Representation embeddings, interpretation functors and controlled wild algebras

Gregory, Lorna ORCID: and Prest, Mike (2016) Representation embeddings, interpretation functors and controlled wild algebras. Journal of the London Mathematical Society, 94 (3). pp. 747-766. ISSN 0024-6107

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We establish a number of results which say, roughly, that interpretation functors preserve algebraic complexity. First, we show that representation embeddings between categories of modules of finitedimensional algebras induce embeddings of lattices of pp formulas, and hence are non-decreasing on Krull-Gabriel dimension and uniserial dimension. A consequence is that the category of modules of any wild finite-dimensional algebra has width ∞, and hence, if the algebra is countable, there is a superdecomposable pure-injective representation. It is conjectured that a stronger result is true: That a representation embedding from Mod-S to Mod-R admits an inverse interpretation functor from its image, and hence that, in this case, Mod-R interprets Mod-S. This would imply, for instance, that every wild category of modules interprets the (undecidable) word problem for (semi)groups. We show that the conjecture holds for finitely controlled representation embeddings. Finally, we prove that if R, S are finite-dimensional algebras over an algebraically closed field and I : Mod-R → Mod-S is an interpretation functor such that the smallest definable subcategory containing the image of I is the whole of Mod-S, then if R is tame, so is S and, similarly, if R is domestic, then S also is domestic.

Item Type: Article
Additional Information: Publisher Copyright: © 2016 London Mathematical Society.
Uncontrolled Keywords: mathematics(all) ,/dk/atira/pure/subjectarea/asjc/2600
Faculty \ School: Faculty of Science > School of Mathematics
Related URLs:
Depositing User: LivePure Connector
Date Deposited: 17 Oct 2022 12:30
Last Modified: 25 Oct 2022 00:17
DOI: 10.1112/jlms/jdw055

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