The Stacks project

Typo in (2): it should be “let $\mathcal{L}$ be an invertible $\mathcal{O}_Y$-module.”

Part (1) can be stated more generally as:

Let $f:X\to Y$ be a morphism of locally ringed spaces. Assume pullbacks of meromorphic functions are defined for $f$. Let $\mathcal{G}\in Mod(\mathcal{O}_X)$, $\mathcal{F}\in Mod(\mathcal{O}_Y)$ and suppose we have a morphism $\mathcal{F}\to f_*\mathcal{G}$ of $\mathcal{O}_Y$-modules. Then there is a canonical morphism $\mathcal{K}_Y(\mathcal{F})\to f_*\mathcal{K}_X(\mathcal{G})$ of $\mathcal{K}_Y$-modules. (In particular, setting $\mathcal{G}=f^*\mathcal{F}$, the map $\mathcal{F}\to f_*f^*\mathcal{F}$ to be the unit of $f_*\dashv f^*$ and taking global sections gives the pullback map pullback map $f^* : \Gamma(Y, \mathcal{K}_Y(\mathcal{F})) \to \Gamma(X, \mathcal{K}_X(f^*\mathcal{F}))$.)

The construction is the composite of canonical maps:

$$\mathcal{F}\otimes_{\mathcal{O}_Y}\mathcal{K}_Y \to f_*\mathcal{G}\otimes_{f_*\mathcal{O}_X}f_*\mathcal{K}_X \to f_*(\mathcal{G}\otimes_{\mathcal{O}_X}\mathcal{K}_X).$$

I regarded this more general form of part (1) useful for instance if one is working with the sheaf of meromorphic differentials on a $k$-variety $X$, i.e., $\mathcal{K}_X(\Omega_{X/k})$. Pullbacks of meromorphic functions are defined for the normalization $\nu:X'\to X$. Hence one obtains a morphism $\mathcal{K}_X(\Omega_{X/k})\to\nu_*\mathcal{K}_{X'}(\Omega_{X'/k})$. When $X$ is a curve, this last morphism is used by Serre in Algebraic Groups and Class Fields to define the “sheaf of regular meromorphic differentials.”

The distinguished triangle is backwards (wrong truncations). So we need to rewrite the proof.

I think some $R$'s should be $A$'s.

Typo in the second equation in the proof: omitted closing bracket in $[m+m']$.

A small typo in remark 35.8.4, after "Hence, given S′/S we let", f_{sites, *}O(S'/S) should equal to O(T\times_{S}S'/T)

Does part (2) of this lemma need a flatness hypothesis as in Tag 20.11.11?

It doesn’t seem to be clear that $H^0 \circ \mathrm{Can} = id$ suffices to prove that $\mathrm{Can}$ is fully faithful. I thought about using Tag 05ER? But in general, \varphi is not flat.

In (1) we may replace '$\mathcal{E}_i$ is strictly perfect' into '$\mathcal{E}^\bullet_i$ is strictly perfect'.

Oh I am sorry. "Apply to x_i$means replace$n$by$x_i$ and use the same argument.

I am sorry that I cannot get $\sum_j\sigma(a_{ij})y_j\in M$ from the fact $\sum_i\sigma(a_i)x_i\in M$. I know that $\varphi(\sum_i \sigma(a_i)x_i\otimes 1) = \sum_i 1\otimes\sigma(a_i)x_i$ from the conclusion by (1). And as $\varphi(\sum_i \sigma(a_i)x_i\otimes 1)=\sum_{i,j}\sigma(a_i)a_{ij}\otimes y_j$, I only can conclude that $\sum_{i,j}\sigma(a_i)\sigma(a_{ij})y_j\in M$.

To Comment #8193: Then for a topological space $X$, the category $\mathrm{Sh}(X)$ of abelian sheaves on $X$ is an abelian subcategory of the category $\mathrm{PSh}(X)$ of abelian presheaves on $X$ in the comment's sense. In general, the inclusion functor $\mathrm{Sh}(X)\to \mathrm{PSh}(X)$ is not exact, so $\mathrm{Sh}(X)$ is not an abelian subcategory of $\mathrm{PSh}(X)$ in the sense of p.7 of "Introduction to Homological Algebra" by Weibel.

In the first paragraph: ``Note that by the previous section this functor commutes with all limits. " Does $u^p$ commute with colimits as well?

Proof of Lemma 009F (2), end of paragraph: typo, we need phi_{i'i''}(s') two times (instead of phi_{i'i''}(s))

On lemma 03AJ we have a bijection between two abelian groups. It would be nice to say that this is indeed an isomorphism. (Said otherwise, it would be nice to explain that the contracted product of torsors "is" the addition on H^1.)

Do we have any explicit example of a point $p$ of the site $\mathcal{C}$? I thought it should be the skyscraper sheaf, but the skyscraper sheaf is a contravariant functor instead of a covariant functor.

There is a typo "$H^ 0(\Gamma (U, \mathop{\mathcal{H}\mathit{om}} \nolimits ^\bullet (\mathcal{L}^ \bullet , \mathcal{I} ^ \bullet ))) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O} _ U)}(L| _ U, M| _ U)$" should be "$H^ 0(\Gamma (U, \mathop{\mathcal{H}\mathit{om}} \nolimits ^\bullet (\mathcal{L}^ \bullet , \mathcal{I} ^ \bullet ))) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O} _ U)}(L| _ U, \mathcal{I} ^ \bullet| _ U)$"

Is here $( \coprod_{i\in I} h_{U_i} )^\sharp = \coprod_{i\in I} h_{U_i} ^\sharp$?

@#8671 No, simple misread on my part.

In the proof of Lemma 12.3.4: "It follows that $i \circ p + j \circ q : z \to z$ is the identity since..."

I think that this should read "It follows that $p \circ i + q \circ j : z \to z$ is the identity since..." Otherwise the map in question is not well-defined. The rest of the argument works fine.

Typo: "In Situation 0G8G the psuhforward" should be "... pushforward".