The Stacks project

78.5 Formal glueing of quasi-coherent modules

This section is the analogue of More on Algebra, Section 15.82. In the case of morphisms of schemes, the result can be found in the paper by Joyet [Joyet]; this is a good place to start reading. For a discussion of applications to descent problems for stacks, see the paper by Moret-Bailly [MB]. In the case of an affine morphism of schemes there is a statement in the appendix of the paper [Ferrand-Raynaud] but one needs to add the hypothesis that the closed subscheme is cut out by a finitely generated ideal (as in the paper by Joyet) since otherwise the result does not hold. A generalization of this material to (higher) derived categories with potential applications to nonflat situations can be found in [Section 5, Bhatt-Algebraize].

We start with a lemma on abelian sheaves supported on closed subsets.

Lemma 78.5.1. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let $Z \subset X$ closed subspace such that $f^{-1}Z \to Z$ is integral and universally injective. Let $\overline{y}$ be a geometric point of $Y$ and $\overline{x} = f(\overline{y})$. We have

\[ (Rf_*Q)_{\overline{x}} = Q_{\overline{y}} \]

in $D(\textit{Ab})$ for any object $Q$ of $D(Y_{\acute{e}tale})$ supported on $|f^{-1}Z|$.

Proof. Consider the commutative diagram of algebraic spaces

\[ \xymatrix{ f^{-1}Z \ar[r]_{i'} \ar[d]_{f'} & Y \ar[d]_ f \\ Z \ar[r]^ i & X } \]

By Cohomology of Spaces, Lemma 66.9.4 we can write $Q = Ri'_*K'$ for some object $K'$ of $D(f^{-1}Z_{\acute{e}tale})$. By Morphisms of Spaces, Lemma 64.53.7 we have $K' = (f')^{-1}K$ with $K = Rf'_*K'$. Then we have $Rf_*Q = Rf_*Ri'_*K' = Ri_*Rf'_*K' = Ri_*K$. Let $\overline{z}$ be the geometric point of $Z$ corresponding to $\overline{x}$ and let $\overline{z}'$ be the geometric point of $f^{-1}Z$ corresponding to $\overline{y}$. We obtain the result of the lemma as follows

\[ Q_{\overline{y}} = (Ri'_*K')_{\overline{y}} = K'_{\overline{z}'} = (f')^{-1}K_{\overline{z}'} = K_{\overline{z}} = Ri_*K_{\overline{x}} = Rf_*Q_{\overline{x}} \]

The middle equality holds because of the description of the stalk of a pullback given in Properties of Spaces, Lemma 63.19.9. $\square$

Lemma 78.5.2. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let $Z \subset X$ closed subspace such that $f^{-1}Z \to Z$ is integral and universally injective. Let $\overline{y}$ be a geometric point of $Y$ and $\overline{x} = f(\overline{y})$. Let $\mathcal{G}$ be an abelian sheaf on $Y$. Then the map of two term complexes

\[ \left(f_*\mathcal{G}_{\overline{x}} \to (f \circ j')_*(\mathcal{G}|_ V)_{\overline{x}}\right) \longrightarrow \left(\mathcal{G}_{\overline{y}} \to j'_*(\mathcal{G}|_ V)_{\overline{y}}\right) \]

induces an isomorphism on kernels and an injection on cokernels. Here $V = Y \setminus f^{-1}Z$ and $j' : V \to Y$ is the inclusion.

Proof. Choose a distinguished triangle

\[ \mathcal{G} \to Rj'_*\mathcal{G}|_ V \to Q \to \mathcal{G}[1] \]

n $D(Y_{\acute{e}tale})$. The cohomology sheaves of $Q$ are supported on $|f^{-1}Z|$. We apply $Rf_*$ and we obtain

\[ Rf_*\mathcal{G} \to Rf_*Rj'_*\mathcal{G}|_ V \to Rf_*Q \to Rf_*\mathcal{G}[1] \]

Taking stalks at $\overline{x}$ we obtain an exact sequence

\[ 0 \to (R^{-1}f_*Q)_{\overline{x}} \to f_*\mathcal{G}_{\overline{x}} \to (f \circ j')_*(\mathcal{G}|_ V)_{\overline{x}} \to (R^0f_*Q)_{\overline{x}} \]

We can compare this with the exact sequence

\[ 0 \to H^{-1}(Q)_{\overline{y}} \to \mathcal{G}_{\overline{y}} \to j'_*(\mathcal{G}|_ V)_{\overline{y}} \to H^0(Q)_{\overline{y}} \]

Thus we see that the lemma follows because $Q_{\overline{y}} = Rf_*Q_{\overline{x}}$ by Lemma 78.5.1. $\square$

Lemma 78.5.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism. Let $\overline{x}$ be a geometric point of $X$ and let $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}) \to X$ be the canonical morphism. For a quasi-coherent module $\mathcal{G}$ on $Y$ we have

\[ f_*\mathcal{G}_{\overline{x}} = \Gamma (Y \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}), p^*\mathcal{F}) \]

where $p : Y \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}) \to Y$ is the projection.

Proof. Observe that $f_*\mathcal{G}_{\overline{x}} = \Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}), h^*f_*\mathcal{G})$ where $h : \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}) \to X$. Hence the result is true because $h$ is flat so that Cohomology of Spaces, Lemma 66.11.2 applies. $\square$

Lemma 78.5.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $i : Z \to X$ be a closed immersion of finite presentation. Let $Q \in D_\mathit{QCoh}(\mathcal{O}_ X)$ be supported on $|Z|$. Let $\overline{x}$ be a geometric point of $X$ and let $I_{\overline{x}} \subset \mathcal{O}_{X, \overline{x}}$ be the stalk of the ideal sheaf of $Z$. Then the cohomology modules $H^ n(Q_{\overline{x}})$ are $I_{\overline{x}}$-power torsion (see More on Algebra, Definition 15.81.1).

Proof. Choose an affine scheme $U$ and an étale morphism $U \to X$ such that $\overline{x}$ lifts to a geometric point $\overline{u}$ of $U$. Then we can replace $X$ by $U$, $Z$ by $U \times _ X Z$, $Q$ by the restriction $Q|_ U$, and $\overline{x}$ by $\overline{u}$. Thus we may assume that $X = \mathop{\mathrm{Spec}}(A)$ is affine. Let $I \subset A$ be the ideal defining $Z$. Since $i : Z \to X$ is of finite presentation, the ideal $I = (f_1, \ldots , f_ r)$ is finitely generated. The object $Q$ comes from a complex of $A$-modules $M^\bullet $, see Derived Categories of Spaces, Lemma 72.4.2 and Derived Categories of Schemes, Lemma 36.3.5. Since the cohomology sheaves of $Q$ are supported on $Z$ we see that the localization $M^\bullet _ f$ is acyclic for each $f \in I$. Take $x \in H^ p(M^\bullet )$. By the above we can find $n_ i$ such that $f_ i^{n_ i} x = 0$ in $H^ p(M^\bullet )$ for each $i$. Then with $n = \sum n_ i$ we see that $I^ n$ annihilates $x$. Thus $H^ p(M^\bullet )$ is $I$-power torsion. Since the ring map $A \to \mathcal{O}_{X, \overline{x}}$ is flat and since $I_{\overline{x}} = I\mathcal{O}_{X, \overline{x}}$ we conclude. $\square$

Lemma 78.5.5. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let $Z \subset X$ be a closed subspace. Assume $f^{-1}Z \to Z$ is an isomorphism and that $f$ is flat in every point of $f^{-1}Z$. For any $Q$ in $D_\mathit{QCoh}(\mathcal{O}_ Y)$ supported on $|f^{-1}Z|$ we have $Lf^*Rf_*Q = Q$.

Proof. We show the canonical map $Lf^*Rf_*Q \to Q$ is an isomorphism by checking on stalks at $\overline{y}$. If $\overline{y}$ is not in $f^{-1}Z$, then both sides are zero and the result is true. Assume the image $\overline{x}$ of $\overline{y}$ is in $Z$. By Lemma 78.5.1 we have $Rf_*Q_{\overline{x}} = Q_{\overline{y}}$ and since $f$ is flat at $\overline{y}$ we see that

\[ (Lf^*Rf_*Q)_{\overline{y}} = (Rf_*Q)_{\overline{x}} \otimes _{\mathcal{O}_{X, \overline{x}}} \mathcal{O}_{Y, \overline{y}} = Q_{\overline{y}} \otimes _{\mathcal{O}_{X, \overline{x}}} \mathcal{O}_{Y, \overline{y}} \]

Thus we have to check that the canonical map

\[ Q_{\overline{y}} \otimes _{\mathcal{O}_{X, \overline{x}}} \mathcal{O}_{Y, \overline{y}} \longrightarrow Q_{\overline{y}} \]

is an isomorphism in the derived category. Let $I_{\overline{x}} \subset \mathcal{O}_{X, \overline{x}}$ be the stalk of the ideal sheaf defining $Z$. Since $Z \to X$ is locally of finite presentation this ideal is finitely generated and the cohomology groups of $Q_{\overline{y}}$ are $I_{\overline{y}} = I_{\overline{x}}\mathcal{O}_{Y, \overline{y}}$-power torsion by Lemma 78.5.4 applied to $Q$ on $Y$. It follows that they are also $I_{\overline{x}}$-power torsion. The ring map $\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{Y, \overline{y}}$ is flat and induces an isomorphism after dividing by $I_{\overline{x}}$ and $I_{\overline{y}}$ because we assumed that $f^{-1}Z \to Z$ is an isomorphism. Hence we see that the cohomology modules of $Q_{\overline{y}} \otimes _{\mathcal{O}_{X, \overline{x}}} \mathcal{O}_{Y, \overline{y}}$ are equal to the cohomology modules of $Q_{\overline{y}}$ by More on Algebra, Lemma 15.82.2 which finishes the proof. $\square$

Situation 78.5.6. Here $S$ is a base scheme, $f : Y \to X$ is a quasi-compact and quasi-separated morphism of algebraic spaces over $S$, and $Z \to X$ is a closed immersion of finite presentation. We assume that $f^{-1}(Z) \to Z$ is an isomorphism and that $f$ is flat in every point $x \in |f^{-1}Z|$. We set $U = X \setminus Z$ and $V = Y \setminus f^{-1}(Z)$. Picture

\[ \xymatrix{ V \ar[r]_{j'} \ar[d]_{f|_ V} & Y \ar[d]^ f \\ U \ar[r]^ j & X } \]

In Situation 78.5.6 we define $\textit{QCoh}(Y \to X, Z)$ as the category of triples $(\mathcal{H}, \mathcal{G}, \varphi )$ where $\mathcal{H}$ is a quasi-coherent sheaf of $\mathcal{O}_ U$-modules, $\mathcal{G}$ is a quasi-coherent sheaf of $\mathcal{O}_ Y$-modules, and $\varphi : f^*\mathcal{H} \to \mathcal{G}|_ V$ is an isomorphism of $\mathcal{O}_ V$-modules. There is a canonical functor

78.5.6.1
\begin{equation} \label{spaces-pushouts-equation-formal-glueing-modules} \mathit{QCoh}(\mathcal{O}_ X) \longrightarrow \textit{QCoh}(Y \to X, Z) \end{equation}

which maps $\mathcal{F}$ to the system $(\mathcal{F}|_ U, f^*\mathcal{F}, can)$. By analogy with the proof given in the affine case, we construct a functor in the opposite direction. To an object $(\mathcal{H}, \mathcal{G}, \varphi )$ we assign the $\mathcal{O}_ X$-module

78.5.6.2
\begin{equation} \label{spaces-pushouts-equation-reverse} \mathop{\mathrm{Ker}}(j_*\mathcal{H} \oplus f_*\mathcal{G} \to (f \circ j')_*\mathcal{G}|_ V) \end{equation}

Observe that $j$ and $j'$ are quasi-compact morphisms as $Z \to X$ is of finite presentation. Hence $f_*$, $j_*$, and $(f \circ j')_*$ transform quasi-coherent modules into quasi-coherent modules (Morphisms of Spaces, Lemma 64.11.2). Thus the module (78.5.6.2) is quasi-coherent.

Proof. This follows easily from the adjointness of $f^*$ to $f_*$ and $j^*$ to $j_*$. Details omitted. $\square$

Lemma 78.5.8. In Situation 78.5.6. Let $X' \to X$ be a flat morphism of algebraic spaces. Set $Z' = X' \times _ X Z$ and $Y' = X' \times _ X Y$. The pullbacks $\mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_{X'})$ and $\mathit{QCoh}(Y \to X, Z) \to \mathit{QCoh}(Y' \to X', Z')$ are compatible with the functors (78.5.6.2) and 78.5.6.1).

Proof. This is true because pullback commutes with pullback and because flat pullback commutes with pushforward along quasi-compact and quasi-separated morphisms, see Cohomology of Spaces, Lemma 66.11.2. $\square$

Proof. We first treat the special case where $X$ and $Y$ are affine schemes and where the morphism $f$ is flat. Say $X = \mathop{\mathrm{Spec}}(R)$ and $Y = \mathop{\mathrm{Spec}}(S)$. Then $f$ corresponds to a flat ring map $R \to S$. Moreover, $Z \subset X$ is cut out by a finitely generated ideal $I \subset R$. Choose generators $f_1, \ldots , f_ t \in I$. By the description of quasi-coherent modules in terms of modules (Schemes, Section 26.7), we see that the category $\textit{QCoh}(Y \to X, Z)$ is canonically equivalent to the category $\text{Glue}(R \to S, f_1, \ldots , f_ t)$ of More on Algebra, Remark 15.82.10 such that the functors (78.5.6.1) and (78.5.6.2) correspond to the functors $\text{Can}$ and $H^0$. Hence the result follows from More on Algebra, Proposition 15.82.15 in this case.

We return to the general case. Let $\mathcal{F}$ be a quasi-coherent module on $X$. We will show that

\[ \alpha : \mathcal{F} \longrightarrow \mathop{\mathrm{Ker}}\left(j_*\mathcal{F}|_ U \oplus f_*f^*\mathcal{F} \to (f \circ j')_*f^*\mathcal{F}|_ V\right) \]

is an isomorphism. Let $(\mathcal{H}, \mathcal{G}, \varphi )$ be an object of $\mathit{QCoh}(Y \to X, Z)$. We will show that

\[ \beta : f^*\mathop{\mathrm{Ker}}\left( j_*\mathcal{H} \oplus f_*\mathcal{G} \to (f \circ j')_*\mathcal{G}|_ V \right) \longrightarrow \mathcal{G} \]

and

\[ \gamma : j^*\mathop{\mathrm{Ker}}\left( j_*\mathcal{H} \oplus f_*\mathcal{G} \to (f \circ j')_*\mathcal{G}|_ V \right) \longrightarrow \mathcal{H} \]

are isomorphisms. To see these statements are true it suffices to look at stalks. Let $\overline{y}$ be a geometric point of $Y$ mapping to the geometric point $\overline{x}$ of $X$.

Fix an object $(\mathcal{H}, \mathcal{G}, \varphi )$ of $\mathit{QCoh}(Y \to X, Z)$. By Lemma 78.5.2 and a diagram chase (omitted) the canonical map

\[ \mathop{\mathrm{Ker}}(j_*\mathcal{H} \oplus f_*\mathcal{G} \to (f \circ j')_*\mathcal{G}|_ V)_{\overline{x}} \longrightarrow \mathop{\mathrm{Ker}}( j_*\mathcal{H}_{\overline{x}} \oplus \mathcal{G}_{\overline{y}} \to j'_*\mathcal{G}_{\overline{y}} ) \]

is an isomorphism.

In particular, if $\overline{y}$ is a geometric point of $V$, then we see that $j'_*\mathcal{G}_{\overline{y}} = \mathcal{G}_{\overline{y}}$ and hence that this kernel is equal to $\mathcal{H}_{\overline{x}}$. This easily implies that $\alpha _{\overline{x}}$, $\beta _{\overline{x}}$, and $\beta _{\overline{y}}$ are isomorphisms in this case.

Next, assume that $\overline{y}$ is a point of $f^{-1}Z$. Let $I_{\overline{x}} \subset \mathcal{O}_{X, \overline{x}}$, resp. $I_{\overline{y}} \subset \mathcal{O}_{Y, \overline{y}}$ be the stalk of the ideal cutting out $Z$, resp. $f^{-1}Z$. Then $I_{\overline{x}}$ is a finitely generated ideal, $I_{\overline{y}} = I_{\overline{x}}\mathcal{O}_{Y, \overline{y}}$, and $\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{Y, \overline{y}}$ is a flat local homomorphism inducing an isomorphism $\mathcal{O}_{X, \overline{x}}/I_{\overline{x}} = \mathcal{O}_{Y, \overline{y}}/I_{\overline{y}}$. At this point we can bootstrap using the diagram of categories

\[ \xymatrix{ \mathit{QCoh}(\mathcal{O}_ X) \ar[r]_-{(0AEW)} \ar[d] & \mathit{QCoh}(Y \to X, Z) \ar[d] \ar@/_2pc/[l]^{(0AEX)} \\ \text{Mod}_{\mathcal{O}_{X, \overline{x}}} \ar[r]^-{\text{Can}} & \text{Glue}(\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{Y, \overline{y}}, f_1, \ldots , f_ t) \ar@/^2pc/[l]_{H^0} } \]

Namely, as in the first paragraph of the proof we identify

\[ \text{Glue}(\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{Y, \overline{y}}, f_1, \ldots , f_ t) = \mathit{QCoh}(\mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}}) \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}), V(I_{\overline{x}})) \]

The right vertical functor is given by pullback, and it is clear that the inner square is commutative. Our computation of the stalk of the kernel in the third paragraph of the proof combined with Lemma 78.5.3 implies that the outer square (using the curved arrows) commutes. Thus we conclude using the case of a flat morphism of affine schemes which we handled in the first paragraph of the proof. $\square$

Lemma 78.5.10. In Situation 78.5.6 the functor $Rf_*$ induces an equivalence between $D_{\mathit{QCoh}, |f^{-1}Z|}(\mathcal{O}_ Y)$ and $D_{\mathit{QCoh}, |Z|}(\mathcal{O}_ X)$ with quasi-inverse given by $Lf^*$.

Proof. Since $f$ is quasi-compact and quasi-separated we see that $Rf_*$ defines a functor from $D_{\mathit{QCoh}, |f^{-1}Z|}(\mathcal{O}_ Y)$ to $D_{\mathit{QCoh}, |Z|}(\mathcal{O}_ X)$, see Derived Categories of Spaces, Lemma 72.6.1. By Derived Categories of Spaces, Lemma 72.5.5 we see that $Lf^*$ maps $D_{\mathit{QCoh}, |Z|}(\mathcal{O}_ X)$ into $D_{\mathit{QCoh}, |f^{-1}Z|}(\mathcal{O}_ Y)$. In Lemma 78.5.5 we have seen that $Lf^*Rf_*Q = Q$ for $Q$ in $D_{\mathit{QCoh}, |f^{-1}Z|}(\mathcal{O}_ Y)$. By the dual of Derived Categories, Lemma 13.7.2 to finish the proof it suffices to show that $Lf^*K = 0$ implies $K = 0$ for $K$ in $D_{\mathit{QCoh}, |Z|}(\mathcal{O}_ X)$. This follows from the fact that $f$ is flat at all points of $f^{-1}Z$ and the fact that $f^{-1}Z \to Z$ is surjective. $\square$

Lemma 78.5.11. In Situation 78.5.6 there exists an fpqc covering $\{ X_ i \to X\} _{i \in I}$ refining the family $\{ U \to X, Y \to X\} $.

Proof. For the definition and general properties of fpqc coverings we refer to Topologies, Section 34.9. In particular, we can first choose an étale covering $\{ X_ i \to X\} $ with $X_ i$ affine and by base changing $Y$, $Z$, and $U$ to each $X_ i$ we reduce to the case where $X$ is affine. In this case $U$ is quasi-compact and hence a finite union $U = U_1 \cup \ldots \cup U_ n$ of affine opens. Then $Z$ is quasi-compact hence also $f^{-1}Z$ is quasi-compact. Thus we can choose an affine scheme $W$ and an étale morphism $h : W \to Y$ such that $h^{-1}f^{-1}Z \to f^{-1}Z$ is surjective. Say $W = \mathop{\mathrm{Spec}}(B)$ and $h^{-1}f^{-1}Z = V(J)$ where $J \subset B$ is an ideal of finite type. By Pro-étale Cohomology, Lemma 59.5.1 there exists a localization $B \to B'$ such that points of $\mathop{\mathrm{Spec}}(B')$ correspond exactly to points of $W = \mathop{\mathrm{Spec}}(B)$ specializing to $h^{-1}f^{-1}Z = V(J)$. It follows that the composition $\mathop{\mathrm{Spec}}(B') \to \mathop{\mathrm{Spec}}(B) = W \to Y \to X$ is flat as by assumption $f : Y \to X$ is flat at all the points of $f^{-1}Z$. Then $\{ \mathop{\mathrm{Spec}}(B') \to X, U_1 \to X, \ldots , U_ n \to X\} $ is an fpqc covering by Topologies, Lemma 34.9.2. $\square$


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