ERRATUM TO: CONNECTIVITY AND PURITY FOR LOGARITHMIC MOTIVES

The proof of [1, Lemma 7.2] contains a gap: the equality $\omega _{\sharp } h_{0}(\Lambda _{\mathrm {ltr}}(\eta ,\mathrm {triv})) = \omega _{\sharp } h_{0}(\omega ^{*}\Lambda _{\mathrm {tr}}(\eta ))$ is false. Indeed one can check that for $X\in \mathbf {Sm}(k)$ proper,

$$ \begin{align*} \operatorname{Hom}( \omega_{\sharp} h_{0}(\Lambda_{\mathrm{ltr}} (\eta_{X}, \mathrm{triv})), \mathbf{G}_{a}) \neq \operatorname{Hom}( \omega_{\sharp} h_{0} (\omega^{*} \Lambda_{{\mathrm{tr}}}( \eta_{X})) , \mathbf{G}_{a}), \end{align*} $$

as the left-hand side is $\mathbf {G}_{a}(\eta _{X})$ , whereas the right-hand side is $\mathbf {G}_{a}(X)$ . For now, we can give a proof only of a weaker version of [1, Proposition 7.3]:

Proposition 0.1. Let k be a perfect field. Then the compositions

$$ \begin{align*} \mathbf{CI}^{\log}_{{\mathrm{dNis}}}\xrightarrow{i}\mathbf{Shv}^{\log}_{{\mathrm{dNis}}}\xrightarrow{\omega_{\sharp}^{\log}}\mathbf{Shv}_{{\mathrm{Nis}}},\qquad \mathbf{CI}^{\mathrm{ltr}}_{{\mathrm{dNis}}}\xrightarrow{i^{\mathrm{tr}}}\mathbf{Shv}^{\mathrm{ltr}}_{{\mathrm{dNis}}}\xrightarrow{\omega_{\sharp}^{\mathrm{ltr}}}\mathbf{Shv}_{{\mathrm{Nis}}}^{{\mathrm{tr}}} \end{align*} $$

are faithful and exact. In particular, both functors are conservative.

Proof. Exactness follows from the exactness of i and $\omega _{\sharp }^{\log }$ (resp., $i^{\mathrm {tr}}$ and $\omega _{\sharp }^{\mathrm {ltr}}$ ). To show faithfulness, it is enough to show that for all $F \in \mathbf {CI}^{\log }_{{\mathrm {dNis}}}$ (resp., $\mathbf {CI}^{\mathrm {ltr}}_{{\mathrm {dNis}}}$ ), the unit map

$$ \begin{align*} F\to \omega^{\mathbf{CI}}_{\log }\omega_{\sharp}^{\log} F\quad (\text{resp., } F\to \omega^{\mathbf{CI}}_{\mathrm{ltr}}\omega_{\sharp}^{\mathrm{ltr}} F) \end{align*} $$

is injective. By [1, Theorem 5.10], we have that for all $X \in \mathbf {SmlSm}(k)$ ,

$$ \begin{align*} F(X)\hookrightarrow F\left({\underline{X}}-\lvert\partial X\rvert\right) = \omega^{*}_{\log}\omega_{\sharp}^{\log} F\quad (\text{resp., } \omega^{*}_{\mathrm{ltr}}\omega_{\sharp}^{\mathrm{ltr}} F), \end{align*} $$

and hence $u\colon F\hookrightarrow \omega ^{*}_{\log }\omega _{\sharp }^{\log } F$ (resp., $u^{{\mathrm {tr}}}\colon F\hookrightarrow \omega ^{*}_{\mathrm {ltr}}\omega _{\sharp }^{\mathrm {ltr}} F$ ) is injective. Because F is ${\overline {\square }}$ -local, the map u (resp., $u^{{\mathrm {tr}}}$ ) factors through $\omega ^{\mathbf {CI}}_{\log }\omega _{\sharp }^{\log } F$ (resp., $\omega ^{\mathbf {CI}}_{\mathrm {ltr}}\omega _{\sharp }^{\mathrm {ltr}} F$ ), which concludes the proof.

We believe that the full statement of a more general version of [1, Proposition 7.3] holds:

Conjecture 0.2. The functors of Proposition 0.1 are full.

We stress that the previous statement does not assume (RS), nor transfers. We cannot give a proof of [1, Lemma 7.2] at the moment, but we expect the statement to hold as a consequence of the following conjecture:

Conjecture 0.3. The inclusion $\iota ^{\mathrm {tr}}\colon \mathbf {CI}^{\mathrm {ltr}}_{{\mathrm {dNis}}}(k,\Lambda )\subseteq \mathbf {Shv}_{\mathrm {dNis}}^{\mathrm {ltr}}(k,\Lambda )$ is Serre – that is, for all $F\in \mathbf {CI}^{\mathrm {ltr}}_{{\mathrm {dNis}}}(k,\Lambda )$ , if $G\subseteq F$ is a subsheaf with log stransfers, then G is strictly ${\overline {\square }}$ -invariant – that is, G lies in $\mathbf {CI}^{\mathrm {ltr}}_{{\mathrm {dNis}}}(k,\Lambda )$ .

If Conjecture 0.3 holds, then the counit map $\iota ^{\mathrm {tr}} h^{0}_{\mathrm {ltr}}G\to G$ is a monomorphism for all $G\in \mathbf {Shv}_{{\mathrm {dNis}}}(k,\Lambda )$ . In particular, this would imply that the natural map

$$ \begin{align*} \omega^{\mathbf{CI}}_{\mathrm{ltr}}\omega_{\mathbf{CI}}^{\mathrm{ltr}}F = \iota^{\mathrm{tr}}h^{0}_{\mathrm{ltr}}\omega^{*}_{\mathrm{tr}}\omega_{\sharp}^{\mathrm{ltr}}\iota^{\mathrm{tr}}F\hookrightarrow \omega^{*}_{\mathrm{ltr}}\omega_{\sharp}^{\mathrm{ltr}}\iota^{\mathrm{tr}}F \end{align*} $$

is injective, so we could proceed as in [1, Proposition 7.3.] to prove Conjecture 0.2 in the case with transfers. On the other hand, we do not expect Conjecture 0.3 to hold for $\mathbf {CI}^{\log }_{{\mathrm {dNis}}}(k,\Lambda )$ , as its counterpart is already false for the category of $\mathbf {A}^{1}$ -local sheaves without transfers.

All the results of [1, §7] must be considered conjectural as well.