Tame algebras and degenerations of modules

Grzegorz Zwara

Let A be a finite dimensional associative algebra over algebraically closed field k.

This talk will be divided in two parts.

1) Tame algebras and generic modules.

Definition 1 We say that A is of tame representation type if for all d ≥ 1 there are a finite number of A-k[X]-bimodules M_{1}, …, M_{n} which are free of rank d as right k[X]-modules, and such that all but finitely many isomorphism classes of indecomposable A-modules of dimension d are of the form M_{i} ⊗_{k[X]} k[X]/(X − λ) for some 1 ≤ i ≤ k and λ ∈ k.

By Drozd’s Tame and Wild Theorem it is known that the class of finite dimensional associative k-algebras may be didided in two disjoint classes. One class consists of tame algebras and the second class is formed by the wild algebras whose representation theory is as complicated as the study of finite dimensional vector spaces together with two non-commuting endomorphisms, for which the classification is a well-known unsolved problem. Hence, we can realistically hope to classify modules only for tame algebras.

Definition 2 If G is an A-module, then we call its length as an End_{A}(G)-module the endolength of G. We say that that G is a generic module if it is indecomposable, of infinite length over A, but finite endolength.

Theorem 3 ([2]) The following are equivalent:

1. A has tame representation type;

2. For each d ≥ 1 there are only finitely many isomorphism classes of generic left Amodules of endolength d.

Moreover, generic modules are related to one-parameter families of indecomposable modules as follows.

Theorem 4 ([2]) Assume that A is tame. Then for any generic module G, End_{A}(G)/radEnd_{A}(G) is a rational function field in one variable over k and we can choose an A-R_{G}-bimodule M_{G}, such that the following hold.

1. R_{G} is a finitely generated localization of k[X]. As a right R_{G}-module, M_{G} is free of rank equal to the endolength of G. If K_{G} is the quotient field of R_{G}, then M_{G} ⊗_{R_{G}} K_{G} ≃ G.

2. The functor M_{G} ⊗_{R_{G}} − from R_{G}-modules to A-modules preserves isomorphism classes, indecomposability and Auslander-Reiten sequences.

3. For each d ≥ 1, all but finitely many isomorphism classes of d-dimensional A-modules arise as M_{G} ⊗_{R_{G}} R_{G}/(r) for some G and some 0 ≠ r ∈ R_{G}.

By condition (2) almost all indecomposable d-dimensional A-modules lie in homogeneous tubes (see [CB1]). For any such tube T = (T_{⟩})_{⟩≥∞} we define a direct limit T of modules T_{i}, i ≥ 1. Then the following holds.

Theorem 5 ([3]) Module T is Σ-pure-injective, indecomposable, but not generic. The set of generic modules which belong to the Ziegler closure of T is non-empty and finite.

These modules are direct summands of infinite product of T. We shall present a more general construction of generic modules (in fact endofinite modules) given in the proof of the following theorem due to Ringel.

Theorem 6 ([5]) Let X be an infinite dimensional A-module, let φ be a locally nilpotent endomorphism of X with finite dimensional kernel. Let I be some infinite set. Then the product X^{I} is the direct sum of a non-zero module P of finite endolength and of copies of X. The endolength of P is bounded by the minimum of the dimension of the vector spaces Kerφ^{t}/Kerφ^{t−1}, with t ≥ 1.

2) Degenerations of finite dimensional modules.

If a_{1} = 1, …, a_{α} is a basis of A over k, we have the structure constants a_{ijk} defined by a_{i}a_{j} = ∑ a_{ijk}a_{k}. The affine variety mod_{A}^{d}(k) of A-module structures on k^{d} consists of α-tuples m = (m_{1}, …, m_{α}) of d × d-matrices with coefficients in k such that m_{1} is the identity matrix and m_{i}m_{j} = ∑ a_{ijk}m_{k} holds for all indices i and j. Any such α-tuple m corresponds to a d-dimensional A-module M in the obvious way. The general linear group Gl_{d}(k) acts on mod_{A}^{d}(k) by conjugation, and the orbits correspond to the isomorphism classes of d-dimensional A-modules. By abuse of notation N is a degeneration of M if n belongs to the Zariski closure \overline{Gl_{d}(k)(m)} of Gl_{d}(k)(m) in mod_{A}^{d}(k), and we denote this fact by M ≤_{deg} N.

Theorem 7 ([4]) Let M, N, Z be finite dimensional A-modules such that there is an exact sequence of one of the forms

0 → N → M ⊕ Z → Z → 0  or  0 → Z → Z ⊕ M → N → 0.

Then M ≤_{deg} N.

Let m, n be two points in mod_{A}^{d}(k) such that M ≤_{deg} N. Then there is a non-singular curve C, a point c_{0} in C and a morphism μ : C → \overline{G↕_{⌈}(∥)⇕} such that μ(c_{0}) = n and μ(c) ∈ Gl_{d}(k)m for all c ≠ c_{0}. Let R be the local ring of the point c_{0} in C with the maximal ideal I. Then R is a discrete valuation k-algebra and μ corresponds to A-R-bimodule Y which is as R-module free of rank d. We shall show the following proposition:

Proposition 8 ([6]) Assume that there is a monomorphism φ : M ⊗_{k} R → Y of A-R-bimodules, for some d-dimensional A-module M. Then, for a sufficiently large h, there is an A-module isomorphism

Y/Y I^{h+1} ≃ Y/Y I^{h} ⊕ M.

We apply the above two facts in the proof of the following characterization of degenerations:

Theorem 9 ([6]) Let m, n be points in mod_{A}^{d}(k), d ≥ 1. Then the following conditions are equivalent:

1. M ≤_{deg} N.

2. There is a short exact sequence 0 → N → M ⊕ Z → Z → 0 for some finite dimensional A-module Z.

3. There is a short exact sequence 0 → Z → Z ⊕ M → N → 0 for some finite dimensional A-module Z.

References

[1] W. Crawley-Boevey, ”On tame algebras and bocses”, Proc. London Math. Soc. 56 (1988), 451–483 [2] W. Crawley-Boevey, ”Tame algebras and generic modules”, Proc. London Math. Soc. 63 (1991), 241–

265

[3] H. Krause, ”Generic modules over artin algebras”, (Preprint 35pp)

[4] C. Riedtmann, ”Degenerations for representations of quivers with relations”, Ann. Sci. École Normale Sup. 4 (1986), 275–301

[5] C.M. Ringel, ”A construction of endofinite modules”, in: Proc. Algebra and model theory, (to appear).

[6] G. Zwara, ”Degenerations of finite dimensional modules are given by extensions”, (Preprint 13pp)

Grzegorz Zwara, Faculty of Mathematics and Informatics, Nicholas Copernicus University, Chopina 12/18, 87–100 Toruń (Poland) E–mail address: gzwara@mat.uni.torun.pl