nLab mixed Tate motive

Idea

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? $\mathbf{Z}(n)$.

Properties

• $DTM(S)$ 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).

Related concepts

• mixed motives
• mixed elliptic motives?
• multiple zeta values

References

• 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.