Let be a field, be a finitely generated associative -algebra and Rep be the functor Fields Sets, which sends a field containing to the set of isomorphism classes of representations of of dimension at most . We study the asymptotic behavior of the essential dimension of this functor, i.e., the function (Rep, as . In particular, we show that the rate of growth of determines the representation type of . That is, is bounded from above if is of finite representation type, grows linearly if is of tame representation type, and grows quadratically if is of wild representation type. Moreover, allows us to construct invariants of algebras which are finer than the representation type.
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Federico Scavia, Essential dimension of representations of algebras. Comment. Math. Helv. 95 (2020), no. 4, pp. 661–702DOI 10.4171/CMH/500