Complexes with homology of finite length and Frobenius functors (Q908974)

Let R be a Noetherian local ring of prime characteristic p with residue field k. For any R-module M by \(M_{(n)}\) is denoted the R-module on M with \(r.m=r^{p^ n}m\), \(r\in R\), \(m\in M\); \(F^ n_ R\) denotes the n-th Frobenius functor \(F^ n_ R(M)=M\otimes_ RR_{(n)}\) with \(r(m\otimes s)=m\otimes rs\), \(r,s\in R\), \(m\in M.\) Main results: 1. Assume that \(R_{(1)}\) is a finite R-module and \(a=[k:k^ p]\). Let \({\mathcal C}\) be a family of finite R-modules with dimension \(\geq j\) and such that for any exact sequence of R-modules \(0\to M'\to M\to M''\to 0\) we have \(M\in {\mathcal C}\Leftrightarrow M',M''\in {\mathcal C}\). Then \(M\in {\mathcal C}\) implies \(M_{(n)}\in {\mathcal C}\) and for any integer-valued function g which is defined on \({\mathcal C}\) and is additive on exact sequences one has \(a^{-n}g(M_{(n)})=b_ 0+b_ 1p^ j+...+b_ jp^{jn}\) for some rational \(b_ 0,...,b_ j.\) 2. If M, N are finite R-modules, \(pd_ RM<\infty\) and \(\ell_ R(M\otimes_ RN)<\infty\) then \(\ell_ R(Tor_ i(F^ n_ R(M),N))<\infty\) and \(\sum^{\dim (R)}_{i=0}(-1)^ i\ell_ R(Tor_ i(F^ n_ M(M),N))=b_ 0+b_ 1p^ n+...+b_{\nu}p^{\nu n}\) for some rational \(b_ 0,...,b_{\nu}\), \(\nu =\dim (N)\).