Frobenius powers of complete intersections

Abstract

Let R be a commutative noetherian local ring of characteristic p > 0, and let φ : R → R be the Frobenius endomorphism, φ(a) = a. Each iteration φ defines on R a new structure of R-module, denoted φ r R, for which a · b = aprb. In 1969 Kunz [7, (3.3)] discovered that if R is regular, then φ r R is flat for all r ≥ 0 and, conversely, that if R is reduced and φrR is flat for some r ≥ 1, then R is regular. Regularity is equivalent to the finiteness of the projective dimension of the R-module k = R/m, where m is the maximal ideal of R, so Kunz’s theorem connects the homological properties of k and those of φ. To summarize further results along these lines, we let c(R) denote the least integer s such that (y : m) * m for some maximal R-regular sequence y (such an s exists by the Artin-Rees Lemma). For a finitely generated R-module M the following conditions are equivalent. (i) M has finite projective dimension. (ii) Torn (M, φrR) = 0 for all n, r ≥ 1. (iii) Torn (M, φrR) = 0 for all n ≥ 1 and infinitely many r. (iv) Torn (M, φrR) = 0 for j ≤ n ≤ j + depthR+ 1 where j, r are fixed integers satisfying j ≥ 1 and r > logp(c(R)). The implication (i) =⇒ (ii) is a fundamental theorem of Peskine and Szpiro [9, (1.7)]. An early converse, (iii) =⇒ (i), was given by Herzog [4, (3.1)]. Recently, Koh and Lee [6, (2.6)] proved (iv) =⇒ (i) (but stated a weaker result). The local ring R is a complete intersection if in some (equivalently, in each) Cohen presentation of its m-adic completion as a homomorphic image of a regular local ring, the defining ideal is generated by a regular sequence. When R has this property and the length `R(M) is finite, a sharpening of Herzog’s theorem is proved (but is not stated explicitly) in [8, (2.4)]: If n ≥ 1 and M has infinite projective dimension, then 0 < limr→∞ ( `R ( Torn (M, φrR) ) /p dimR ) <∞. Our main result links, qualitatively and quantitatively, the homology of Frobenius powers of a complete intersection and the homology of the residue field.