FINITE VERSUS INFINITE DIMENSIONAL REPRESENTATIONS – A NEW DEFINITION OF TAMENESS

HENNING KRAUSE

The connection between finite and infinite dimensional representations of a finite dimensional algebra Λ will be the main theme of my talk. First I shall explain the concept of a definable subcategory of the category Mod Λ of all Λ-modules. A classical result says that a definable subcategory X of Mod Λ is determined by its intersection X ∩Ind Λ with the set Ind Λ of isomorphism classes of indecomposable pure-injective Λ-modules. We present another description, using only the category mod Λ of finite dimensional Λ-modules. Here, we denote for any class X of Λ-modules by [X] the class of maps in mod Λ which factor through a module in X. Furthermore, viewing mod Λ as a ring with several objects, we say that an ideal ℑ in mod Λ is fp-idempotent if the finitely presented functors mod Λ → Ab vanishing on ℑ are closed under forming extensions.

Theorem. The map X ↦ [X] defines a bijection between the set of definable subcategories of Mod Λ and the set of fp-idempotent ideals in mod Λ.

In the second part, I shall apply this result in order to give a new characterization of tameness. Recall that an algebra Λ has tame representation type if there exists for every n ∈ ℕ a finite number of continuous 1-parameter families which parametrize all n-dimensional indecomposable Λ-modules. Let us denote by ind_{n} Λ the isomorphism classes of n-dimensional indecomposables. We need also the Ziegler topology on Ind Λ; a subset U is Ziegler-closed if U = X ∩ Ind Λ for some definable subcategory X of Mod Λ.

Theorem. The following conditions are equivalent:

(1) the algebra Λ is of tame representation type;

(2) for every n ∈ ℕ the Ziegler closure of ind_{n} Λ contains only finitely many modules which do not belong to ind_{n} Λ;

(3) for every n ∈ ℕ there are only finitely many fp-idempotent and nilpotent ideals in mod Λ which are contained in the ideal [ind_{n} Λ].

Note that the characterization (3) involves only finite dimensional modules, although the original definition of tameness is based on infinite dimensional modules. Using this example, I shall demonstrate that the Ziegler spectrum Ind Λ and the fp-idempotent ideals in mod Λ provide an efficient link between infinite and finite dimensional representations.

References

[1] W.W. Crawley-Boevey, Tame algebras and generic modules, Proc. London Math. Soc. 63 (1991) 241–264.

[2] W.W. Crawley-Boevey, Infinite dimensional modules in the representation theory of finite dimensional algebras, preprint (1996).

[3] I. Herzog, The Ziegler spectrum of a locally coherent Grothendieck category, Proc. London Math. Soc. 74 (1997) 503–558.

[4] H. Krause, Generic modules over artin algebras, Proc. London Math. Soc. 76 (1998) 276–306.

[5] H. Krause, The spectrum of a module category, Habilitationsschrift, Universität Bielefeld (1998).

Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

  E-mail address: henning@mathematik.uni-bielefeld.de