Coherent functors and asymptotic properties

In 1998, Hartshorne proved that half exact coherent functors over a discrete valuation ring ?? are direct sums of the identity functor, Hom-functors of quotient modules of ?? and tensor products of quotient modules of ??. In our first article (Paper A), we obtain a similar characterization for half exact coherent functors over a much wider class of rings: Dedekind domains. This fact allows us to classify half exact coherent functors over Dedekind domains.

In our second article (Paper B), coherent functors over noetherian rings are considered. We study asymptotic properties of sets of prime ideals connected with coherent functors applied to artinian modules or finitely generated modules. Also considering quotient modules M /anM, where an is the nthpower of an ideal ??, one obtains that the Betti and Bass numbers of the images under a coherent functor of the quotient modules above are polynomials in n for large n. Furthermore, the lengths of these image modules are polynomial in ??, for large ??, under the condition that the image modules have finite length.