Motives of Azumaya algebras

Abstract

Abstract. We study the slice filtration for the K-theory of a sheaf of Azumaya algebras A, and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson-Lichtenbaum conjecture, we apply our results to show the vanishing of SK 2 (A) for a central simple algebra A of square-free index.

Birational Motives, II: Triangulated Birational Motives

“…As was already observed in [20] and [24], this implies that the terms of the "associated graded of the slice filtration" on an object of DM eff are twists of birational motives.…”

“…Given the long period of gestation of this work, there have been other expositions of triangulated birational motives, notably in [20] and [24]: they are essentially independent from the present one. We would like to finish this introduction by pointing a mistake in the initial version:…”

On a theorem of Hazrat and Hoobler

“…The following is an interesting corollary of our approach: there are natural filtrations of length d on G i (X) and G i (X; A) coming from [11]. The map B A : G i (X) → G i (X; A) respects the filtrations.…”

“…As in Kahn-Levine [11], define the cycle complex of X with coefficients in A as follows. Let S X (s) (t) denote the set of closed subsets W ⊂ X × k ∆ t such that…”

Motivic Homotopy Theory

“…Together with B. Kahn [18], we have examined the layers and spectral sequence for the K-theory of a central simple algebra A over K, as well as for the motive of the Severi-Brauer variety X = SB(A) associated to A. We compute the homotopy motives for K A as π µ n (K A ) = Z A where Z A ⊂ Z is the subsheaf with value on a field F the ideal in Z generated by the index of A ⊗ k F .…”