Let $C\subset \mathbb{P}^2$ be a smooth conic without $k$-points.
Call the Chow $k$-motive in *zero-dimensional* if it is a sum of $M\mathbb{L}^n$ where $M$ is an Artin motive, i.e. a part of a motive of zero-dimensional scheme. 


**Q.** How to see that a motive of $C$ is not (or is) zero-dimensional?


**P.S.** By Chow $k$-motive I understand the category of motives $Chow\_{\mathbb{Q}}$ defined by rational equivalence.



Motive of a conic without points
This is true with $\mathbb Q$-coefficients; see the proposition below. The reason it's difficult is that it's hard to compute Chow groups over non-algebraically closed fields. However, we have the following:



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> > **Lemma 1.** *Let $k \subseteq \ell$ be a separable algebraic field extension, and let $X$ be a proper geometrically integral $k$-scheme. Then the pullback map $\operatorname{Pic}(X) \to \operatorname{Pic}(X\_\ell)$ is injective.*
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*Proof.* See e.g. [this post](https://mathoverflow.net/a/180188/82179). $\square$



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> > **Lemma 2.** *Let $k \subseteq \ell$ be a finite field extension, and let $X$ be any $k$-scheme. Then the pullback map $\operatorname{CH}\_\*(X) \otimes\_\mathbb Z \mathbb Q \to \operatorname{CH}\_\*(X\_\ell) \otimes\_\mathbb Z \mathbb Q$ is injective.*
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*Proof.* If $\pi \colon X\_\ell \to X$ is the map induced by $k \to \ell$, then $\pi\_\* \pi^\*$ is multiplication by $[\ell:k]$; see e.g. [Ful, Ex. 1.7.4]. (This is in fact true for *any* finite flat morphisms of schemes.) $\square$


However, there exist Severi-Brauer varieties $X$ for which the pullback $\operatorname{CH}\_\*(X) \to \operatorname{CH}\_\*(X\_\ell)$ (with integral coefficients) is *not* injective, as is explained in [this post](https://mathoverflow.net/a/289803/82179).


Let $C$ be a smooth conic without rational points, and let $k \to \ell$ be the minimal splitting field (which is a separable quadratic extension of $k$).



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> > **Lemma 3.** *We have $\operatorname{CH}^\*(C) = \mathbb Z[2h] \subseteq \mathbb Z[h]/(h^2) = \operatorname{CH}^\*(C\_\ell)$.*
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*Proof.* The pullback $\operatorname{CH}^\*(C) \to \operatorname{CH}^\*(C\_\ell)$ is injective by **Lemma 1** above. The line bundle $\omega\_C^{-1}$ has class $2h$. If $\mathscr L$ is a line bundle with class $h$, then Riemann-Roch gives an isomorphism $C \stackrel\sim\to \mathbb P^1$, contradicting the choice of $C$. $\square$



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> > **Lemma 4.** *The image of the pullback $\operatorname{CH}^\*(C \times C) \to \operatorname{CH}^\*((C\times C)\_\ell)$ equals*
> > $$\mathbb Z[h\_1+h\_2,2h\_1] \subseteq \mathbb Z[h\_1,h\_2]/(h\_1^2,h\_2^2).$$
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*Proof.* Clearly $h\_1 + h\_2 = [\Gamma\_{\operatorname{id}\_C}]$ and $2h\_1 = \omega\_C^{-1} \boxtimes \mathcal O\_C$ are contained in $\operatorname{CH}^\*(C \times C)$. Moreover, they span $\operatorname{CH}^1(C \times C) = \operatorname{Pic}(C \times C)$, since $h\_1$ is not defined on $C \times C$. Indeed, if $\mathscr L$ is a line bundle with class $h\_1$, then it defines a map $C \times C \to \mathbb P^1$, whose image under the $2$-uple embedding is the first projection $C \times C \to C \subseteq \mathbb P^2$. This again contradicts the choice of $C$.


Finally, the intersection $(h\_1 + h\_2) \cdot 2h\_1 = 2h\_1h\_2$ is a degree $2$ zero-cycle on $C \times C$. There are no zero-cycles of odd degree, for their pushforward under either projection would be a divisor of odd degree on $C$. $\square$


**Remark.** I don't know if the pullback $\operatorname{CH}\_\*(C \times C) \to \operatorname{CH}^\*((C \times C)\_\ell)$ is *injective*. This is true for divisors (and for dimension $2$ cycles), and it is true rationally. However, there could be torsion zero-cycles that disappear on $(C \times C)\_\ell$.


**Remark.** We will use the ring isomorphism $\operatorname{CH}^\*(X \times \mathbb P^1) \cong \operatorname{CH}^\*(X) \otimes \mathbb Z[h]/h^2$ of [Ful, Ex. 8.3.4] for any smooth $k$-variety $X$ (it is true as groups in much greater generality). Then the pushfoward $\operatorname{CH}^\*(X \times \mathbb P^1) \to \operatorname{CH}^\*(X)$ is given by $\alpha \otimes 1 \mapsto 0$ and $\alpha \otimes h \mapsto \alpha$ for $\alpha \in \operatorname{CH}^\*(X)$.




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> > **Proposition.** *The motives $C$ and $\mathbb P^1$ are isomorphic rationally.*
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*Proof.* With rational coefficients, we have $\operatorname{CH}\_\mathbb Q^\*(C) \cong \operatorname{CH}\_\mathbb Q^\*(\mathbb P^1) \cong \mathbb Q[h]/(h^2)$, and similarly for all finite products involving $C$ and $\mathbb P^1$. The class $\phi = \tfrac{1}{2}[\omega\_C^{-1} \boxtimes \mathcal O\_{\mathbb P^1}(2)] \in \operatorname{Corr}(C, \mathbb P^1)$ corresponds to $h\_1+h\_2$. We now easily compute
\begin{align\*}
\phi \circ \phi^\top &= \pi\_{13,\*}(\pi\_{12}^\* \phi^\top \cdot \pi\_{23}^\* \phi) = \pi\_{13,\*}((h\_1+h\_2)(h\_2+h\_3)) \\
&= \pi\_{13,\*}(h\_1h\_3 + h\_2(h\_1 + h\_3) ) = h\_1 + h\_3 \\
&= [\Gamma\_{\operatorname{id}\_{\mathbb P^1}}] \in \operatorname{CH}^\*\_\mathbb Q(\mathbb P^1 \times \mathbb P^1),
\end{align\*}
and similarly for the other composition $\phi^\top \circ \phi$. $\square$


**Remark.** I think that the motive $C$ does not occur integrally as a direct summand of $\mathbb P^n \times Z$ for any $n \in \mathbb Z\_{\geq 0}$ and any zero-dimensional smooth $k$-scheme $Z$.


For example, if $Z = \operatorname{Spec} m$ for a Galois extension $k \subseteq m$ with group $G$, then $H^0(C\_\bar k, \mathbb F\_p)$ should occur as a direct summand of
$$H^0((\operatorname{Spec}\_\ell)\_\bar k, \mathbb F\_p) = \mathbb F\_p[G]$$
for any prime $p \not\mid \operatorname{char} k$.


But if $p \mid \#G$, then the unique trivial subrepresentation $\mathbb F\_p \cdot (\sum\_{\sigma \in G} \sigma) \subseteq \mathbb F\_p[G]$ does not split as a direct summand. If $\operatorname{char} k = 0$, this forces $G = 1$, i.e. $Z = \operatorname{Spec} k$. Then it remains to compute $\operatorname{CH}^\*(C \times \mathbb P^n)$ as above and see that no $\phi$ in there satisfies $\phi \circ \phi^\top = 1$, even modulo projectors.


This is not a very good argument, so I would be interested to see if someone has a cleaner obstruction for the integral version of the question.




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**References.**


[Ful] W. Fulton, *Intersection theory*, second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. **2**. Springer-Verlag (1998). [ZBL0885.14002](https://zbmath.org/?q=an:0885.14002).


