TY - JOUR

T1 - Representation embeddings, interpretation functors and controlled wild algebras

AU - Gregory, Lorna

AU - Prest, Mike

N1 - Publisher Copyright:
© 2016 London Mathematical Society.

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85014996929&partnerID=8YFLogxK

U2 - 10.1112/jlms/jdw055

DO - 10.1112/jlms/jdw055

M3 - Article

AN - SCOPUS:85014996929

VL - 94

SP - 747

EP - 766

JO - Journal of the London Mathematical Society-Second Series

JF - Journal of the London Mathematical Society-Second Series

SN - 0024-6107

IS - 3

ER -