The Stacks project

Nice. The condition that $M$ has an inclusion-minimal generating set is only used to ensure that $M$ either is trivial or has a nontrivial cyclic quotient (hence a simple quotient), no?

One way to understand the geometric content of Nakayama's lemma is as saying that (1) the support of a finite module is the vanishing locus of its annihilator and (2) the support of a (finitary) tensor product of finite modules is the intersection of their supports. From that perspective, the finiteness hypothesis arises naturally from that finite modules are precisely those which can be built up via extensions from cyclic modules (the proof of (1) and (2) proceeding inductively along such extensions).

It's not mine to say whether your proposed generalization is within the scope of Stacks. That said, it's interesting to note that neither (1) nor (2) above remains valid in the generality of $M$ merely having an inclusion-minimal generating set.

(A counterexample to (1) is $R := \mathbb{Z}$ and $M := \bigoplus _ { \text{positive prime } p \in \mathbb{Z}} \mathbb{Z} / (p)$; a counterexample to (2) is $R := k [ x , y ] _ { ( x , y ) }$ and $M := \left\langle \left( \frac { y } { x } \right) ^ n \right\rangle _ { n \in \mathbb{N} } \subset R [ x ^ { - 1 } ] / R$, with $\text{supp} ( M ) = \mathcal{V} ( (x) )$ but $\text{supp} ( M \otimes _ R M ) = \mathcal{V} ( ( x , y ) )$.)

After the proof of Lemma 65.14.3 and after Definition 65.14.4 it is claimed that condition (*) from the statement of Lemma 65.14.3 is equivalent to Groupoids, Definition 39.10.2. For the record, this is proven in #11588.

Another tentative addition to this section:

Lemma. Let $G$ be a group. Let $X$ be an $S$-scheme equipped with a $G_S$-action $a:G_S\times_SX\to X$. Then the action is free if and only if the following condition holds:

(*) For any point $x \in X$ and $g\in G$, if $g(x)=x$ and $g$ induces the trivial automorphism on $\kappa(x)$, then $g=e$ is the identity.

Proof. We will use the identification of $a$ with $G\to\operatorname{Aut}_S(X)$ explained in #11587.

($\Leftarrow$). It's done in the proof of Algebraic Spaces, Lemma 65.14.3.

($\Rightarrow$). Let $x$ and $g$ be as in the statement and consider the morphisms $\operatorname{Spec}(\kappa(x))\overset{g}{\underset{e}{\rightrightarrows}} G_S\times_SX\cong_S\coprod_{g'\in G}X$ given by the canonical morphism $\operatorname{Spec}(\kappa(x))\to X$ followed by the insertion into the $g$-th and $e$-th components. These two morphisms become equal after postcomposition by $G_S\times_SX\to X\times_SX$. But the latter morphism is monic (Lemma 39.10.3), hence $g=e$. $\square$

Maybe the following is worth to mention as a comment/remark/lemma. Let $G$ be a group. Let $S$ be a scheme. Let $X$ be an $S$-scheme. There is a bijective correspondence between $G_S$-actions on $X$ and group homomorphisms $G\to\operatorname{Aut}_S(X)$. Namely, since $G_S\cong\coprod_{g\in G}S$, giving a morphism $G_S\times_SX\to X$ is the same as giving a morphism $\coprod_{g\in G}X\to X$ over $S$, and the latter is the same as giving a map $G\to\operatorname{Hom}_S(X,X)$. It is not difficult to see that commutativity of the diagrams 39.10.1.1 amounts to the induced map $G\to\operatorname{Hom}_S(X,X)$ defining a group homomorphism $G\to\operatorname{Aut}_S(X)\subset\operatorname{Hom}_S(X,X)$.

In the comments after the proof of Lemma 65.14.3, to give a reference for the notation $G_S$ one could maybe link Groupoids, Example 39.5.6.

Regarding the terminology "localization functor" (appearing once in the statement and once in the proof): This is not used (nor defined) anywhere else, right? I don't know if "morphism of topoi" would be preferred. With the identification $j=j_{V/U}$ of Sites, Lemma 7.25.8 the most similar thing that comes up is "localization morphism" of Sites, Definition 7.25.1.

In the introduction, after reading "hence by the results of the previous sections we obtain a morphism of topoi $$j_U:Sh(\mathcal{C}/U)\to Sh(\mathcal{C}/U)$$ given by $j_U^{-1}$ and $j_{U*}$ as well as a functor $j_{U!}$" it took me a while to find out which are such results. Maybe one could say something like "see Lemma 7.21.5"? This is (I think) the first time lower shriek appears.

then the following are equivalent

If we assume condition (2) holds, then condition (1') is equivalent to "$u^p$ sends sheaves into sheaves." See SGA 4, Exp. III, Proposition 1.6.

One could also mention that by Lemma 7.13.3 the condition is equivalent to $u_s$ being left exact.

I agree with Haohao Liu one could insert a link to Categories, Definition 4.23.1.

For the reader's benefit, I want to record that given functors $u:\mathcal{C}\to\mathcal{D}$ and $\mathcal{F}:\mathcal{C}\to Set$, the functor $u_p\mathcal{F}:\mathcal{D}\to Set$ is the left Kan extension of $\mathcal{F}$ along $u$. The formula $u_p \mathcal{F}(V)=\operatorname{colim}_{\mathcal{I}_V^{\text {opp }}} \mathcal{F}_V$ is exactly [Rie, eq. (6.2.2)], and Lemma 7.5.4 is [Rie, Proposition 6.1.6].

References

[Rie] Emily Riehl, Category Theory in Context

This is EGA IV, 17.16.3 (ii).

In regards to:

Historically the first proof of the semistable reduction theorem for curves can be found in the paper [DM] by Deligne and Mumford. It proves the theorem by reducing the problem to the case of Abelian varieties which was already known at the time thanks to Grothendieck and others, see [SGA7-I] and [SGA7-II]). The semistable reduction theorem for abelian varieties uses the theory of Néron models which in turn rests on a treatment of birational group laws over a base.

This doesn' appear to be entirely accurate. The paper of Deligne–Mumford claims the result was known to Grothendieck and Mumford, but actually supplies no reference. SGA7 doesn't really do it either. The heart of the proof in SGA7-I, Exp IX, Thm 3.6, appearing at the top of p.35 (just above Cor 3.8) provides the isomorphism of the dual Tate module with étale cohomology, the latter for which "cette assertion a été prouvée dans III (en utilisant le résolution des singularités des schémas excellents de dimension 2)." This appears to me to be circling back to curves! And although Néron models are in the picture, they don't seem to be they key ingredient. FWIW, Exp III just referenced doesn't even appear in the volume, and SGA7-II does not seem to cover related material.

Is there some other known (published) argument of Grothendieck that works directly on the level of abelian varieties and does not go through curves?

It would be nice if it was stated somewhere that the map $r_{\mathcal{L}, \psi}$ coincides with the map $r_{\psi}$ in https://stacks.math.columbia.edu/tag/07ZG . Namely, https://stacks.math.columbia.edu/tag/01O8 gives us a canonical morphism

$$r_{\mathcal{L}, \psi}\colon U(\psi) \to \underline{\text{Proj}}_S(\mathcal{A})$$

and https://stacks.math.columbia.edu/tag/07ZG gives us a canonical morphism

$$r_{\psi}\colon U(\psi)' \to \underline{\operatorname{Proj}}_X(f^* \mathcal{A}).$$

I think $U(\psi) = U(\psi)'$ and it would be nice if there was a lemma saying that $r_{\mathcal{L}, \psi}$ coincides with the composition of $r_{\psi}$ and the projection

$$\underline{\operatorname{Proj}}_X(f^* \mathcal{A}) \to \underline{\operatorname{Proj}}_S(\mathcal{A}).$$

It seems that this might be implicitly used in the proof of https://stacks.math.columbia.edu/tag/01VR part (4).

Strictly speaking, the first sentence of the section is incorrect: namely, G-unramified maps are called "unramified" in EGA.

More generally: If $X$ is étale over $S$ and $Y$ is unramified over $S$, then $X\to Y$ is étale. See Görtz, Wedhorn, Algebraic Geometry II, Springer, Remark 18.30.(2).

In the first paragraph of the proof of Lemma 37.43.3, should “finite quasi-coherent O_X-algebras” be “finite quasi-coherent O_S-algebras”?

I think there is a shorter proof of a more general statement.

Theorem: Let $R$ be a ring. Suppose an $R$-module $M$ admits a minimal generating set (with respect to inclusion). If $IM=M$ for every annihilator $I$ of a simple $R$-module, then $M=0$.

Proof: By contradiction. If $M\neq 0$, the minimal generating set contains at least one element $m\neq 0$. By Zorn there exists a maximal submodule $N\subset M$ not containing $m$ but containing all other generators from our minimal generating set. The quotient $M/N$ has to be simple. If we let $I$ be its annihilator, then $IM\subset N\neq M$. QED

I think there is a shorter proof of a more general statement.

Theorem: Let $R$ be a ring. Suppose an $R$-module $M$ admits a minimal generating set (with respect to inclusion). If $IM=M$ for every annihilator $I$ of a simple $R$-module, then $M=0$.

Proof: If $M\neq 0$, the minimal generating contains at least one element $m\neq 0$. By Zorn there exists a maximal submodule $N\subset M$ not containing $m$. The quotient $M/N$ has to be simple. If we let $I$ be its annihilator, then $IM\subset N\neq M$. Contradiction. QED