The Frobenius endomorphism and homological dimensions

Details The Frobenius endomorphism and homological dimensions The Frobenius endomorphism and homological dimensions

2003-01-20

arxiv.org/abs/math/0301208v3

Mathematics

Commutative Algebra

28 pages, uses conm-p-l.sty. To be published in Contemporary Mathematics series volume (Conference Proceedings for Summer 2001

Scientific paper

This is a survey on the relation between homological properties of the Frobenius endomorphism and finiteness of various homological dimensions of the ring or of modules over it, such as global dimension and projective dimension. We begin with Kunz's surprising result in 1969 that the regularity of a Noetherian local ring is equivalent to the flatness of its Frobenius endomorphism, as well as the subsequent generalizations to the module setting by Peskine and Szpiro and continue up through the recent flurry of results in the last five years. An attempt is made to include proofs whenever feasible.

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