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new page mixed Tate motive, mostly to record some references for now
The existence of a left adjoint is related to the Tate-Beilinson conjecture.
I donâ€™t know what this refers to, but Totaro proved that a left adjoint does not exist for most algebraically closed fields: http://arxiv.org/pdf/1502.05079.pdf
In the introduction of that paper, Totaro also writes
By contrast, the Tate-Beilinson conjecture would imply that the inclusion of DMT(k; Q) into DM(k; Q) is a Frobenius functor when k is algebraic over a finite field (Theorem 8.1). This is the strong property that the right adjoint to the inclusion is also left adjoint to the inclusion (and so there is an infinite sequence of adjoints). It is not clear what to expect when k is a number field, or when k is replaced by a regular scheme of finite type over Z.
I edited the page to add the fact you mentioned as well.
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