The Frobenius Endomorphism and Multiplicities.

Abstract

The dissertation utilizes the Frobenius endomorphism in positive characteristic to attack several problems in local cohomology and multiplicities.

The first main result proves that local cohomology modules (supported at the maximal ideal) of an F-pure local ring have only finitely many F-stable submodules. Recent applications of this result such as special cases of the deformation of F-injectivity are discussed. The second main result shows that the category of Lyubeznik's F-modules has finite global dimension one more than the dimension of the ring.

In the final part we use Cohen factorization coupled with the Frobenius endomorphism to attack the long standing Lech's conjecture on multiplicities of local rings. We obtain some positive results in low dimension and extend many known results.