The Stacks project

Proof. Recall that the support of a section is closed (Étale Cohomology, Lemma 59.31.4) hence the material in Cohomology of Schemes, Section 30.26 applies. By the lemma above and Cohomology of Schemes, Lemma 30.26.6 we find that our subset of $f_*\mathcal{F}(V)$ is a subgroup. By Cohomology of Schemes, Lemma 30.26.4 we see that our rule defines a sub presheaf. Finally, suppose that we have $s \in f_*\mathcal{F}(V)$ and an étale covering $\{ V_ i \to V\} $ such that $s|_{V_ i}$ has support proper over $V_ i$. Observe that the support of $s|_{V_ i}$ is the inverse image of the support of $s|_ V$ (use the characterization of the support in terms of stalks and Étale Cohomology, Lemma 59.36.2). Whence the support of $s$ is proper over $V$ by Descent, Lemma 35.25.5. This proves that our rule satisfies the sheaf condition. $\square$

Proof. The construction of $j_!\mathcal{F} \to j_*\mathcal{F}$ in the proof of Étale Cohomology, Lemma 59.70.6 is via the construction of a map $j_{p!}\mathcal{F} \to j_*\mathcal{F}$ of presheaves whose image is clearly contained in the subsheaf of Lemma 63.3.1. Hence since $j_!\mathcal{F}$ is the sheafification of $j_{p!}\mathcal{F}$ we conclude the image of $j_!\mathcal{F} \to j_*\mathcal{F}$ is contained in this subsheaf. Conversely, let $s \in j_*\mathcal{F}(V)$ have support $Z$ proper over $V$. Then $Z \to V$ is finite with closed image $Z' \subset V$, see More on Morphisms, Lemma 37.44.1. The restriction of $s$ to $V \setminus Z'$ is zero and the zero section is contained in the image of $j_!\mathcal{F} \to j_*\mathcal{F}$. On the other hand, if $v \in Z'$, then we can find an étale neighbourhood $(V', v') \to (V, v)$ such that we have a decomposition $U_{V'} = W \amalg U'_1 \amalg \ldots \amalg U'_ n$ into open and closed subschemes with $U'_ i \to V'$ an isomorphism and with $T_{V'} \subset U'_1 \amalg \ldots \amalg U'_ n$, see Étale Morphisms, Lemma 41.18.2. Inverting the isomorphisms $U'_ i \to V'$ we obtain $n$ morphisms $\varphi '_ i : V' \to U$ and sections $s'_ i$ over $V'$ by pulling back $s$. Then the section $\sum (\varphi '_ i, s'_ i)$ of $j_{p!}\mathcal{F}$ over $V'$, see formula for $j_{p!}\mathcal{F}(V')$ in proof of Étale Cohomology, Lemma 59.70.6, maps to the restriction of $s$ to $V'$ by construction. We conclude that $s$ is étale locally in the image of $j_!\mathcal{F} \to j_*\mathcal{F}$ and the proof is complete. $\square$

Proof. Immediate from the construction of $f_!$. $\square$

Proof. We have $(j_!\mathcal{F})|_ X = \mathcal{F}$, see Étale Cohomology, Lemma 59.70.4. Thus the displayed arrow is the injective map $f_!(\mathcal{G}|_ X) \to \overline{f}_!\mathcal{G}$ of Remark 63.3.5 for $\mathcal{G} = j_!\mathcal{F}$. The explicit nature of this map implies that it now suffices to show: if $V \in Y_{\acute{e}tale}$ and $s \in \overline{f}_!\mathcal{G}(V) = \overline{f}_*\mathcal{G}(V) = \mathcal{G}(\overline{X}_ V)$ is a section, then the support of $s$ is contained in the open $X_ V \subset \overline{X}_ V$. This is immediate from the fact that the stalks of $\mathcal{G}$ are zero at geometric points of $\overline{X} \setminus X$. $\square$

Proof. Immediate from the construction of $H^0_ c$. $\square$

Proof. We have $(j_!\mathcal{F})|_ X = \mathcal{F}$, see Étale Cohomology, Lemma 59.70.4. Thus the displayed arrow is the injective map $H^0_ c(X, \mathcal{G}|_ X) \to H^0_ c(\overline{X}, \mathcal{G})$ of Remark 63.3.9 for $\mathcal{G} = j_!\mathcal{F}$. The explicit nature of this map implies that it now suffices to show: if $s \in H^0(\overline{X}, \mathcal{G})$ is a section, then the support of $s$ is contained in the open $X$. This is immediate from the fact that the stalks of $\mathcal{G}$ are zero at geometric points of $\overline{X} \setminus X$. $\square$

Proof. Recall that $(f_*\mathcal{F})_{\overline{y}} = \mathop{\mathrm{colim}}\nolimits f_*\mathcal{F}(V)$ where the colimit is over the étale neighbourhoods $(V, \overline{v})$ of $\overline{y}$. If $s \in f_*\mathcal{F}(V) = \mathcal{F}(X_ V)$, then we can pullback $s$ to a section of $\mathcal{F}$ over $(X_ V)_{\overline{v}} = X_{\overline{y}}$. Thus we obtain a canonical map

We claim that this map induces a bijection between the subgroups $(f_!\mathcal{F})_{\overline{y}}$ and $H^0_ c(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}})$. The claim implies the lemma, but is a little bit more precise in that it describes the identification of the lemma as given by pullbacks of sections of $\mathcal{F}$ to the geometric fibre of $f$.

Observe that any element $s \in (f_!\mathcal{F})_{\overline{y}} \subset (f_*\mathcal{F})_{\overline{y}}$ is mapped by $c_{\overline{y}}$ to an element of $H^0_ c(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}}) \subset H^0(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}})$. This is true because taking the support of a section commutes with pullback and because properness is preserved by base change. This at least produces the map in the statement of the lemma. To prove that it is an isomorphism we may work Zariski locally on $Y$ and hence we may and do assume $Y$ is affine.

An observation that we will use below is that given an open subscheme $X' \subset X$ and if $f' = f|_{X'}$, then we obtain a commutative diagram

where the horizontal arrows are the maps constructed above and the vertical arrows are given in Remarks 63.3.5 and 63.3.9. The reason is that given an étale neighbourhood $(V, \overline{v})$ of $\overline{y}$ and a section $s \in f_*\mathcal{F}(V) = \mathcal{F}(X_ V)$ whose support $Z$ happens to be contained in $X'_ V$ and is proper over $V$, so that $s$ gives rise to an element of both $(f'_!(\mathcal{F}|_{X'}))_{\overline{y}}$ and $(f_!\mathcal{F})_{\overline{y}}$ which correspond via the vertical arrow of the diagram, then these elements are mapped via the horizontal arrows to the pullback $s|_{X_{\overline{y}}}$ of $s$ to $X_{\overline{y}}$ whose support $Z_{\overline{y}}$ is contained in $X'_{\overline{y}}$ and hence this restriction gives rise to a compatible pair of elements of $H^0_ c(X'_{\overline{y}}, \mathcal{F}|_{X'_{\overline{y}}})$ and $H^0_ c(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}})$.

Suppose $s \in (f_!\mathcal{F})_{\overline{y}}$ maps to zero in $H^0_ c(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}})$. Say $s$ corresponds to $s \in f_*\mathcal{F}(V) = \mathcal{F}(X_ V)$ with support $Z$ proper over $V$. We may assume that $V$ is affine and hence $Z$ is quasi-compact. Then we may choose a quasi-compact open $X' \subset X$ containing the image of $Z$. Then $Z$ is contained in $X'_ V$ and hence $s$ is the image of an element $s' \in f'_!(\mathcal{F}|_{X'})(V)$ where $f' = f|_{X'}$ as in the previous paragraph. Then $s'$ maps to zero in $H^0_ c(X'_{\overline{y}}, \mathcal{F}|_{X'_{\overline{y}}})$. Hence in order to prove injectivity, we may replace $X$ by $X'$, i.e., we may assume $X$ is quasi-compact. We will prove this case below.

Suppose that $t \in H^0_ c(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}})$. Then the support of $t$ is contained in a quasi-compact open subscheme $W \subset X_{\overline{y}}$. Hence we can find a quasi-compact open subscheme $X' \subset X$ such that $X'_{\overline{y}}$ contains $W$. Then it is clear that $t$ is contained in the image of the injective map $H^0_ c(X'_{\overline{y}}, \mathcal{F}|_{X'_{\overline{y}}}) \to H^0_ c(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}})$. Hence in order to show surjectivity, we may replace $X$ by $X'$, i.e., we may assume $X$ is quasi-compact. We will prove this case below.

In this last paragraph of the proof we prove the lemma in case $X$ is quasi-compact and $Y$ is affine. By More on Flatness, Theorem 38.33.8 there exists a compactification $j : X \to \overline{X}$ over $Y$. Set $\mathcal{G} = j_!\mathcal{F}$ so that $\mathcal{F} = \mathcal{G}|_ X$ by Étale Cohomology, Lemma 59.70.4. By the disussion above we get a commutative diagram

By Lemmas 63.3.6 and 63.3.10 the vertical maps are isomorphisms. This reduces us to the case of the proper morphism $\overline{X} \to Y$. For a proper morphism our map is an isomorphism by Lemmas 63.3.4 and 63.3.8 and proper base change for pushforwards, see Étale Cohomology, Lemma 59.91.4. $\square$

Proof. In great generality there is a pullback map $g^{-1}f_*\mathcal{F} \to f'_*(g')^{-1}\mathcal{F}$, see Sites, Section 7.45. We claim that this map sends $g^{-1}f_!\mathcal{F}$ into the subsheaf $f'_!(g')^{-1}\mathcal{F}$ and induces the isomorphism in the lemma.

Choose a geometric point $\overline{y}': \mathop{\mathrm{Spec}}(k) \to Y'$ and denote $\overline{y} = g \circ \overline{y}'$ the image in $Y$. There is a commutative diagram

where the horizontal maps were used in the proof of Lemma 63.3.11 and the vertical maps are the pullback maps above. The diagram commutes because each of the four maps in question is given by pulling back local sections along a morphism of schemes and the underlying diagram of morphisms of schemes commutes. Since the diagram in the statement of the lemma is cartesian we have $X'_{\overline{y}'} = X_{\overline{y}}$. Hence by Lemma 63.3.11 and its proof we obtain a commutative diagram

where the horizontal arrows of the inner square are isomorphisms and the two right vertical arrows are equalities. Also, the se, sw, ne, nw arrows are injective. It follows that there is a unique bijective dotted arrow fitting into the diagram. We conclude that $g^{-1}f_!\mathcal{F} \subset g^{-1}f_*\mathcal{F} \to f'_*(g')^{-1}\mathcal{F}$ is mapped into the subsheaf $f'_!(g')^{-1}\mathcal{F} \subset f'_*(g')^{-1}\mathcal{F}$ because this is true on stalks, see Étale Cohomology, Theorem 59.29.10. The same theorem then implies that the induced map is an isomorphism and the proof is complete. $\square$

Proof. We strongly urge the reader to prove this for themselves. Let $W \in Z_{\acute{e}tale}$ and $s \in (g \circ f)_*\mathcal{F}(W) = \mathcal{F}(X_ W)$. Denote $T \subset X_ W$ the support of $s$; this is a closed subset. Observe that $s$ is a section of $(g \circ f)_!\mathcal{F}$ if and only if $T$ is proper over $W$. We have $f_!\mathcal{F} \subset f_*\mathcal{F}$ and hence $g_!f_!\mathcal{F} \subset g_!f_*\mathcal{F} \subset g_*f_*\mathcal{F}$. On the other hand, $s$ is a section of $g_!f_!\mathcal{F}$ if and only if (a) $T$ is proper over $Y_ W$ and (b) the support $T'$ of $s$ viewed as section of $f_!\mathcal{F}$ is proper over $W$. If (a) holds, then the image of $T$ in $Y_ W$ is closed and since $f_!\mathcal{F} \subset f_*\mathcal{F}$ we see that $T' \subset Y_ W$ is the image of $T$ (details omitted; look at stalks).

The conclusion is that we have to show a closed subset $T \subset X_ W$ is proper over $W$ if and only if $T$ is proper over $Y_ W$ and the image of $T$ in $Y_ W$ is proper over $W$. Let us endow $T$ with the reduced induced closed subscheme structure. If $T$ is proper over $W$, then $T \to Y_ W$ is proper by Morphisms, Lemma 29.41.7 and the image of $T$ in $Y_ W$ is proper over $W$ by Cohomology of Schemes, Lemma 30.26.5. Conversely, if $T$ is proper over $Y_ W$ and the image of $T$ in $Y_ W$ is proper over $W$, then the morphism $T \to W$ is proper as a composition of proper morphisms (here we endow the closed image of $T$ in $Y_ W$ with its reduced induced scheme structure to turn the question into one about morphisms of schemes), see Morphisms, Lemma 29.41.4. $\square$

Proof. It suffices to show that the canonical map from right to left is a bijection when evaluated on a quasi-compact object $V$ of $Y_{\acute{e}tale}$. Observe that the colimit on the right hand side is directed and has injective transition maps. Thus we can use Sites, Lemma 7.17.7 to evaluate the colimit. Hence, the statement comes down to the observation that a closed subset $Z \subset X_ V$ proper over $V$ is quasi-compact and hence is contained in $X_{i, V}$ for some $i$. $\square$

Proof. Let $\mathcal{F} = \bigoplus \mathcal{F}_ i$. To show that the map $\bigoplus f_!\mathcal{F}_ i \to f_!\mathcal{F}$ is an isomorphism, it suffices to show that these sheaves have the same sections over a quasi-compact object $V$ of $Y_{\acute{e}tale}$. Replacing $Y$ by $V$ it suffices to show $H^0(Y, f_!\mathcal{F}) \subset H^0(X, \mathcal{F})$ is equal to $\bigoplus H^0(Y, f_!\mathcal{F}_ i) \subset \bigoplus H^0(X, \mathcal{F}_ i) \subset H^0(X, \bigoplus \mathcal{F}_ i)$. In this case, by writing $X$ as the union of its quasi-compact opens and using Lemma 63.3.15 we reduce to the case where $X$ is quasi-compact as well. Then $H^0(X, \mathcal{F}) = \bigoplus H^0(X, \mathcal{F}_ i)$ by Étale Cohomology, Theorem 59.51.3. Looking at supports of sections the reader easily concludes. $\square$

Proof. The functor $f_!$ is left exact by construction. Right exactness may be checked on stalks (Étale Cohomology, Theorem 59.29.10). Thus it suffices to prove part (1).

Let $\overline{y} : \mathop{\mathrm{Spec}}(k) \to Y$ be a geometric point. The scheme $X_{\overline{y}}$ has a discrete underlying topological space (Morphisms, Lemma 29.20.8) and all the residue fields at the points are equal to $k$ (as finite extensions of $k$). Hence $\{ \overline{x} : \mathop{\mathrm{Spec}}(k) \to X : f(\overline{x}) = \overline{y}\} $ is equal to the set of points of $X_{\overline{y}}$. Thus the computation of the stalk follows from the more general Lemma 63.3.11. $\square$