Frobenius Powers of Non-complete Intersections

The purpose of this paper is to address a number of issues raised by Avramov and Miller in a recent paper [1]. Let (R,m, k) be a Noetherian local ring of characteristic p > 0 with residue field k, and let φ : R → R be the the Frobenius homomorphism defined by φ(a) = a. For r ≥ 1, we denote by φrR the R-module structure on R via φ. That is, for a ∈ R and b ∈ φ r R, a · b = a r b. When R is a regular ring, φ r R is flat; in fact, this condition characterizes regular rings [4]. When R is a complete intersection, Avramov and Miller [1] proved that Tor∗ (−, φrR) is rigid in the following sense.