Abstract
Let A be an abelian category. The Ziegler spectrum of A is used to introduce the notion of a locally simple object of A, which is analyzed in terms of the transitive relation A ⊏ B on A, defined to hold whenever the object A occurs at least twice as a subquotient of B. It is shown that a nonzero object is locally simple if and only if it is minimal with respect to ⊏. By analogy to the notion of simplicity, the Serre subcategory generated by the locally simple objects is used to define local Krull-Gabriel dimension. It is proved that an object has local Krull-Gabriel dimension if and only if it satisfies the ⊏-descending chain condition and that the Serre subcategory of objects with local Krull-Gabriel dimension corresponds in the Ziegler spectrum to the interior of the set of points E satisfying Prest’s condition (A): The localization A/S(E) is local, where S(E) denotes the Serre annihilator of E in A. We also show that a locally finite length object satisfies Fitting’s Lemma.
Supported by NSF Grant DMS96-26708.
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References
M. Auslander, Coherent functors, in Proceedings of the Conference on Categorical Algebra, La Jolla 1965, (S. Eilenberg, D. K. Harrison, S. MacLane, and H. Röhrl, eds.) Springer, New York (1966), 189–231.
M. Auslander, Large modules over artin algebras, in Algebra, Topology and Category Theory. A collection of papers in honor of Samuel Eilenberg, (A. Heller and M. Tierney, eds.), Academic Press, New York (1976), 1–17.
K. Burke and M. Prest, Ziegler spectra of some domestic string algebras, in preparation, University of Manchester.
W. W. Crawley-Boevey, Modules of finite length over their endomorphism ring, in Representations of Algebras and Related Topics, (H. Tachikawa and Sh. Brenner, eds.), London Mathematical Society Lecture Notes Series 168, Cambridge University Press (1992), 127–184.
P. Freyd, Representations in abelian categories, in Proceedings of the Conference on Categorical Algebra, La Jolla, 1965, (S. Eilenberg, D.K. Harrison, S. MacLane and H. Röhrl, eds.), Springer-Verlag, New York/Berlin (1966).
P. Gabriel, Des Catégories Abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.
K. Goodearl, Von Neumann Regular Rings, Pitman, London (1979).
I. Herzog, Elementary duality of modules, Transactions of the American Mathematical Society 340 (1993), 37–69.
I. Herzog, The Ziegler spectrum of a locally coherent Grothendieck category, Proceedings of the London Mathematical Society (3) 74 (1997), 503–558.
H. Krause, The spectrum of a locally coherent Grothendieck category, Journal of Pure and Applied Algebra 114 (1997), 259–271.
M. Prest, Model Theory and Modules, London Mathematical Society Lecture Notes Series 130, Cambridge University Press (1988).
J. Schröer, On the Krull-Gabriel dimension of an algebra, preprint, Universität Bielefeld.
B. Stenström, Rings of Quotients, Springer, New York (1975).
J. Trlifaj, Two problems of Ziegler and uniform modules over regular rings, in Abelian Groups and Modules, Marcel Dekker (1996), 373–383.
M. Ziegler, Model theory of modules, Annals of Pure and Applied Logic 26 (1984), 149–213.
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Herzog, I. (1999). Locally simple objects. In: Eklof, P.C., Göbel, R. (eds) Abelian Groups and Modules. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7591-2_28
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DOI: https://doi.org/10.1007/978-3-0348-7591-2_28
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7593-6
Online ISBN: 978-3-0348-7591-2
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