Covariant Functors and Asymptotic Stability

TL;DR

This paper investigates the asymptotic behavior of associated primes and depth of modules under covariant functors, establishing stability results for coherent functors and exploring cases beyond coherence.

Contribution

It proves asymptotic stability of associated primes and depths for coherent functors and extends the analysis to certain non-coherent functors like local cohomology.

Findings

Associated primes stabilize for large n under coherent functors.

Stability of J-depths is established for large n.

Non-coherent functors can also exhibit asymptotic stability in some cases.

Abstract

Let R be a commutative Noetherian ring, I and J ideals of R and M a finitely generated R-module. Let F be a covariant R-linear functor from the category of finitely generated R-modules to itself. We first show that if F is coherent, then the sets of associated primes of F(M/I^n M) and F(I^n-1 M / I^n M) and the J-depths of F(M/I^n M) and F(I^n-1 M / I^n M) become independent of n for large n. Next, we consider several examples in which F is a rather familiar functor, but is not coherent or not even finitely generated in general. In these cases, the set of associated primes of F(M/I^n M) still becomes independent of n for large n. We then show one negative result where F is not finitely generated. Finally, we give a positive result where F belongs to a special class of functors which are not finitely generated in general, an example of which is the zeroth local cohomology functor.