February 27, 2016
CONJECTURES ON TAME AND WILD ALGEBRAS
JAN SCHRÖER
Contents
1. Tame algebras
2. Wild algebras
3. Hierarchy of wild algebras
4. Conjectures
References
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Let K be a field, and let A be a finite-dimensional K-algebra.
1. Tame algebras
Let K be algebraically closed. The algebra A is tame or of tame representation
type if A is not representation-finite, and if for each d there exist finitely many AK[X]-bimodules M1 , . . . , Mt , which are free of finite rank as right K[X]-modules,
such that (up to isomorphism) all but finitely many indecomposable d-dimensional
A-modules are isomorphic to a module of the form Mi ⊗K[X] S with S a simple
K[X]-module. In this case, let µ(d) be the minimal number of such bimodules.
(Recall that the simple K[X]-modules are of the form Sλ := K[X]/(X − λ) with
λ ∈ K, and Sλ ∼
= Sµ if and only if λ = µ.)
Assume now that A is tame. We say that A is
• of finite growth (or domestic or of domestic representation type) if
there exists some N ≥ 1 with µ(d) ≤ N for all d.
• of linear growth if there exists some N ≥ 1 such that µ(d) ≤ N d for all d.
• of polynomial growth if there exists some N ≥ 1 such that µ(d) ≤ N d for
all d.
• of exponential growth if for each N ≥ 1 there exists some d ≥ 1 such that
µ(d) > N d .
Conjecture 1.1. For a tame algebra A the following are equivalent:
(i) A is of linear growth.
(ii) A is of polynomial growth.
2. Wild algebras
Most of this section is just a reformulation of [S]. The K-algebra A is
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• wild if there exists a faithful exact K-functor
mod(Khx, yi) → mod(A)
which respects isomorphism classes and indecomposables.
• fully wild a.k.a. strictly wild there exists a fully faithful exact K-linear
functor
mod(Khx, yi) → mod(A).
• Wild if there exists a faithful exact K-linear functor
Mod(Khx, yi) → Mod(A)
which respects isomorphism classes and indecomposables.
• Fully Wild a.k.a. Strictly Wild if there exists a fully faithful exact Klinear functor
Mod(Khx, yi) → Mod(A).
• controlled wild if there exists a faithful exact K-linear functor
F : mod(Khx, yi) → mod(A)
and a class C of modules in mod(A) such that for all M, N ∈ mod(Khx, yi)
we have
HomA (F (M ), F (N )) = F (HomKhx,yi (M, N )) ⊕ C(F (M ), F (N ))
•
•
•
•
where C(F (M ), F (N )) is the subspace of HomA (F (M ), F (N )) consisting of
all homomorphisms factoring through a finite direct sum of modules in C.
endo-wild if for each finite-dimensional K-algebra B there exists some M ∈
mod(A) with EndA (M ) ∼
= B.
Endo-Wild if for each K-algebra B there exists some M ∈ Mod(A) with
EndA (M ) ∼
= B.
Corner endo-wild if for each finite-dimensional K-algebra B there exists
some M ∈ mod(A) and a nilpotent ideal I of EndA (M ) with EndA (M )/I ∼
=
B.
Corner Endo-Wild if for each K-algebra B there exists some M ∈ Mod(A)
and a nilpotent ideal I of EndA (M ) with EndA (M )/I ∼
= B.
One can show that A is wild if and only if there exists an A-KhX, Y i-bimodule
M , which is free of finite rank as a right K[X]-module, such that the functor
M ⊗KhX,Y i − : mod(KhX, Y i) → mod(A)
respects isomorphism classes and indecomposables.
Theorem 2.1 (Drozd). Let K be algebraically closed, and let A be a finite-dimensional
representation infinite K-algebra. Then A is tame or wild, but not both.
All wild path algebras are fully wild.
The m-Kronecker algebra with m ≥ 3 is fully wild and Fully Wild. The 2Kronecker algebra is Fully Wild, but not wild, see Ringel [R].
The algebra A has enough large indecomposable modules if for each infinite
cardinal λ there exists an indecomposable A-module of cardinality ≥ λ.
CONJECTURES ON TAME AND WILD ALGEBRAS
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3. Hierarchy of wild algebras
Most of the following implications are proved in [S].
+3
fully wild
&.
endo-wild
Fully Wild
Endo-Wild
controlled wild
px
Corner endo-wild
wild
&.
Wild
rep. infinite ks
s{
Corner Endo-Wild
ow
enough large indec.
4. Conjectures
Conjecture 4.1. For a finite-dimenional K-algebra A the following are equivalent:
(i) A is wild.
(ii) A is controlled wild.
Conjecture 4.2. For a finite-dimenional K-algebra A the following are equivalent:
(i) A is wild.
(ii) A is Corner endo-wild.
Let M ∈ Mod(A) be an A-module. Then M is also a B-module with B :=
EndA (M ). Let endolength(M ) be the length of M as a B-module. The length of M
as an A-module is denoted by length(M ). Following Crawley-Boevey [CB1, CB2],
M is a generic module if the following hold:
(i) M is indecomposable;
(ii) length(M ) = ∞;
(iii) endolength(M ) < ∞.
The algebra A is generically tame if for each d there are only finitely many (up
to isomorphism) generic A-modules of endolength d. This version of tameness has
the advantage that it does not rely on any assumptions on the ground field K.
The next conjecture has been proved by Crawley-Boevey [CB2] for algebraically
closed fields.
Conjecture 4.3 (Crawley-Boevey [CB1]). The following are equivalent:
(i) A is not wild;
(ii) A is generically tame.
References
[CB1] W. Crawley-Boevey, Modules of finite length over their endomorphism rings, In: Representations of Algebras and Related Topics, London Mathematical Society, Lecture Note, vol.
168 (1992) 127–184.
[CB2] W. Crawley-Boevey, Tame algebras and generic modules, Proc. London Math. Soc., 63
(1991) 241–265.
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JAN SCHRÖER
[R]
[S]
C.M. Ringel, Tame algebras are Wild, Algebra Colloq. 6 (1999) 473–490.
D. Simson, On Corner type endo-wild algebras, J. Pure Appl. Algebra 202 (2005), no. 1-3,
118–132.
Jan Schröer
Mathematisches Institut
Universität Bonn
Endenicher Allee 60
53115 Bonn
Germany
E-mail address: [email protected]