Tannakian categories: origins and summary
We describe the origins of the theory of tannakian categories, and summarize its main results.
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arXiv subjects
James S Milne
Publications and source records attributed to James S Milne.
Descent for Algebraic Schemes
This is an elementary exposition of the basic descent theorems for algebraic schemes over fields (Grothendieck, Weil, ...).
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Algebraic groups as automorphism groups of algebras
We show that every algebraic group scheme over a field with at least 8 elements can be realized as the group of automorphisms of a nonassociative algebra. This is only a modest improvement of the theorem of Gordeev and Popov (2003), but it allows us to give a new characterization of algebraic Lie algebras and to simplify the standard descriptions of Mumford--Tate domains and Shimura varieties as moduli spaces. Once the original argument of Gordeev and Popov has been rewritten in the language of schemes, we find that it also applies to algebraic groups over discrete valuation rings.
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On the Tate and Standard Conjectures over Finite Fields
For an abelian variety over a finite field, Clozel (1999) showed that l-homological equivalence coincides with numerical equivalence for infinitely many l, and the author (1999) gave a criterion for the Tate conjecture to follow from Tate's theorem on divisors. We generalize both statements to motives, and apply them to other varieties including K3 surfaces.
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