The Frobenius endomorphism and homological dimensions

In 1969 Kunz [Ku] proved a fundamental result, connecting the regularity of a local ring of positive characteristic with the flatness of its Frobenius endomorphism φ. This was a first indication of the important role that φ would play in homological commutative algebra, especially in reflecting basic homological properties of the ring. Some results in Peskine and Szpiro’s groundbreaking thesis, announced in [PS1] in 1969 and published with full proofs in [PS2] in 1973, carry this idea further. Peskine and Szpiro discovered that base change along φ preserves resolutions of modules of finite projective dimension and then applied this result as a main ingredient in their proof of the Intersection Theorem. Thus they introduced the use of the Frobenius map as a tool in solving homological problems in positive characteristic. This approach has been developed and used to prove some major theorems in homological commutative algebra in the last quarter century. Herzog in 1974, and others more recently, established various converses of Peskine and Szpiro’s theorem. Such results extend another aspect of Kunz’s theorem, namely the fact that the Frobenius endomorphism detects finite homological dimensions. As yet, the full extent of its power to do so is not completely understood. In this article we survey what is known about this property of the Frobenius endomorphism and discuss some questions that remain. Another major theme of this survey is the behavior of numerical functions defined by the Frobenius endomorphism that are closely tied to its homological properties. The study of such functions, once again, originated in Kunz’s Theorem. In this survey we give, whenever possible, proofs or motivations of the results we discuss, we record the most precise conclusions that can be drawn from the proofs, and we recast a few arguments into a shorter or more illuminating form by using more recent techniques, sometimes obtaining a new result.