Representation Type Of Algebras Research Articles

Tensor product and theories of modules

Modules over a ring R, when tensored with an (R, S)-bimodule, are converted to S-modules. Here I consider, from the standpoint of the model theory of modules, the effect of this operation. The primary motivation arises from questions concerning representation type of algebras and interpretability of modules, where such tensor functors play a key role, but this paper is devoted to more general considerations. For instance, the elementary duality of [2] and [1] is generalised here. It is also shown that, although tensor product does not preserve elementary equivalence, one can define the tensor product of two complete theories of R-modules. The results in Section 1 grew out of a number of discussions with T. Kucera.This is our generic situation. We have an (R, S)-bimodule B and we are considering the functor –⊗RBS from Mod – R (the category of right R-modules) to Mod – S which is given on objects by MR ↦ (M ⊗RB)s and has the obvious action on morphisms. There is a somewhat more general situation: namely we may consider the effect of tensoring (T, R)-bimodules over R with B to obtain (T, S)-bimodules. For the intended applications this case is not needed. Moreover, although some results extend to this more general case, there are some which definitely do not (see the example after 2.1). Therefore we confine ourselves to the case first described. We will also consider B itself as a variable and so ask “What is the effect of tensoring R-modules with (R, S)-bimodules?”.

Homological dimension and representation type of algebras under base field extension

We prove that for finite-dimensional associative algebras the finitistic global dimension in the sense of H. Bass and also the self-injective dimension are preserved under arbitrary extensions of the base field. Further, the global dimension, and as a consequence also finite representation type, are preserved under base field extensions which are separable in the sense of S. MacLane. The results are derived with the aid of ultraproduct-techniques and are applied to the model theory of finite dimensional algebras.

The Representation Type of Algebras and Subalgebras

For A an associative algebra with identity over a field K, [A : K] < ∞, and d an integer, we define g Λ(d) to be the number of inequivalent indecomposable Λ-modules of degree d over K. Following (6), we define Λ to be of finite representation type if . Λis said to be of bounded representation type if there exists d 0 such that g Λ(d) = 0 for d ⩾ d 0; Λ is of unbounded representation type if not of bounded type.