С Herrmann et al

tic , only. Denoting by F^ the algebraic closure of the prime field F^, we get C^n^p/^' for each nontrivial ultrafilter В on any infinite set of prime numbers by Steinitz's theorem. Therefore

Corollary 4.9. For algebraically closed fields the numbers k^{K) only depend on the characteristic p (0 or prime) of K. Thus к^{К) = к^{р). Further, up to the possible exception of finitely many prime numbers (depending on d) к^{0) = к^(р), where k^ stands for n^, t^, g^ or a^, respectively

Remark 4.10 The ultraproduct proof above gives no indication on the size of the set of exceptional prime numbers p. However, as is shown m §6, there is an explicit first order characterization of the numbers k^ in question. Therefore, the methods of Brown [5] allow to bound the exceptional set

5 . The Number of Algebras of Finite Representation Type

For every field К and integer d let J^j(X), Em^iK), Graph^(X) denote the class of all d-dimensional X-algebras of finite representation type, the set of phism classes of i^(K), the set of isomorphism classes of Auslander-Reiten graphs r{R) of algebras R of ^di^)^ respectively.

Theorem 5.1. Graphj(X) is finite for every infinite field К and any dimension d. Moreover, if K'^=Y\KJS^ is an ultraproduct of infinite fields, there is a cal bijection

cp : ПОгарЬ,(Х,)Э->ОгарЬ,(Х*), IFJ^-^Urj^.

Proof Graphd(K) is always finite as a consequence of Corollary 3.4. Since a X*-algebra R is just an ultraproduct R^YlRJB of K„-algebras R^, the proof of the last assertion is an easy consequence of Corollary 3.2 and Theorem 2 2.

The same proof will also work m the case of Fin^(iC).

Theorem 5.2. // K'^^flKJS' is an ultraproduct of infinite fields, there is a canonical bijection

ф : nFin,(K.)/^-Fin,(X*), IRJb^YiRJ^'

Corollary 5.3. // К is an algebraically closed field, either Fin^{K) is finite or |Fin^(K)| = |K|. // FindiK) is finite, then iFind(X)| = |Fin^(L)|/or any cally closed field of the same characteristic. Further \Е1щ{€)\=п is finite if and only if there is a finite set i of exceptional prime numbers, such that |Fin^(K)| = nfor any algebraically closed field of characteristic рфе.

Proof For the first assertion we may argue as in the proof of Proposition 4.1, replacing Proposition 1.4 by Theorem 5.2. The rest of the proof is tic transfer via ultraproducts (cf §4).

Remark 5.4. As was pointed out to the authors by CM. Ringel, the first assertion of Corollary 5.3 (a similar argument applies to Proposition 4.1) may