UC Berkeley

The goal of this Thesis is to develop the theory of Picard and Albanese 1-motives attached to a variety X defined over a perfect field of positive characteristic, and to relate these 1-motives to the etale cohomology groups of X. This should be viewed as a generalization of the classical theory of Picard and Albanese varieties attached to a smooth and proper variety X. Moreover, giving such a theory allows us to relate the dimension- and codimension-one etale cohomology groups in the most natural (`motivic') way possible; in particular, independence-of-l type results in dimension- and codimension-one are automatic once one has developed such a theory.

In the case of a base field of characteristic zero, the corresponding 1-motives have been constructed and studied in previous work of Barbieri-Viale and Srinivas. When one deals with a positive-characteristic base field, new difficulties arise due to the fact that resolution of singularities in positive characteristic is still an open problem. This forces us to introduce new methods, especially a strong form of de Jong's results that allows us to resolve (in a weak sense) an arbitrary separated finite type k-scheme by a smooth Deligne-Mumford stack. A large part of this thesis is devoted to preliminary results on divisors and cycle class maps for Deligne-Mumford stacks that we need when applying the methods of Barbieri-Viale and Srinivas with stacks rather than schemes.

In the end, we manage to construct the Picard 1-motives of an arbitrary separated finite type k-scheme with no additional assumptions, and prove various functoriality and compatibility properties for these 1-motives. The situation with the Albanese 1-motives is more complicated; over an arbitrary perfect field, we only manage to show that the Albanese 1-motives of X exist after possibly base extending X via a finite field extension. We show, however, that in the case that k is a finite field or an algebraically closed field, no such field extension is necessary. The case of a finite field uses a method for descending 1-motives along an extension of finite fields when the 1-motive is only given up to isogeny. This method may be of some independent interest.