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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
Posted:
November 30, 2007
MathSciNet review:
2361847
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Additional Information
Abstract: Given a commutative coherent ring  , a bijective correspondence between the thick subcategories of perfect complexes  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.
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- M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass, 1969. MR 0242802 (39:4129)
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- P. Gabriel, Des catégories abeliénnes, Bull. Soc. Math. France 90 (1962), 323-448.MR 0232821 (38:1144)
- 3.
- I. Herzog, The Ziegler spectrum of a locally coherent Grothendieck category, Proc. London Math. Soc. 74(3) (1997), 503-558. MR 1434441 (98j:18017)
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- M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43-60. MR 0251026 (40:4257)
- 5.
- M. J. Hopkins, Global methods in homotopy theory, Homotopy theory (Durham, 1985), London Math. Soc. Lecture Note Ser. 117, Cambridge Univ. Press, Cambridge, 1987, pp. 73-96.MR 0932260 (89g:55022)
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- M. Hovey, Classifying subcategories of modules, Trans. Amer. Math. Soc. 353(8) (2001), 3181-3191.MR 1828603 (2002i:13007)
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- H. Krause, The spectrum of a locally coherent category, J. Pure Appl. Algebra 114(3) (1997), 259-271. MR 1426488 (98e:18006)
- 8.
- A. Neeman, The chromatic tower for
 , Topology 31(3) (1992), 519-532.MR 1174255 (93h:18018)
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- M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984), 149-213.MR 0739577 (86c:03034)
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Additional Information
Grigory Garkusha
Affiliation:
Department of Mathematics, Swansea University, SA2 8PP Swansea, United Kingdom
Email:
G.Garkusha@swansea.ac.uk
Mike Prest
Affiliation:
Department of Mathematics, University of Manchester, Oxford Road, M13 9PL Manchester, United Kingdom
Email:
mprest@maths.man.ac.uk
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08844-2
PII:
S 0002-9939(07)08844-2
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
Posted:
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
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