The Stacks project

Thanks for the suggestion. I added and proved the lemma along the lines you suggested. Of course there are generalizations of this lemma long the lines of 33.7.13 where you have a "dense family" of morphisms $T_i \to X$ with $T_i$ geometrically irreducible. See changes in this commit.

Thanks for pointing this out. I only found one place where the arrow was in the wrong direction but it was confusing ideed. See changes in this commit.

Thanks and fixed here.

Thanks and fixed here.

Thanks and fixed here.

I think you are right about the bidegree of $d_r$ being $(-r , r - 1)$ but I think you swapped the roles of $i$ and $j$. So I think the differential for the first spectral squence of the example on the $E_2$-page is given by maps $Tor_j(H_i, M) \to Tor_{j - 2}(H_{i + 1}, M)$ which is what it says.

OK, I think this isn't necessary because we mention in the introduction 92.1 to this chapter that we are working without any separation assumptions. Moreover, just the fact that you are referring to Definition 4.1 of their book (which is their definition of algebraic stacks and not the definition of a quasi-separated morphism of algebraic stacks) seems to suggest that things are quite a bit different.

True but the proof as given is fine too.

A simple lemma that for some reason no one points out is the following: if $X$ is an irreducible scheme of finite type over $k$ and $X(k)$ is dense, then $X$ is geometrically irreducible. This follows from the similar statement 04KV for geometrically connected schemes by considering $X\setminus D$ where $D$ is the image of the pairwise intersections of irreducible components of $X_{\bar{k}}$. Obviously it is possible to relax the finite type hypothesis.

By the definition of morphism of families, 00VT, seems incompatible with the lemmas later on where $\alpha$ seems to be a map from $J \rightarrow I$ rather than $I \rightarrow J$, i.e. 0G1L. Or am I missing something?

@#5799: If $X$ is defined over a field $k$ and $\mathcal{L}$ is relatively very ample on $X$ over $k$, then it follows from the definition that there is a $k$-vector space $E$ and a surjective module map $E \otimes_k \mathcal{O}_X \to \mathcal{L}$. Sure this does not mean that $E \to f_*\mathcal{L}$ is surjective, but we never use this in the argument. (The argument actually shows you can take $E$ to correspond to $f_*\mathcal{L}$ and then the surjectivity also holds on the base but it doesn't always need to be true.)

The last step in proving (4) should be modified, since even in the case when base is a field there are examples in which $E \to f_\star L$ is not surjective, therefore the second map defined only on an open. Lukely we can chose the open as the complement of $P(coker\ E_j \to f_\star L)$, then show that image of map $r$ is contained in the chosen open.

I think "By Conditions (2), (3), and (4) imply that $B_\mathfrak{q}/\mathfrak{p}B_\mathfrak{q}=\kappa(\mathfrak{p})$. Algebra, Lemma..." should read instead "Conditions (2), (3), and (4) imply that $B_\mathfrak{q}/\mathfrak{p}B_\mathfrak{q}=\kappa(\mathfrak{p})$. By Algebra, Lemma..." -- I guess the "By" got cycled around.

What I meant is that to prove theorem tag. 05A9, one can directly prove that being a direct sum of countably generated modules if fpqc local, so this would shorten the argument. The link to Drinfeld's paper is not the right one, I've sent the right version to stacks.project@gmail.com

It appears to me that there is a typo. In the displayed equation in the paragraph just above Lemma 068K, the Tor's subscrip indices of the 2nd and the 3rd term should be n+m, not n.

There's a small error in the sketch: it says "Thus $\mathcal F$ if non-empty has maximal elements", but it should be "Thus if the set of ideals not in $\mathcal F$ is non-empty, it has maximal elements".

Is there a typo in the indices for Example 061Z? Homological spectral sequences have $d_r$ with bidegree $(-r,r-1)$, so $d_2$ should go from $E_{p,q}^2$ to $E_{p-2,q+1}^2$. This would be a map $Tor_q( H_p(K_\bullet), M) \rightarrow Tor_{q+1}(H_{p-2}(K_\bullet),M)$. It seems that the $E_1$ page in the same example does have the correct indices.

It may be worth pointing out that Laumon-Moret-Bailly seem to define quasiseparated to have separated diagonal (Definition 4.1). Some other places in the literature follow this convention, for example in Olsson's Log Geometry and Algebraic Stacks Remark 3.17, he explicitly states $Log_S$ is not quasiseparated even though his Theorem 3.2 shows that its diagonal is quasicompact and quasiseparated (a consequence of his notion of "locally separated," which is the stacks project's notion + quasiseparated).

I believe the first assertion in 00L3 was already shown in 0BUR.

In this section, it would be helpful to add the construction of the Karoubi envelope of a category.