation:
Department of Mathematics, University of Manchester, Oxford Road, M13 9PL Manchester, United Kingdom
MR Author ID:
141975
Email:
mprest@maths.man.ac.uk
Keywords:
Thick subcategories,
perfect complexes,
Ziegler and Zariski spectra
Received by editor(s):
May 23, 2006
Received by editor(s) in revised form:
July 5, 2006
Published electronically:
November 30, 2007
Additional Notes:
This paper was written during the visit of the first author to the University of Manchester, which was supported by the MODNET Research Training Network in Model Theory. He would like to thank the University for its kind hospitality.
Communicated by:
Paul Goerss
Article copyright:
© Copyright 2007
American Mathematical Society
