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Maybe this is a question to naive for the MO community! For a projective smooth variety $X$ defined over a field $F$, and for simplicity let's assume $F$ is a number field. One way to define motivic cohomology is through $K$-theory \begin{equation} H^i_M(X,\mathbb{Q}(j))=K_{2j-i}(X)^{(j)}_{\mathbb{Q}} \end{equation} where $(j)$ means the eigenspace where Adam operators $\psi^k$ act as multiplication by $k^j$.

From the construction of higher $K$-theory and Adam operators, for a ring $A$, $K_l(A)^{(j)}$ only depends on the ring structure of $A$, and it has no dependence on whether we consider $A$ and an $L_1$-algebra through an embedding $L_1 \rightarrow A$ or $L_2$-algebra through $L_2 \rightarrow A$, where $L_1$ and $L_2$ are two different fields. So I guess this definition of motivic cohomology has no dependence on whether we consider $X$ as a variety over $F$ or a variety over a subfield (restriction of scalar), e.g. $\mathbb{Q}$.

On the other hand there are other constuctions of motivic cohomology of $X$, e.g. Bloch’s higher Chow groups and Voevodsky's construction, which indeed use the field $F$ in its constructions. I could not see whether $H^i_M(X,\mathbb{Q}(j))$ from these constructions has dependence on $X$ being considered as a variety over $F$ or over $\mathbb{Q}$. Anyone who can explain why?

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  • $\begingroup$ I don't think looking at the composite $X \to \mathrm{Spec} F \to \mathrm{Spec} Q $ is the good notion of restriction of scalars. It is rather defined by a functor, and it is a theorem that for smooth projective varieties this functor is representable. As an example, restriction of scalars of an elliptic curve E/F is an abelian variety A/Q of dimension [F:Q]. $\endgroup$
    – François Brunault
    Commented Apr 12, 2018 at 9:12
  • $\begingroup$ Could I bother you with a reference about the definition of restriction of scalars given by a functor and the theorem you have mentioned? $\endgroup$
    – Wenzhe
    Commented Apr 12, 2018 at 9:25
  • $\begingroup$ See Poonen's book "Rational points on varieties" section 4.6. Also see the book "Néron models" by Bosch-Lütkebommert-Raynaud for the proof that restriction of scalars preserve being smooth projective. The following MO discussion should also be useful mathoverflow.net/questions/212989/… $\endgroup$
    – François Brunault
    Commented Apr 12, 2018 at 10:24
  • $\begingroup$ I don't think that Bloch's definition actually depends on the base field.:) $\endgroup$
    – Mikhail Bondarko
    Commented Apr 12, 2018 at 17:05
  • $\begingroup$ @MikhailBondarko But the definition of algebraic simplex $\Delta_n$ involves the base field, and also in the definition of complex, it makes use of algebraic subvarieties of $X \times \Delta_n$, which also involves the base field, how to see Bloch's definition actually is independent of the base field? $\endgroup$
    – Wenzhe
    Commented Apr 12, 2018 at 17:25

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By the work of Voevodsky, A^1-homotopy theory, or Cisinski-Déglise, Triangulated categories of mixed motives, we have motivic cohomology of arbitrary schemes (maybe noetherian and finite dimensional, but this is the case here) with integral coefficients. For regular schemes, motivic cohomology with Q-coefficients is isomorphic to the K-theoretic version you mentioned, see Cisinski-Déglise, Introduction, Thm 10, footnote 12. So you get independence of the base field. See also their computation in Example 11.2.3.

Presumably this independence can be checked directly with Voevodsky's definition but I haven't tried to do it.

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  • $\begingroup$ The reference is very helpful. $\endgroup$
    – Wenzhe
    Commented Apr 12, 2018 at 9:23

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