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?



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).

  • $\begingroup$ 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. $\endgroup$ – Evgeny Shinder Jan 6 '10 at 5:18

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