Abstract
Let Λ be a finite dimensional algebra over some algebraically closed field k. In this note I discuss the relationship between finite and infinite dimensional modules over Λ. This discussion is based on the following three fundamental concepts:
-
fp-idempotent ideals in the category mod Λ of finite dimensional Λ-modules
-
endofinite modules in the category Mod Λ of all Λ-modules
-
coherent functors Mod Γ→ Mod Λ between two module categories
This article is dedicated to Professor Herbert Kupisch on the occasion of his 70th birthday.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
W.W. Crawley-Boevey, On tame algebras and bocses, Proc. London Math. Soc. 56 (1988), 451–483.
W.W. Crawley-Boevey, Tame algebras and generic modules, Proc. London Math. Soc. 63 (1991), 241–264.
W.W. Crawley-Boevey, Modules of finite length over their endomorphism ring, in: Representations of algebras and related topics, eds. S. Brenner and H. Tachikawa, London Math. Soc. Lec. Note Series 168 (1992), 127–184.
W.W. Crawley-Boevey, Infinite dimensional modules in the representation theory of finite dimensional algebras, preprint (1996).
Yu.A. Drozd, Tame and wild matrix problems, Representations and quadratic forms, Institute of Mathematics, Academy of Sciences, Ukrainian SSR, Kiev (1979), 39–74, Amer. Math. Soc. Transl. 128 (1986), 31–55.
I. Herzog, The Ziegler spectrum of a locally coherent Grothendieck category, Proc. London Math. Soc. 74 (1997), 503–558.
C.U. Jensen and H. Lenzing, Model theoretic algebra, Gordon and Breach, New York (1989).
H. Krause, The spectrum of a locally coherent category, J. Pure Appl. Algebra 114 (1997), 259–271.
H. Krause, Generic modules over artin algebras, Proc. London Math. Soc. 76 (1998), 276–306.
H. Krause, Exactly definable categories, J. Algebra 201 (1998), 456–492.
H. Krause, Functors on locally finitely presented categories, Colloq. Math. 75 (1998), 105–131.
H. Krause, The spectrum of a module category, Habilitationsschrift, Universität Bielefeld (1998), Mem. Amer. Math. Soc., to appear.
M. Prest, Model theory and modules, London Math. Soc. Lec. Note Series 130 (1988).
C.M. Ringel, Recent advances in the representation theory of finite dimensional algebras, in: Representations of finite groups and finite dimensional algebras, Birkhauser-Verlag, Basel (1991), 141–192.
M. Ziegler, Model theory of modules, Ann. of Pure and Appl. Logic 26 (1984), 149–213.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Basel AG
About this paper
Cite this paper
Krause, H. (2000). Finite Versus Infinite Dimensional Representations — A New Definition of Tameness. In: Krause, H., Ringel, C.M. (eds) Infinite Length Modules. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8426-6_20
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8426-6_20
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9562-0
Online ISBN: 978-3-0348-8426-6
eBook Packages: Springer Book Archive