Intersection multiplicities over Gorenstein rings

Abstract

Let R be a complete local ring of dimension d over a perfect field of prime characteristic p, and let M be an R-module of finite length and finite projective dimension. S. Dutta showed that the equality \(\lim_{n\to\infty}\frac{\ell(F^n_R(M))}{p^{nd}} = \ell(M)\) holds when the ring R is a complete intersection or a Gorenstein ring of dimension at most 3. We construct a module over a Gorenstein ring R of dimension five for which this equality fails to hold. This then provides an example of a nonzero Todd class \(\tau_3(R)\), and of a bounded free complex whose local Chern character does not vanish on this class.