The Stacks project

This is probably because I first looked at EGA I and it wasn't in there (it got introduced in the second edition of EGA I and I prefer to use the first edition). Then I looked in some of the other books in my office and settled on this choice. Your and Grothendieck's choice would have been better (fitting better with the philosphy in the stacks project), but now it's too late as I have found a few references to this section in arXiv papers, at least 1 of which is about to be published.

OK, I replaced the current link on the bibiography page by the archived link you sent as it seems the most stable. However, I do not know if the authors are happy to have this version of their work posted or made public to a wider audience so I inserted a note to that effect. Thanks!

Thanks and fixed here.

Same remark for Lemma 05NJ.

For example one could replace "injective" by "$F$-acyclic" in Definition 13.21.1. Then a different mechanism/property of the situation would be needed to prove the existence. For example, note that in the steps (2), (3) we need something slightly weaker than Lemma 13.18.9. So I think having a collection of objects as in Lemma 13.15.6 would work.

I do not understand your comment, because we always have $\{\text{id} : U \to U\}$ and this is a covering by definition.

Thanks! Fixed here.

Sorry for the late realization that you pointed out an actual error in the proof. I've fixed it without adding too much text by choosing the "correct" isomorphisms of $G(A)[1]$ with $G(A[1])$. See this commit.

I think in the very last sentence it should say every nonzero element of $A$ can be written as $u \pi^n$...

Dear Jay, I think the missing part is in Expose I of SGA 7-I, especially in remarque 3.6. The circling back to curves is still true, but it is only using resolution of models of curves over dvrs and not semi-stable reduction. So it is not a loop.

I think using https://stacks.math.columbia.edu/tag/0D60 you can directly get that $H^q(U,K) = \lim H^0(U,H^q(K_n))$ using that $H^q(U,K_n) =H^0(U,H^q(K_n))$.

I think there may be a typo in the statement of Lemma 0520.

The conclusion currently reads

$$\varphi^{-1}(\mathfrak{q}S_{g})=(\mathfrak{p},x_{r+1},\ldots, x_{n})R_{f}[x_{r+1},\ldots, x_{n}].$$

Since the left-hand side is an ideal of $R_{f}[x_1,\ldots,x_n]$, shouldn't the right-hand side instead be

$$(\mathfrak{p},x_{r+1},\ldots, x_{n})R_{f}[x_{1},\ldots, x_{n}]?$$

I would understand the geometric content slightly differently. Suppose $M$ is a finitely generated $R$-module and $I M=M$ for an ideal $I\subset R$. Then $\mathfrak p M_{\mathfrak p}=M_{\mathfrak p}$ for every prime ideal $\mathfrak p$ above $I$ and hence $M_{\mathfrak p}=0$, since otherwise $M_{\mathfrak p}$ would have a simple quotient which necessarily is killed by ${\mathfrak p}$. So the support of $M$ is disjoint to the closed set of prime ideals above $a$. Again by finite generation, the support is the set of prime ideals containing the annihilator $\operatorname{Ann}(M)$. So there are no prime ideals above $I$ and $\operatorname{Ann}(M)$, which means their sum is the whole ring $R$. In particular, $1=f+a$ with $a\in I$ and $f$ in the annihilator of $M$.

The following result might fit to this section (it is the analogous result to Properties of Algebraic Spaces, Lemma 66.7.1, but for properties of morphisms). It is stated without proof in Olsson's Algebraic Spaces and Stacks, 5.4.3.

Lemma. Let $\mathcal{P}$ be a property of morphisms of schemes satisfying:

1. is preserved under any base change, see Schemes, Definition 26.18.3, and

2. is étale local on the base, see Descent, Definition 35.22.1.

(This is almost like the hypotheses from Algebraic Spaces, Definition 65.5.1 but we require étale locality on the base, rather than fppf locality.) Let $S$ be a scheme and let $f:X\to Y$ be a representable morphism of algebraic spaces over $S$. The following are equivalent:

1. For some scheme $U$ and surjective étale morphism $U\to Y$, the morphism of schemes $X\times_YU\to U$ has property $\mathcal{P}$.

2. For every scheme $U$ and every morphism $U\to Y$, the morphism of schemes $X\times_YV\to V$ has property $\mathcal{P}$. (That is, $f$ has $\mathcal{P}$ as in Algebraic Spaces, Definition 65.5.1.)

Proof. The implication (2) $\Rightarrow$ (1) is immediate. For the converse, choose a surjective étale morphism $U\to Y$ with $U$ a scheme such that $X\times_YU\to U$ has $\mathcal{P}$ and let $V$ be a $Y$-scheme. Let $W=V\times_YX\times_YU$. That is, every face in this commutative cube is cartesian. Note that every vertex of the cube other than $X$ and $Y$ is a scheme. In the cube diagram, since the top square is cartesian and $\mathcal{P}$ is preserved under base-change, $W\to U\times_YV$ has $\mathcal{P}$. On the other hand, $U\times_YV\to V$ is étale surjective. Thus, since $\mathcal{P}$ is étale local on the base, we conclude that $X\times_YV\to V$ has $\mathcal{P}$, as desired. $\square$

In item (3) of the lemma, $R$-module maps $\varphi_\lambda, \psi_\lambda: M_\lambda \rightarrow N_\lambda$ should be $R_\lambda$-module maps $\varphi, \psi: M_\lambda \rightarrow N_\lambda$.

Point 5 is "add more here", and should probably be removed.

The notation $G_1\times G_2$ here should be $(G_1,G_2)$

In loc. cit. we find a fifth equivalent condition: for every field $K$, there is an algebraically closed field extension $K'$ of $K$ such that the induced map $\operatorname{Mor}(\operatorname{Spec}(K'),X) \to\operatorname{Mor}(\operatorname{Spec}(K'),Y)$ is injective.

Also, in Altman, Kleiman, Introduction to Grothendieck Duality Theory, p. 119, there is yet another additional equivalent condition: for any field $K$ and any morphism $\operatorname{Spec}(K)\to Y$, the morphism $f_K:X\times_Y\operatorname{Spec}(K)\to\operatorname{Spec}(K)$ is injective.

This result is EGA I, 2nd ed., Ch. I, Proposition 3.7.1.

Is the third sentence of the last lemma missing the rank "$r$"?