The Stacks project

Proof. Consider the commutative diagram of algebraic spaces

By Cohomology of Spaces, Lemma 69.9.4 we can write $Q = Ri'_*K'$ for some object $K'$ of $D(f^{-1}Z_{\acute{e}tale})$. By Morphisms of Spaces, Lemma 67.53.7 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

The middle equality holds because of the description of the stalk of a pullback given in Properties of Spaces, Lemma 66.19.9. $\square$

Proof. Choose a distinguished triangle

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

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

We can compare this with the exact sequence

Thus we see that the lemma follows because $Q_{\overline{y}} = Rf_*Q_{\overline{x}}$ by Lemma 81.10.1. $\square$

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 69.11.2 applies. $\square$

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 75.4.2 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. $\square$

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 81.10.1 we have $Rf_*Q_{\overline{x}} = Q_{\overline{y}}$ and since $f$ is flat at $\overline{y}$ we see that

Thus we have to check that the canonical map

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 81.10.4 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 15.89.2 which finishes the proof. $\square$

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

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 69.11.2. $\square$

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 15.89.10 such that the functors (81.10.6.1) and (81.10.6.2) correspond to the functors $\text{Can}$ and $H^0$. Hence the result follows from More on Algebra, Proposition 15.89.15 in this case.

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

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

and

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 81.10.2 and a diagram chase (omitted) the canonical map

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

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

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 81.10.3 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. $\square$

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 75.6.1. By Derived Categories of Spaces, Lemma 75.5.5 we see that $Lf^*$ maps $D_{\mathit{QCoh}, |Z|}(\mathcal{O}_ X)$ into $D_{\mathit{QCoh}, |f^{-1}Z|}(\mathcal{O}_ Y)$. In Lemma 81.10.5 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. $\square$

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 61.5.1 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. $\square$