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Denote $CHM(F)$ to be the category of Chow motives over a field $F$.

Let's consider an algebraic exension $E/F$, then there is a natural extension of scalars functor $CHM(F) \to CHM(E)$.

I was wondering if this functor is conservative, i.e. if a morphism $f: M \to N$ becomes an isomorphism after a field extension, does it imply $f$ is an isomorphism itself?

A related question is: if a motive $M$ becomes zero after a field extension, does it imply that $M = 0$? I believe this question is weaker, than that of being conservative.

Merkurjev-Gille-Chernousov (Corollary 8.4) prove this for motives of homogenous spaces for algebraic group actions (so-called Rost Nilpotence theorem, since it was originally prove by Rost for quadrics).

Do people believe that this holds in general? Is it related to some standard motivic conjectures?

Thanks.

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With rational coefficients, the answer is yes.

The first case to understand is when $E$ is a finite algebraic extension of $F$. In the case when moreover $E$ is purely inseparable, then the extension of scalars functors $CHM(F)\to CHM(E)$ is fully faithful, and if $E$ is Galois of degree $d$, then the extension of scalars functor $$\pi^\star:CHM(F)\to CHM(E)$$ has a right adjoint $\pi_\star$, and for any motive $M$ over $F$, there is a trace map $$tr_M : \pi_\star \pi^\star(M)\to M$$ whose composition with the unit map $$M\to \pi_\star \pi^\star(M)$$ is multiplication by $d$. If you work with rational coefficients, this implies that $\pi^\star$ is then conservative and faithful.

From there, to prove the general case, we may assume that $E$ is a filtered colimit of smooth $F$-algebras $A_i$. But then, for any index $i$, possibly after taking a finite extension of $F$, the map $F\to A_i$ has a retraction, so that, writing $CHM(E)$ as the $2$-colimit of the categories $CHM(A_i)$, we see easily that the extension of scalars functors is again faithful and conservative (for Chow motives over a smooth $F$-algebra, see for instance Definition 5.16 in Levine's paper arXiv:0807.2265).

If you really want Chow motives with integral coefficients, you mays still have to invert the (exponential) characteristic of $F$. Then, assuming furthermore that $F$ and $E$ are algebraically closed, the extension of scalars functors will be conservative again (this uses rigidity theorems; see O. Röndigs and P. A. Østvær, Rigidity in motivic homotopy theory, Math. Ann. 341 (2008), 651-675).

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Thanks for your answer and the references. However, I'm mostly interested in the case of a Galois extension for motives with integer coefficients. It seems to me that the answer is unknown in this case, but I'm curious if there is a belief that this statement must be true. –  Evgeny Shinder Jan 6 '10 at 5:18
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