Existence of generalized Frobenius decomposition limits

Establish whether the limit defining the asymptotic multiplicity of a finitely generated module as a direct summand of iterated Frobenius pushforwards exists in full generality, namely whether, for finitely generated modules M and N over an F-finite local ring, the limit of the normalized maximum integer n such that N^{\oplus n} is a direct summand of F_*^eM exists without assuming finite Frobenius representation type.

Background

The paper recalls a broader class of asymptotic invariants of the form involving the maximal number of copies of a module N occurring as a direct summand of an iterated Frobenius pushforward F_*^eM. Existence of these limits was known under the finite Frobenius representation type hypothesis and in certain special cases, but the authors explicitly identify the unrestricted case as unresolved.

The paper’s main results establish existence for surjective F-signatures, which are defined using surjections rather than direct-summand decompositions. Thus, the unresolved question concerns the more general direct-summand limit and is not resolved by the results presented here.