344 research outputs found
A Tannakian classification of torsors on the projective line
In this small note we present a Tannakian proof of the theorem of
Grothendieck-Harder on the classification of torsors under a reductive group on
the projective line over a field.Comment: 13 pages; any comments or hints to existing literature welcom
On Harder-Narasimhan filtrations and their compatibility with tensor products
We attach buildings to modular lattices and use them to develop a metric
approach to Harder-Narasimhan filtrations. Switching back to a categorical
framework, we establish an abstract numerical criterion for the compatibility
of these filtrations with tensor products. We finally verify our criterion in
three cases, one of which is new
Noncommutative numerical motives, Tannakian structures, and motivic Galois groups
In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the change-of-coefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schur-finiteness, we prove that the category NNum(k)_F of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum(k)_F is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric monoidal functor HP* on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues C_(NC) and D_(NC) of Grothendieck's standard conjectures C and D. Assuming C_(NC), we prove that NNum(k)_F can be made into a Tannakian category NNum (k)_F by modifying its symmetry isomorphism constraints. By further assuming D_(NC), we neutralize the Tannakian category Num (k)_F using HP*. Via the (super-)Tannakian formalism, we then obtain well-defined noncommutative motivic Galois (super-)groups. Finally, making use of Deligne-Milne's theory of Tate triples, we construct explicit morphisms relating these noncommutative motivic Galois (super-)groups with the classical ones as suggested by Kontsevich
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