Intersection multiplicity of modules in the positive characteristics

We introduce an asymptotic length criteria on Tor0 and Tor1 of a pair of modules, via the Frobenius map, to study positivity and non-negativity of Serre-intersection multiplicity over local complete intersections and Gorenstein rings in the positive characteristics, when both modules have finite projective dimension. We use this criteria to derive a sufficient condition for (a) non-negativity over complete intersections when one of the modules has dimension 2, and(b) non-negativity over Gorenstein rings of dimension 5. We also show that the problem of intersection multiplicity on local rings (complete intersection, Gorenstein, Cohen–Macaulay) in characteristic 0 can be reduced to local rings (complete intersection, Gorenstein, Cohen–Macaulay respectively) in the positive characteristics.