The upper bound of Frobenius related length functions

In this paper, we study the asymptotic behavior of lengths of Tor modules of homologies of complexes under the iterations of the Frobenius functor in positive characteristic. We first give upper bounds to this type of length functions in lower dimensional cases and then construct a counterexample to the general situation. The motivation of studying such length functions arose initially from an asymptotic length criterion given in [S.P. Dutta, Intersection multiplicity of modules in the positive characteristics, J. Algebra 280 (2004) 394-411] which is a sufficient condition to a special case of nonnegativity of χ∞. We also provide an example to show that this sufficient condition does not hold in general.