Abelian Groups and Modules pp 341-351 | Cite as
Locally simple objects
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.
Keywords
Simple Object Endomorphism Ring Abelian Category London Mathematical Society Lecture Note Injective EnvelopePreview
Unable to display preview. Download preview PDF.
References
- [1]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.CrossRefGoogle Scholar
- [2]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.Google Scholar
- [3]K. Burke and M. Prest, Ziegler spectra of some domestic string algebras, in preparation, University of Manchester.Google Scholar
- [4]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.Google Scholar
- [5]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).Google Scholar
- [6]P. Gabriel, Des Catégories Abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.MathSciNetMATHGoogle Scholar
- [7]K. Goodearl, Von Neumann Regular Rings, Pitman, London (1979).MATHGoogle Scholar
- [8]I. Herzog, Elementary duality of modules, Transactions of the American Mathematical Society 340 (1993), 37–69.MathSciNetMATHCrossRefGoogle Scholar
- [9]I. Herzog, The Ziegler spectrum of a locally coherent Grothendieck category, Proceedings of the London Mathematical Society (3) 74 (1997), 503–558.MATHCrossRefGoogle Scholar
- [10]H. Krause, The spectrum of a locally coherent Grothendieck category, Journal of Pure and Applied Algebra 114 (1997), 259–271.MathSciNetMATHCrossRefGoogle Scholar
- [11]M. Prest, Model Theory and Modules, London Mathematical Society Lecture Notes Series 130, Cambridge University Press (1988).Google Scholar
- [12]J. Schröer, On the Krull-Gabriel dimension of an algebra, preprint, Universität Bielefeld.Google Scholar
- [13]B. Stenström, Rings of Quotients, Springer, New York (1975).MATHCrossRefGoogle Scholar
- [14]J. Trlifaj, Two problems of Ziegler and uniform modules over regular rings, in Abelian Groups and Modules, Marcel Dekker (1996), 373–383.Google Scholar
- [15]M. Ziegler, Model theory of modules, Annals of Pure and Applied Logic 26 (1984), 149–213.MathSciNetMATHCrossRefGoogle Scholar