The Stacks project

Typo in the section introduction : replace "spaces" by stacks" in the sentence "we may define what it means for a morphism of algebraic spaces..."

May need add comma before "in other words".

Yes, this is very confusing! Thanks and fixed here.

The categories in the statement are claimed to be triangulated, but it needs abelian categories (and this is what it used in the proof anyway).

Suggested slogan: Quasi-finite, separated morphisms are quasi-affine

Also, why is there an "(!)" in the proof?

If $f: Sh(\mathcal C) \to Sh(\mathcal D)$ is a morphism of topoi, then $f_\ast(\mathcal F)$ might just be a sheaf of sets, not a sheaf of abelian groups, even if $\mathcal F$ is. Then how do we define $Rf_\ast(\mathcal F)$?

Typo: last paragraph $M\in D(\mathcal O_Y)$ not $\mathcal O_U$

The meaning of the statement is not completely formal. To make sense of "the identity on cohomology" we need to show that we can identify $\mathcal{F}$ with $g_*(\mathcal{F})$ and/or $g^{-1}(\mathcal{F})$, for any $\mathcal{F}$. This is of course the case in subsequent lemmas where $\mathcal{F}$ is a constant sheaf.

It seems that in the statement of 53.20.12, we could add "the image of $a$ vanishes at the base point $v$ of $V$, and the base point $u$ of $U$ maps to the node of the fiber $W_v$", right?

@James A. Myer: No. Every closed set with $\geq 2$ elements is reducible.

Typo: in (03SX) the first $\pi_X$ should be $\pi_x$.

g should be from Sh(C) to Sh(C')

Dear Shiji, thanks for finding this error. There is a way of fixing this using limit arguments, but that would mean pushing this much later in the project. Another fix is the following.

We have to show: Given a quasi-compact open immersion $j : V \to Y$ of a quasi-separated and quasi-compact algebraic spaces and an integral morphism $\pi : Z \to V$ we can extend $\pi$ to an integral morphism $\pi' : Z' \to Y$ in the sense that $Z$ is isomorphic to the inverse image of $V$ in $Z'$. To do this, let $\mathcal{A}$ be the quasi-coherent sheaf of $\mathcal{O}_V$-algebras on $V$ corresponding to $\pi : Z \to V$, in other words, $\mathcal{A} = \pi_*\mathcal{O}_Z$. Pushforward along $j$ preserves quasi-coherence. Hence $j_*\mathcal{A}$ is a quasi-coherent $\mathcal{O}_Y$-algebra. Now we let $\mathcal{A}' \subset j_*\mathcal{A}'$ be the integral closure of $\mathcal{O}_Y$. By Lemma 29.51.1 (translated over to the category of algebraic spaces; details omitted) we see that $\mathcal{A}'$ is a quasi-coherent $\mathcal{O}_Y$-algebra and we see that $\mathcal{A}'|_V \cong \mathcal{A}$ because the integral closure of $\mathcal{O}_V$ in $mathcal{A}$ is $\mathcal{A}$ as $\pi$ is integral! Thus $Z' = \underline{\text{Spec}}_Y(\mathcal{A}') \to Y$ is the answer to our problem.

In the last paragraph we applied Lemma 0ABS to the morphism $Z \to Y$. However, it seems that this morphism is not necessarily quasi-finite since the morphism $Z \to V$ is just integral, not finite. Is there a way to resolve this?

Typo: in the proof of 53.19.13, the map $\mathcal{O}_{X, x}^\wedge \to \mathcal{O}_{Y, y}$ should be $\mathcal{O}_{X, x}^\wedge \to \mathcal{O}_{Y, y}^\wedge$.

In (3), element' should beelements'. In (2) it would be good to parenthesize the argument of dim, even though one could argue that no confusion is possible and that this is a matter of taste.

If $x/1\in A[S^{-1}]$ is prime then doesn't ideal correspondence for localizations imply $x \in A$ is prime?

There is a nice discussion of the exact sequences using only the universal properties in a down-to-Earth way in the book "Commutative Algebra" by Singh on page 249.

I think the method of proof here works immediately for Koszul regular sequences (and is even a little simpler since you don't need the induction and you don't have to argue flat locally in the next lemma). Just compute $L_{B/A}$ for $A = \mathbf{Z} [x_1, \dots , x_r]$, and then if $(f_1, \dots , f_r)$ is a Koszul regular sequence on $C$, then $C / (f_1, \dots , f_r) = B \otimes _A ^{\mathbf{L}} C$.

Argh! It seems you are correct. This just is a terrible lemma. Luckily it seems we only use it once!