Lemma 61.19.3. Let $X$ be a scheme. Let $Y = \mathop{\mathrm{lim}}\nolimits Y_ i$ be the limit of a directed inverse system of quasi-compact and quasi-separated objects of $X_{pro\text{-}\acute{e}tale}$ with affine transition morphisms. For any sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have

$\epsilon ^{-1}\mathcal{F}(Y) = \mathop{\mathrm{colim}}\nolimits \epsilon ^{-1}\mathcal{F}(Y_ i)$

Moreover, if $Y_ i$ is in $X_{\acute{e}tale}$ we have $\epsilon ^{-1}\mathcal{F}(Y) = \mathop{\mathrm{colim}}\nolimits \mathcal{F}(Y_ i)$.

Proof. By the description of $\epsilon ^{-1}\mathcal{F}$ in Lemma 61.19.1, the displayed formula is a special case of Étale Cohomology, Theorem 59.51.3. (When $X$, $Y$, and the $Y_ i$ are all affine, see the easier to parse Étale Cohomology, Lemma 59.51.5.) The final statement follows immediately from this and Lemma 61.19.2. $\square$

Comment #2044 by Dragos Fratila on

To finish the proof don't we need to show also that $Y\mapsto colim_i F(Y_i)$ satisfies the sheaf axiom? I understand that the presheaf pullback $F'=\epsilon_p(F)$ of $F$ satisfies the conclusion $F'(Y)=colim_{i} F(Y_i)$ (since coequalisers in presheaves are computed pointwise) but it's not obvious to me why should the sheafification of $F'$ satisfy this equality too.

Comment #2082 by on

No because we are directly computing the values of the sheaves! I tried to clarify this in the following edit.

Comment #6318 by Owen on

if F is a sheaf on $X_{\text{étale}}$ then shouldn't the $Y_i$ be objects of $X_{\text{étale}}$ if we are to evaluate $F(Y_i)$?

Comment #6321 by Owen on

I think I understand what the statement should be (as needed to prove 61.19.4):

Let $X$ be a scheme. Let $Y=\lim Y_i$ be the limit of a directed inverse system of quasi-compact and quasi-separated objects of $X_{\text{pro-étale}}$ with affine transition morphisms. For any sheaf $F$ on $X_{\text{étale}}$ we have $\epsilon^{-1}F(Y)=\operatorname{colim}_i\epsilon^{-1}F(Y_i)$. If the $Y_i$ are moreover objects of $X_{\text{étale}}$, we have $\epsilon^{-1}F(Y)=\operatorname{colim}_i F(Y_i)$.

The proof is the same and uses that $\epsilon^{-1}$ commutes with colimits and colimits commute with colimits; i.e. if $C$ denotes $X_{\text{étale}}$, once we have $\epsilon^{-1}h_U(Y)=\operatorname{colim}\epsilon^{-1}h_U(Y_i)$, we write

$\epsilon^{-1}F(Y)=(\epsilon^{-1}\operatorname{colim}_{C/F}h_U)(Y)=\operatorname{colim}_{C/F}\epsilon^{-1}h_U(Y)=\operatorname{colim}_{C/F}\operatorname{colim}_i\epsilon^{-1}h_U(Y_i)$ $=\varinjlim_i\operatorname{colim}_{C/F}\epsilon^{-1}h_U(Y_i)=\operatorname{colim}_i\epsilon^{-1}F(Y_i).$

When the $Y_i$ are in $X_{\text{étale}}$, we have $\epsilon^{-1}h_U(Y)=\operatorname{colim}h_U(Y_i)$ and the above simplifies.

Comment #6427 by on

Thanks very much since this was quite a mess. But I could not make your fix work because we don't know that taking sections over Y commutes with the colimits in question (they aren't filtered). In fact, my first fix was wrong for this reason. I fixed it instead by characterizing the pullback sheaf first (and this should have been done a long time ago --- I was just lazy) and then using the same result for etale sheaves which already was in the stacks project. Corresponding commits on github are: wrong fix and correct fix.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).