The Stacks project

@#9911: As explained in [GW, 21.129], this is [Lip, 3.7.2.i; Del, 2.1.3].

References

[GW]. Ulrich Görtz and Torsten Wedhorn. Algebraic geometry II: cohomology of schemes. With examples and exercises. Springer Stud. Math. – Master. Wiesbaden: Springer Spektrum, 2023

[Lip]. Joseph Lipman. Notes on derived functors and Grothendieck duality. In Foundations of Grothendieck duality for diagrams of schemes, volume 1960 of Lect. Notes Math. Berlin: Springer, 2009

[Del]. Pierre Deligne. “Cohomologie à supports propres”. In: Théorie de Topos et Cohomologie Étale des Schémas (SGA4). Vol. III, Exp. XVII. Lecture Notes in Math. 305. Springer, 1971, pp. 250–461

Lemma 002K (lemma-functorial-colimit) is somewhat overkill here. Suggested proof (to get rid of the "omit"):

1. Wait for material on Karoubi envelopes, in particular their (2-)universal property, to be added to Section 09SF (section-karoubian) (cf. Comment 8065).

2. Observe, as a consequence of the aforementioned (2-)universal property, that if $\mathcal{C} \subset \mathcal{D}$ is fully faithful and $\mathcal{C}$ is Karoubian, then a (finite) biproduct of objects in $\mathcal{D}$ lies in $\mathcal{C}$ iff each biproductand lies in $\mathcal{C}$. (This probably also belongs in Section 09SF (section-karoubian).)

3. Observe that $\text{coCones} ( c ; F \oplus G ) \simeq \text{coCones} ( c ; F ) \oplus \text{coCones} ( c ; G )$ where $\text{coCones} ( - ; X ) : \mathcal{C} \to \textit{Ab}$ is the functor mapping $c \in \text{Ob} (\mathcal{C})$ to the Abelian group of cones under $X$ with vertex $c$.

4. Conclude by that $\mathcal{C} ^ {\text{opp}} \subset \text{Fun} ( \mathcal{C} , \textit{Ab} )$ is a Karoubian fully faithful subcategory (implicitly using that Karoubianness is a self-dual notion) that $\text{coCones} ( c ; F \oplus G )$ is "representable" iff $\text{coCones} ( c ; F )$ and $\text{coCones} ( c ; G )$ are "representable", hence that $\text{colim} ( F \oplus G )$ exists iff $\text{colim} (F)$ and $\text{colim} (G)$ exist.

("Representable" is in scare quotes because it seems that representability/Yoneda aren't treated within Stacks separately in the $\textit{Ab}$-enriched setting. Of course, the relevant results/proofs are vritually the same as in the case of $\textit{Sets}$. As an expositional question I'm not sure what the most parsimonious way to navigate this is.)

An underrated lemma imho. Below is a suggested "abstract abstract nonsense nonsense" style proof. I.e., rather than explicitly deal in "abstract nonsense" (the data of cocones etc.), we instead reduce the claim to symbol-pushing in $\textit{Cat}$ (without invoking $\textit{Cat}$ or any other $2$-categorical notions explicitly).

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Dependencies:

The following would ideally be included as definitions/lemmata in an earlier section, presumably Section 0013 (section-definition-categories).

1. Given categories $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$, define the usual bifunctor and show the bifunctoriality of $$( \circ , \star ) : \text{Fun}( \mathcal{Y} , \mathcal{Z} ) \times \text{Fun}( \mathcal{X} , \mathcal{Y} ) \to \text{Fun}( \mathcal{X} , \mathcal{Z} ) .$$

2. Given categories $\mathcal{X}$ and $\mathcal{Y}$, define the constant functor $$\text{const} : \mathcal{Y} \to \text{Fun}( \mathcal{X} , \mathcal{Y} )$$ as $$\text{const} : y \mapsto ( x \mapsto y )$$ $$\text{const} : \psi \mapsto ( x \mapsto \psi )$$ on objects and morphisms respectively and show that for any functor $F$ and natural transformation $f$ for which the respective compositions are well-defined, $$\text{const} (y) \circ F = \text{const} (y)$$ $$\text{const} (\psi) \star t = \text{const} (\psi) .$$ (There are also formulae for composition on the other side, but they aren't relevant to what follows.)

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Argument:

Given category $\mathcal{X}$ and diagam $F : \mathcal{X} \to \mathcal{C}$, define the (covariant) functor $$\text{coCones}( - ; F) : \mathcal{C} \to \textit{Sets}$$ as $$c \mapsto \text{Mor} _ { \text{Fun}( \mathcal{X} , \mathcal{C} ) } ( F , \text{const} (c) )$$ $$\phi \mapsto \text{Mor} _ { \text{Fun}( \mathcal{X} , \mathcal{C} ) } ( \text{id} _ F , \text{const} (\phi) ) .$$ I.e., $\text{coCones}( c ; F)$ is the set of cocones under $F$ with vertex $c$ equipped with the corresponding functorial action by morphisms of $\mathcal{C}$ realized as the set of natural transformations from $F$ to the constant functor at $c$.

By the prior discussion in Section 002D (section-limits), $\text{colim} (F)$ is defined as the corepresenting object of $\text{coCones}( - ; F)$ (precisely) when the latter exists. Hence by (the formal dual of) Lemma 001P (lemma-yoneda), it suffices in the context of the current lemma to construct a natural transformation $$\text{coCones}( - ; N) \to \text{coCones}( - ; M)$$ (note the implicit contravariance).

Indeed, the formula $$k \mapsto ( k \star \text{id} _ H ) \circ t$$ prescribes the components (the parametrizing $c \in \text{Ob} (\mathcal{C})$ implicit in the type of $k : N \to \text{const} (c)$) of such a natural transformation; naturality is given by the equation $$( k \mapsto ( ( \text{const} (\phi) \circ k ) \star \text{id} _ H ) \circ t ) = ( k \mapsto \text{const} (\phi) \circ ( k \star \text{id} _ H ) \circ t )$$ or equivalently that (for all $k$) $$( ( \text{const} (\phi) \circ k ) \star \text{id} _ H ) \circ t = \text{const} (\phi) \circ ( k \star \text{id} _ H ) \circ t$$ where the latter follows purely symbolically from the interchange law and the relevant identity above.

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Note/strenghtening:

As currently formulated, this lemma is prima facie "evil" in the sense that it constructs a morphism (distinguishable up to equality) in terms of a functor (distinguishable only up to isomorphism). The underlying phenomenon is that there is, given $\mathcal{C}$, a (strict) $2$-category $\text{coDiag} _ {\mathcal{C}}$ defined so that

• the objects of $\text{coDiag} _ {\mathcal{C}}$ are the pairs $( \mathcal{I} : \textit{Cat} , M : \mathcal{I} \to \mathcal{C} )$

• the morphisms from $( \mathcal{I} , M )$ to $( \mathcal{J} , N )$ are the pairs $( H : \mathcal{I} \to \mathcal{J} , t : M \to N \circ H )$

• the morphisms from $( G , s )$ to $( H , t )$ are the pairs $( r : G \to H , ( \text{id} _ N \star r ) \circ s = t )$

and we will have turned out to have, via the above formula, in fact constructed a (strict) $2$-functor $$\text{coCones} : \text{coDiag} _ {\mathcal{C}} \to \text{Fun} ( \mathcal{C} , \textit{Sets} )$$ where the codomain is a $1$-category so all the $2$-morphisms of $\text{coDiag} _ {\mathcal{C}}$ (including the noninvertible ones!) are sent to equalities in $\text{Fun} ( \mathcal{C} , \textit{Sets} )$.

The proof is just more symbol pushing: Given $( r , ( \text{id} _ N \star r ) \circ s = t ) : ( G , s ) \to ( H , t )$ as above, we must show that $$( k \mapsto ( k \star \text{id} _ G ) \circ s ) = ( k \mapsto ( k \star \text{id} _ H ) \circ t )$$ or equivalently that (for all $k$) $$( k \star \text{id} _ G ) \circ s = ( k \star \text{id} _ H ) \circ t .$$ This follows from the equality in the hypothesis, the interchange law, and the equality $$k \star r = ( \text{const} ( \text{id} _ c ) \circ k ) \star ( r \circ \text{id} _ G ) = k \star \text{id} _ G$$ where $c$ is the object of $\mathcal{C}$ at whose component we're implicitly working.

(Of course, it doesn't make expository sense to invoke the language of $2$-categories so early in the chapter just to state all this. But what can be said purely $1$-categorically is that any two $( G , s )$ and $( H , t )$ related by an $(r , ( \text{id} _ N \star r ) \circ s = t )$ induce equal maps between colimits.)

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Ps.:

The above discussion formally dualizes to the corresponding discussion concerning limits (possibly relevant to Stacks) and straightforwardly generalizes to account for weighted (co)limits (presumably outside the scope of Stacks). As always, I am happy to implement and submit the desired subset of the above as a patch/PR.

Just as RHom is being defined, there seems to be a typo: "Then we set Then we set".

Re Comment #12259, the idea is presumably to take the multiplicative closure of the union.

One way to shore up the slight handwave at the end is to observe that given diagrams $F, G : \mathcal{I} \to R \text{-alg}$ and natural transformation $T : F \to G$ such that for all $i \in \text{Ob} (\mathcal{I})$ there exists a (multiplicative) subset $S _ i \subset F (i)$ such that the component $T _ i : F (i) \to G (i)$ is the (up to unique iso) localization of $F (i)$ by $S _ i$, then the map $\text{colim} (T) : \text{colim} (F) \to \text{colim} (G)$ is itself the (up to unique iso) localization of $\text{colim} (F)$ by (the multiplicative subset generated by) the union over $i \in \text{Ob} (\mathcal{I})$ of the image of $S _ i$ under the relevant component $K _ i : F (i) \to \text{colim} (F)$ of the universal cocone of $F$.

This is (assuming I haven't made some mistake) not at all hard to show by straightforward universal property wrangling; at the moment I can't think of any cleverer argument.

05C6 shows this inclusion when $\phi$ is finite. Proposition II/3.2 of Lazard's, Autour de la platitude (https://www.numdam.org/articles/10.24033/bsmf.1675/) also shows that it holds when $\phi$ is flat.

In Definition 4.8.2, it is better to replace $h_U\times_{G}F$ by $h_U\times_{\xi,G,a}F$. Although it is clear from context of the relevence to $\xi$ and $a$, it is pedagogically helpful to see it in the notation. The full notation of fiber product is used rest of this section.

Typo in the third paragraph last line : ...it suffices to prove $R[x] _ \mathfrak q^\wedge \otimes _ {R[x]} \kappa (\mathfrak r)$ is geometrically regular over "$\kappa (\mathfrak r)$" when, in addition...

I don't understand why $O_{X_{k'}, x'} \simeq O_{X, x} \otimes_k k'$.

Why is $\cup S_i$ multiplicatively closed? If $Q_i$ corresponds to the extension $k_i$ and $Q_j$ to $k_j$, it's not obvious that $Q_i Q_j \in \cup S_i$.

This might not be worth the trouble, but since universally closed implies quasicompact, it seems cleaner to phrase Prop01KF as universally closed iff quasicompact + existence part of valuative criterion

Is there a reason these two Lemmas aren't just one Lemma saying separated is equivalent to quasi-separated plus the valuative criterion? (i.e. doesn't separated imply quasiseparated?)

$X$ should be $A$ in the proof.

I suggest replacing "let $C \subset X$ be a proper curve contained in a fiber" with "let $C$ be a proper curve contained in a fiber", as the inclusion $C \subset X$ is, I believe, only meant set-theoretically, but might be perceived as scheme-theoretic (which would merit further justification but is also not needed).

The composition $a\circ b$ is an endomorphism of $B^\bullet$. Therefore, it should require that $a\circ b$ is homotopic to $\mathrm{id}_B$. Similiarly for $b\circ a$.

The composition $a\circ b$ is an endomorphism of $B_\bullet$. Therefore, it should require that $a\circ b$ is homotopic to $\mathrm{id}_B$. Similiarly for $b\circ a$.

↑ sorry $\mathrm{End}_R(E,E)$ ?? typo? $\mathrm{End}_R(E)$ ?? (assertion (3) and l.2 in the proof)

$\End_R(E,E)$ ?? typo? $\End_R(E)$ ?? (assertion (3) and l.2 in the proof)

In the first paragraph of the proof: Is the functor $QCoh(\mathcal{O}_\mathcal{X})\rightarrow QCoh(\mathcal{O}_U)$ really the restriction? If so, I don't think this should be essentially surjective e.g. take $\mathcal{X}$ affine and $j:U = \mathcal{X}\coprod \mathcal{X}\rightarrow \mathcal{X}$. You probably need to require that $j$ is an open immersion.

should say $R\to A$ is filtered colimit of ... instead of $A$ is ...