The Stacks project

40.15 Descending ind-quasi-affine morphisms

Ind-quasi-affine morphisms were defined in More on Morphisms, Section 37.60. This section is the analogue of Descent, Section 35.35 for ind-quasi-affine-morphisms.

Let $X$ be a quasi-separated scheme. Let $E \subset X$ be a subset which is an intersection of a nonempty family of quasi-compact opens of $X$. Say $E = \bigcap _{i \in I} U_ i$ with $U_ i \subset X$ quasi-compact open and $I$ nonempty. By adding finite intersections we may assume that for $i, j \in I$ there exists a $k \in I$ with $U_ k \subset U_ i \cap U_ j$. In this situation we have
\begin{equation} \label{more-groupoids-equation-sections-of-intersection} \Gamma (E, \mathcal{F}|_ E) = \mathop{\mathrm{colim}}\nolimits \Gamma (U_ i, \mathcal{F}|_{U_ i}) \end{equation}

for any sheaf $\mathcal{F}$ defined on $X$. Namely, fix $i_0 \in I$ and replace $X$ by $U_{i_0}$ and $I$ by $\{ i \in I \mid U_ i \subset U_{i_0}\} $. Then $X$ is quasi-compact and quasi-separated, hence a spectral space, see Properties, Lemma 28.2.4. Then we see the equality holds by Topology, Lemma 5.24.7 and Sheaves, Lemma 6.29.4. (In fact, the formula holds for higher cohomology groups as well if $\mathcal{F}$ is abelian, see Cohomology, Lemma 20.19.2.)

Lemma 40.15.1. Let $X$ be an ind-quasi-affine scheme. Let $E \subset X$ be an intersection of a nonempty family of quasi-compact opens of $X$. Set $A = \Gamma (E, \mathcal{O}_ X|_ E)$ and $Y = \mathop{\mathrm{Spec}}(A)$. Then the canonical morphism

\[ j : (E, \mathcal{O}_ X|_ E) \longrightarrow (Y, \mathcal{O}_ Y) \]

of Schemes, Lemma 26.6.4 determines an isomorphism $(E, \mathcal{O}_ X|_ E) \to (E', \mathcal{O}_ Y|_{E'})$ where $E' \subset Y$ is an intersection of quasi-compact opens. If $W \subset E$ is open in $X$, then $j(W)$ is open in $Y$.

Proof. Note that $(E, \mathcal{O}_ X|_ E)$ is a locally ringed space so that Schemes, Lemma 26.6.4 applies to $A \to \Gamma (E, \mathcal{O}_ X|_ E)$. Write $E = \bigcap _{i \in I} U_ i$ with $I \not= \emptyset $ and $U_ i \subset X$ quasi-compact open. We may and do assume that for $i, j \in I$ there exists a $k \in I$ with $U_ k \subset U_ i \cap U_ j$. Set $A_ i = \Gamma (U_ i, \mathcal{O}_{U_ i})$. We obtain commutative diagrams

\[ \xymatrix{ (E, \mathcal{O}_ X|_ E) \ar[r] \ar[d] & (\mathop{\mathrm{Spec}}(A), \mathcal{O}_{\mathop{\mathrm{Spec}}(A)}) \ar[d] \\ (U_ i, \mathcal{O}_{U_ i}) \ar[r] & (\mathop{\mathrm{Spec}}(A_ i), \mathcal{O}_{\mathop{\mathrm{Spec}}(A_ i)}) } \]

Since $U_ i$ is quasi-affine, we see that $U_ i \to \mathop{\mathrm{Spec}}(A_ i)$ is a quasi-compact open immersion. On the other hand $A = \mathop{\mathrm{colim}}\nolimits A_ i$. Hence $\mathop{\mathrm{Spec}}(A) = \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Spec}}(A_ i)$ as topological spaces (Limits, Lemma 32.4.6). Since $E = \mathop{\mathrm{lim}}\nolimits U_ i$ (by Topology, Lemma 5.24.7) we see that $E \to \mathop{\mathrm{Spec}}(A)$ is a homeomorphism onto its image $E'$ and that $E'$ is the intersection of the inverse images of the opens $U_ i \subset \mathop{\mathrm{Spec}}(A_ i)$ in $\mathop{\mathrm{Spec}}(A)$. For any $e \in E$ the local ring $\mathcal{O}_{X, e}$ is the value of $\mathcal{O}_{U_ i, e}$ which is the same as the value on $\mathop{\mathrm{Spec}}(A)$.

To prove the final assertion of the lemma we argue as follows. Pick $i, j \in I$ with $U_ i \subset U_ j$. Consider the following commutative diagrams

\[ \xymatrix{ U_ i \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(A_ i) \ar[d] \\ U_ i \ar[r] & \mathop{\mathrm{Spec}}(A_ j) } \quad \quad \xymatrix{ W \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(A_ i) \ar[d] \\ W \ar[r] & \mathop{\mathrm{Spec}}(A_ j) } \quad \quad \xymatrix{ W \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(A) \ar[d] \\ W \ar[r] & \mathop{\mathrm{Spec}}(A_ j) } \]

By Properties, Lemma 28.18.5 the first diagram is cartesian. Hence the second is cartesian as well. Passing to the limit we find that the third diagram is cartesian, so the top horizontal arrow of this diagram is an open immersion. $\square$

Lemma 40.15.2. Suppose given a cartesian diagram

\[ \xymatrix{ X \ar[d]_ f \ar[r] & \mathop{\mathrm{Spec}}(B) \ar[d] \\ Y \ar[r] & \mathop{\mathrm{Spec}}(A) } \]

of schemes. Let $E \subset Y$ be an intersection of a nonempty family of quasi-compact opens of $Y$. Then

\[ \Gamma (f^{-1}(E), \mathcal{O}_ X|_{f^{-1}(E)}) = \Gamma (E, \mathcal{O}_ Y|_ E) \otimes _ A B \]

provided $Y$ is quasi-separated and $A \to B$ is flat.

Proof. Write $E = \bigcap _{i \in I} V_ i$ with $V_ i \subset Y$ quasi-compact open. We may and do assume that for $i, j \in I$ there exists a $k \in I$ with $V_ k \subset V_ i \cap V_ j$. Then we have similarly that $f^{-1}(E) = \bigcap _{i \in I} f^{-1}(V_ i)$ in $X$. Thus the result follows from equation ( and the corresponding result for $V_ i$ and $f^{-1}(V_ i)$ which is Cohomology of Schemes, Lemma 30.5.2. $\square$

Lemma 40.15.3 (Gabber). Let $S$ be a scheme. Let $\{ X_ i \to S\} _{i\in I}$ be an fpqc covering. Let $(V_ i/X_ i, \varphi _{ij})$ be a descent datum relative to $\{ X_ i \to S\} $, see Descent, Definition 35.31.3. If each morphism $V_ i \to X_ i$ is ind-quasi-affine, then the descent datum is effective.

Proof. Being ind-quasi-affine is a property of morphisms of schemes which is preserved under any base change, see More on Morphisms, Lemma 37.60.6. Hence Descent, Lemma 35.33.2 applies and it suffices to prove the statement of the lemma in case the fpqc-covering is given by a single $\{ X \to S\} $ flat surjective morphism of affines. Say $X = \mathop{\mathrm{Spec}}(A)$ and $S = \mathop{\mathrm{Spec}}(R)$ so that $R \to A$ is a faithfully flat ring map. Let $(V, \varphi )$ be a descent datum relative to $X$ over $S$ and assume that $V \to X$ is ind-quasi-affine, in other words, $V$ is ind-quasi-affine.

Let $(U, R, s, t, c)$ be the groupoid scheme over $S$ with $U = X$ and $R = X \times _ S X$ and $s$, $t$, $c$ as usual. By Groupoids, Lemma 39.21.3 the pair $(V, \varphi )$ corresponds to a cartesian morphism $(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoid schemes. Let $u' \in U'$ be any point. By Groupoids, Lemmas 39.19.2, 39.19.3, and 39.19.4 we can choose $u' \in W \subset E \subset U'$ where $W$ is open and $R'$-invariant, and $E$ is set-theoretically $R'$-invariant and an intersection of a nonempty family of quasi-compact opens.

Translating back to $(V, \varphi )$, for any $v \in V$ we can find $v \in W \subset E \subset V$ with the following properties: (a) $W$ is open and $\varphi (W \times _ S X) = X \times _ S W$ and (b) $E$ an intersection of quasi-compact opens and $\varphi (E \times _ S X) = X \times _ S E$ set-theoretically. Here we use the notation $E \times _ S X$ to mean the inverse image of $E$ in $V \times _ S X$ by the projection morphism and similarly for $X \times _ S E$. By Lemma 40.15.2 this implies that $\varphi $ defines an isomorphism

\begin{align*} \Gamma (E, \mathcal{O}_ V|_ E) \otimes _ R A & = \Gamma (E \times _ S X, \mathcal{O}_{V \times _ S X}|_{E \times _ S X}) \\ & \to \Gamma (X \times _ S E, \mathcal{O}_{X \times _ S V}|_{X \times _ S E}) \\ & = A \otimes _ R \Gamma (E, \mathcal{O}_ V|_ E) \end{align*}

of $A \otimes _ R A$-algebras which we will call $\psi $. The cocycle condition for $\varphi $ translates into the cocycle condition for $\psi $ as in Descent, Definition 35.3.1 (details omitted). By Descent, Proposition 35.3.9 we find an $R$-algebra $R'$ and an isomorphism $\chi : R' \otimes _ R A \to \Gamma (E, \mathcal{O}_ V|_ E)$ of $A$-algebras, compatible with $\psi $ and the canonical descent datum on $R' \otimes _ R A$.

By Lemma 40.15.1 we obtain a canonical “embedding”

\[ j : (E, \mathcal{O}_ V|_ E) \longrightarrow \mathop{\mathrm{Spec}}(\Gamma (E, \mathcal{O}_ V|_ E)) = \mathop{\mathrm{Spec}}(R' \otimes _ R A) \]

of locally ringed spaces. The construction of this map is canonical and we get a commutative diagram

\[ \xymatrix{ & E \times _ S X \ar[rr]_\varphi \ar[ld] \ar[rd]^{j'} & & X \times _ S E \ar[rd] \ar[ld]_{j''} \\ E \ar[rd]^ j & & \mathop{\mathrm{Spec}}(R' \otimes _ R A \otimes _ R A) \ar[ld] \ar[rd] & & E \ar[ld]_ j \\ & \mathop{\mathrm{Spec}}(R' \otimes _ R A) \ar[rd] & & \mathop{\mathrm{Spec}}(R' \otimes _ R A) \ar[ld] \\ & & \mathop{\mathrm{Spec}}(R') } \]

where $j'$ and $j''$ come from the same construction applied to $E \times _ S X \subset V \times _ S X$ and $X \times _ S E \subset X \times _ S V$ via $\chi $ and the identifications used to construct $\psi $. It follows that $j(W)$ is an open subscheme of $\mathop{\mathrm{Spec}}(R' \otimes _ R A)$ whose inverse image under the two projections $\mathop{\mathrm{Spec}}(R' \otimes _ R A \otimes _ R A) \to \mathop{\mathrm{Spec}}(R' \otimes _ R A)$ are equal. By Descent, Lemma 35.10.6 we find an open $W_0 \subset \mathop{\mathrm{Spec}}(R')$ whose base change to $\mathop{\mathrm{Spec}}(A)$ is $j(W)$. Contemplating the diagram above we see that the descent datum $(W, \varphi |_{W \times _ S X})$ is effective. By Descent, Lemma 35.32.13 we see that our descent datum is effective. $\square$

Comments (2)

Comment #2763 by BCnrd on

To prove the instance of ( that is relevant in this proof, namely an inverse system (under reverse inclusions) of quasi-compact open subspaces of a spectral space (rather than some more abstract kind of inverse system), one does not need the extreme generality of the links given above to prove it. Namely, one can given a less notationally intensive argument using basic "spreading out" properties involving quasi-compact open subsets, pro-constructible subsets, and open covers thereof in spectral spaces. I mention this because when trying to follow the links given for the proof of (, one encounters a blizzard of notation which is unpleasant to unravel and too heavy for the actual situation relevant here. I found it easier to make my own proof of the relevant special case than to try to decipher whatever is going on in the given links. How about giving a Lemma here (or Corollary elsewhere) that focuses on the relevant special case of ( with quasi-compact open subsets in a spectral space and gives a direct proof in that case, saving the ready from wading through the extra generality and heavier notation in the links presently given?

Comment #2876 by on

@#2763. Yes, it would be fine to have this special case of Lemma 6.29.4 worked out in this particular case. On the other hand, the whole point of the Stacks project is to have general lemmas such as Lemmas 6.29.4 and 20.19.2 and to use them in multiple ways.

Another direction would be to collect results of the nature: for what kind of subsets of a topological space do we have the formula ? Besides the case mentioned in this section, there is the case discussed in Lemma 20.16.3 (used in the proof of proper base change in topology). Is there a common generalization?

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