Complexes with homology of finite length and Frobenius functors

Abstract

In this paper R is a Noetherian local ring of prime characteristic p, m its maximal ideal, and k its residue field. Let f: R + R(x H xP) be the Frobenius homomorphism. For an R-module M the R-module M,,, is A4 as an additive group with the scalar multiplication given by rOm=f”(r)m=rP”m for rER, REM. F; denotes the nth Frobenius functor, i.e., F”,(M) = MOR R,,, for each R-module M, where the R-module structure of F”,(M) is given by Y(M @ s) = m @ (rs) for r, s E R, m E M. For basic properties of these functors see [2, 5, and 91. Remember that the functors (n)~ (m) resp. F;“Ro Fz and (n + m) resp. Fi+“’ are isomorphic. The length of an R-module M is denoted by &(M), its Krull dimension by dim M or dim, M, and its projective dimension by pd, M. The ith homology of a complex M. will be written as Hi(M.); the support supp M. of M. is the set of all p E Spec R such that M. OR R, is not exact. A complex M. is called bounded if and only if Mi # 0 for only a finite number of iE Z. For a bounded complex M. with homology modules of finite length the Euler characteristic x(M.) = CieZ (l)‘&(Hi(M.)) is defined. The following theorem based on arguments of Dutta [2] and Monsky [8] is the key to our results: