Topics in algebraic geometry: An algebraic Napier-Ramachandran theorem and Steenrod operations on Chow groups.

摘要

This thesis consists of two distinct parts held together by the guiding principle of the interplay between algebraic topology and algebraic geometry. The first part "An Algebraic Napier-Ramachandran Theorem", gives an algebraic proof of a theorem proved by Napier and Ramachandran using the {dollar}Lsp2-bar{lcub}partial{rcub}{dollar} lemma. The main theorem is:; Theorem. Let {dollar}Zto X{dollar} be an unramified map of one smooth, geometrically connected, projective, positive dimensional variety over a field k of finite characteristic to another, and suppose that the normal bundle {dollar}Nsb{lcub}Z/X{rcub}{dollar} is ample. Then the image of the map {dollar}pisbsp{lcub}1{rcub}{lcub}et{rcub}(Z)topisbsp{lcub}1{rcub}{lcub}et{rcub}(X){dollar} has finite index.; The second part is a construction of cohomology operations on Chow groups which commute with Steenrod operations under the cycle class map. It is a direct application of the recent work of D. Edidin and W. Graham on Equivariant Chow Theory.