Motives over finite fields --- J.S. Milne   Top
Expository Notes
A Primer of Commutative Algebra
Motives---Grothendieck's Dream
What is a Shimura Variety?
Introduction to Shimura Varieties
Shimura Varieties and Moduli
Tannakian Categories
The Work of Tate
Errata
Corrected manuscript November 2, 2019.
Original manuscript October 16, 1992.

Abstract

This is the author's manuscript for
Milne, J.S., Motives over finite fields, In: Motives (Eds. U. Jannsen, S. Kleiman, J.-P. Serre), Proc. Symp. Pure Math., AMS, 55, 1994, Part 1, pp. 401--459.
except that the TeX has been updated, some corrections and a few minor editorial changes made, and some footnotes added. Significant changes to the original article have been noted in footnotes. The numbering is unchanged.

The article studies the Tannakian category of motives over a finite field assuming the Tate conjecture. It computes the associated groupoid, the polarizations, studies the reduction functor from the category of CM-motives in characteristic zero. It is partly expository, because many of the results were known to Grothendieck, Langlands, Rapoport, and Deligne, but often not published.

Contents

  1. Construction of the category of motives
  2. Basic properties of the category of motives
  3. Characterizations
  4. The reduction of CM-motives to characteristic p