The stable (infinity,1)-category of mixed Tate motives $DTM(S)$ is the localizing subcategory of the stable (infinity,1)-category of mixed motives $DM(S)$ generated by the twists of the Tate motive? $\mathbf{Z}(n)$.
$DTM(S)$ can be described as a coreflective sub-(infinity,1)-category of $DM(S)$, since it follows from a theorem of Amnon Neeman that the inclusion admits a right adjoint.
The existence of a left adjoint is false for algebraically closed fields that are not the algebraic closure of a finite field, while for fields algebraic over a finite field, the Tate-Beilinson conjecture? would imply that the inclusion is a Frobenius functor, i.e. that the right adjoint is also a left adjoint. See (Totaro 15).
Marc Levine, Mixed Motives, K-theory Handbook?, pdf.
Hélène Esnault, Marc Levine, Tate motives and the fundamental group, arXiv:0708.4034.
Section 7 in
Burt Totaro, The motive of a classifying space, arXiv:1407.1366.
Burt Totaro, Adjoint functors on the derived category of motives, arXiv:1502.05079.
Last revised on March 10, 2015 at 10:58:30. See the history of this page for a list of all contributions to it.