hcju/stacksqa huggingface dataset:
By Algebra, Lemma \ref{algebra-lemma-lci-local} we can write $A$ as the localization at a prime $\mathfrak p$ of a global complete intersection $P$ over $k$. Observe that $\dim(P) = d$ by Algebra, Lemma \ref{algebra-lemma-dimension-at-a-point-finite-type-field}. By Lemma \ref{lemma-smoothing-affine-lci} we can find a flat, finite type ring map $k[[t]] \to Q$ such that $P \cong Q/tQ$ and such that $k((t)) \to Q[1/t]$ is smooth. It follows from the construction of $Q$ in the lemma that $k[[t]] \to Q$ is a relative global complete intersection of relative dimension $d$; alternatively, Algebra, Lemma \ref{algebra-lemma-syntomic} tells us that $Q$ or a suitable principal localization of $Q$ is such a global complete intersection. Hence by Divisors, Lemma \ref{divisors-lemma-d-fitting-ideal-omega-smooth} the $d$th Fitting ideal $I \subset Q$ of $\Omega_{Q/k[[t]]}$ cuts out the singular locus of $\Spec(Q) \to \Spec(k[[t]])$. Thus $t^n \in I$ for some $n$. Let $\mathfrak q \subset Q$ be the inverse image of $\mathfrak p$. Set $B = Q_\mathfrak q$. The lemma is proved.
If $X$ is reduced, then we can choose an embedding $X \subset \overline{X}$ as in Varieties, Lemma \ref{varieties-lemma-reduced-dim-1-projective-completion}. Writing $X = \overline{X} \setminus \{x_1, \ldots, x_n\}$ we see that $\mathcal{O}_{\overline{X}, x_i}$ is a discrete valuation ring and hence in particular a local complete intersection (Algebra, Definition \ref{algebra-definition-lci-local-ring}). Thus $\overline{X}$ is a local complete intersection over $k$ because this holds over the open $X$ and at the points $x_i$ by Algebra, Lemma \ref{algebra-lemma-lci-local}. Thus we may apply Lemma \ref{lemma-smoothing-proper-curve-isolated-lci} to find a projective flat morphism $\overline{Y} \to \Spec(k[[t]])$ whose generic fibre is smooth and whose special fibre is $\overline{X}$. Then we remove $x_1, \ldots, x_n$ from $\overline{Y}$ to obtain $Y$. \medskip\noindent In the general case, write $X = X' \amalg X''$ where with $\dim(X') = 0$ and $X''$ equidimensional of dimension $1$. Then $X''$ is reduced and the first paragraph applies to it. On the other hand, $X'$ can be dealt with as in the proof of Lemma \ref{lemma-smoothing-proper-curve-isolated-lci}. Some details omitted.
Let $S = \bigcup_{k \in K} S_k$ be an affine open covering. Let $X = \bigcup_{i \in I} U_i$, $Y = \bigcup_{j \in J} V_j$ be affine open coverings with $I$, $J$ of cardinality $\leq \text{size}(X), \text{size}(Y)$. For each $i \in I$ there exists a finite set $K_i$ of $k \in K$ such that $f(U_i) \subset \bigcup_{k \in K_i} S_k$. For each $j \in J$ there exists a finite set $K_j$ of $k \in K$ such that $g(V_j) \subset \bigcup_{k \in K_j} S_k$. Hence $f(X), g(Y)$ are contained in $S' = \bigcup_{k \in K'} S_k$ with $K' = \bigcup_{i \in I} K_i \cup \bigcup_{j \in J} K_j$. Note that the cardinality of $K'$ is at most $\max\{\aleph_0, |I|, |J|\}$. Applying Lemma \ref{lemma-bound-size} we see that it suffices to prove that $\text{size}(f^{-1}(S_k) \times_{S_k} g^{-1}(S_k)) \leq \max\{\text{size}(X), \text{size}(Y))\}$ for $k \in K'$. In other words, we may assume that $S$ is affine. \medskip\noindent Assume $S$ affine. Let $X = \bigcup_{i \in I} U_i$, $Y = \bigcup_{j \in J} V_j$ be affine open coverings with $I$, $J$ of cardinality $\leq \text{size}(X), \text{size}(Y)$. Again by Lemma \ref{lemma-bound-size} it suffices to prove the lemma for the products $U_i \times_S V_j$. By Lemma \ref{lemma-bound-affine} we see that it suffices to show that $$ |A \otimes_C B| \leq \max\{\aleph_0, |A|, |B|\}. $$ We omit the proof of this inequality.
We can find a finite affine open covering $X = \bigcup_{i = 1, \ldots n} U_i$ such that each $U_i$ maps into an affine open $S_i$ of $S$. Thus by Lemma \ref{lemma-bound-size} we reduce to the case where both $S$ and $X$ are affine. In this case by Lemma \ref{lemma-bound-affine} we see that it suffices to show $$ |A[x_1, \ldots, x_n]| \leq \max\{\aleph_0, |A|\}. $$ We omit the proof of this inequality.
Let $Y = \bigcup_{j \in J} V_j$ be an affine open covering of $Y$ with $|J| \leq \text{size}(Y)$. By Lemma \ref{lemma-bound-size} it suffices to bound the size of the inverse image of $V_j$ in $X$. Hence we reduce to the case that $Y$ is affine, say $Y = \Spec(B)$. For any affine open $\Spec(A) \subset X$ we have $|A| \leq \max(|B|, \aleph_0) = \text{size}(Y)$, see remark above and Lemma \ref{lemma-bound-affine}. Thus it suffices to show that $X$ has at most $\text{size}(Y)$ affine opens. This is clear if $X$ is quasi-compact, whence case (1) holds. In case (2) the number of isomorphism classes of $B$-algebras $A$ that can occur is bounded by $\text{size}(B)$, because each $A$ is of finite type over $B$, hence isomorphic to an algebra $B[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$ for some $n, m$, and $f_j \in B[x_1, \ldots, x_n]$. However, as $X \to Y$ is a monomorphism, there is a unique morphism $\Spec(A) \to X$ over $Y = \Spec(B)$ if there is one, hence the number of affine opens of $X$ is bounded by the number of these isomorphism classes. \medskip\noindent To prove the final statement of the lemma consider the ring $B = \prod_{n \in \mathbf{N}} \mathbf{F}_2$ and set $Y = \Spec(B)$. For every ultrafilter $\mathcal{U}$ on $\mathbf{N}$ we obtain a maximal ideal $\mathfrak m_\mathcal{U}$ with residue field $\mathbf{F}_2$; the map $B \to \mathbf{F}_2$ sends the element $(x_n)$ to $\lim_\mathcal{U} x_n$. Details omitted. The morphism of schemes $X = \coprod_\mathcal{U} \Spec(\mathbf{F}_2) \to Y$ is a monomorphism as all the points are distinct. However the cardinality of the set of affine open subschemes of $X$ is equal to the cardinality of the set of ultrafilters on $\mathbf{N}$ which is $2^{2^{\aleph_0}}$. We conclude as $|B| = 2^{\aleph_0} < 2^{2^{\aleph_0}}$.
Statements (1) and (2) follow directly from the definitions. Statement (3) follows as the size of an open subscheme $U$ of $S$ is clearly smaller than or equal to the size of $S$. Statement (4) follows from (5). Statement (5) follows from (7). Statement (6) follows as the size of $S$ is $\leq \max\{|I|, \sup_i \text{size}(S_i)\} \leq \text{size}(T)^{\aleph_0}$ by Lemma \ref{lemma-bound-size}. Statement (7) follows from (6). Namely, for any affine open $V \subset T$ we have $\text{size}(V) \leq \text{size}(S)$ by Lemma \ref{lemma-bound-finite-type}. Thus, we see that (6) applies in the situation of (7). Part (8) follows from Lemma \ref{lemma-bound-monomorphism}. \medskip\noindent Statement (9) is translated, via Lemma \ref{lemma-bound-affine}, into an upper bound on the cardinality of the rings $R^*$, $S^{-1}R$, $\overline{\kappa(\mathfrak p)}$, $R'$, etc. Perhaps the most interesting one is the ring $R^*$. As a set, it is the image of a surjective map $R^{\mathbf{N}} \to R^*$. Since $|R^{\mathbf{N}}| = |R|^{\aleph_0}$, we see that it works by our choice of $Bound(\kappa)$ being at least $\kappa^{\aleph_0}$. Phew! (The cardinality of the algebraic closure of a field is the same as the cardinality of the field, or it is $\aleph_0$.)
Choose an affine open covering $X = \bigcup_{a \in A} U_a$ with $|A| \leq \text{size}(X)$. For each $a$ we can choose a finite subset $I_a \subset I$ and for $i \in I_a$ a quasi-compact open $W_{a, i} \subset X_i$ such that $U_a = \bigcup_{i \in I_a} f_i(W_{a, i})$. Then $\text{size}(W_{a, i}) \leq \text{size}(X)$ by Lemma \ref{lemma-bound-finite-type}. We conclude that $\text{size}(\coprod_a \coprod_{i \in I_a} W_{i, a}) \leq \text{size}(X)$ by Lemma \ref{lemma-bound-size}.
Recall that the support of a section is closed (\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-support-section-closed}) hence the material in Cohomology of Schemes, Section \ref{coherent-section-proper-over-base} applies. By the lemma above and Cohomology of Schemes, Lemma \ref{coherent-lemma-union-closed-proper-over-base} we find that our subset of $f_*\mathcal{F}(V)$ is a subgroup. By Cohomology of Schemes, Lemma \ref{coherent-lemma-base-change-closed-proper-over-base} we see that our rule defines a sub presheaf. Finally, suppose that we have $s \in f_*\mathcal{F}(V)$ and an \'etale 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 \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-stalk-pullback}). Whence the support of $s$ is proper over $V$ by Descent, Lemma \ref{descent-lemma-descending-property-proper-over-base}. This proves that our rule satisfies the sheaf condition.
The construction of $j_!\mathcal{F} \to j_*\mathcal{F}$ in the proof of \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-shriek-into-star-separated-etale} 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 \ref{lemma-f-shriek-separated}. 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 \ref{more-morphisms-lemma-characterize-finite}. 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 \'etale 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 \'Etale Morphisms, Lemma \ref{etale-lemma-etale-etale-local-technical}. 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 \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-shriek-into-star-separated-etale}, maps to the restriction of $s$ to $V'$ by construction. We conclude that $s$ is \'etale locally in the image of $j_!\mathcal{F} \to j_*\mathcal{F}$ and the proof is complete.
We have $(j_!\mathcal{F})|_X = \mathcal{F}$, see \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-jshriek-open}. Thus the displayed arrow is the injective map $f_!(\mathcal{G}|_X) \to \overline{f}_!\mathcal{G}$ of Remark \ref{remark-covariance-f-shriek-separated} for $\mathcal{G} = j_!\mathcal{F}$. The explicit nature of this map implies that it now suffices to show: if $V \in Y_\etale$ 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 $\overline{X} \setminus X$.
We have $(j_!\mathcal{F})|_X = \mathcal{F}$, see \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-jshriek-open}. 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 \ref{remark-covariance-compact-support} 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 $\overline{X} \setminus X$.
In great generality there is a pullback map $g^{-1}f_*\mathcal{F} \to f'_*(g')^{-1}\mathcal{F}$, see Sites, Section \ref{sites-section-pullback}. 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. \medskip\noindent Choose a geometric point $\overline{y}': \Spec(k) \to Y'$ and denote $\overline{y} = g \circ \overline{y}'$ the image in $Y$. There is a commutative diagram $$ \xymatrix{ (f_*\mathcal{F})_{\overline{y}} \ar[r] \ar[d] & H^0(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}}) \ar[d] \\ (f'_*(g')^{-1}\mathcal{F})_{\overline{y}'} \ar[r] & H^0(X'_{\overline{y}'}, (g')^{-1}\mathcal{F}|_{X'_{\overline{y}'}}) } $$ where the horizontal maps were used in the proof of Lemma \ref{lemma-stalk-f-shriek-separated} 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 \ref{lemma-stalk-f-shriek-separated} and its proof we obtain a commutative diagram $$ \xymatrix{ (f_*\mathcal{F})_{\overline{y}} \ar[rrr] \ar[ddd] & & & H^0(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}}) \ar[ddd] \\ & (f_!\mathcal{F})_{\overline{y}} \ar[r] \ar@{..>}[d] \ar[lu] & H^0_c(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}}) \ar[d] \ar[ru] \\ & (f'_!(g')^{-1}\mathcal{F})_{\overline{y}'} \ar[r] \ar[ld] & H^0_c(X'_{\overline{y}'}, (g')^{-1}\mathcal{F}|_{X'_{\overline{y}'}}) \ar[rd]\\ (f'_*(g')^{-1}\mathcal{F})_{\overline{y}'} \ar[rrr] & & & H^0(X'_{\overline{y}'}, (g')^{-1}\mathcal{F}|_{X'_{\overline{y}'}}) } $$ 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 \'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-exactness-stalks}. The same theorem then implies that the induced map is an isomorphism and the proof is complete.
We strongly urge the reader to prove this for themselves. Let $W \in Z_\etale$ 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). \medskip\noindent 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 \ref{morphisms-lemma-image-proper-scheme-closed} and the image of $T$ in $Y_W$ is proper over $W$ by Cohomology of Schemes, Lemma \ref{coherent-lemma-functoriality-closed-proper-over-base}. 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 \ref{morphisms-lemma-composition-proper}.
It suffices to show that the canonical map from right to left is a bijection when evaluated on a quasi-compact object $V$ of $Y_\etale$. Observe that the colimit on the right hand side is directed and has injective transition maps. Thus we can use Sites, Lemma \ref{sites-lemma-directed-colimits-sections} 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 $i$.
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 $Y_\etale$. 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 \ref{lemma-colim-f-shriek-separated} 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 \'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-colimit}. Looking at supports of sections the reader easily concludes.
The functor $f_!$ is left exact by construction. Right exactness may be checked on stalks (\'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-exactness-stalks}). Thus it suffices to prove part (1). \medskip\noindent Let $\overline{y} : \Spec(k) \to Y$ be a geometric point. The scheme $X_{\overline{y}}$ has a discrete underlying topological space (Morphisms, Lemma \ref{morphisms-lemma-locally-quasi-finite-fibres}) and all the residue fields at the points are equal to $k$ (as finite extensions of $k$). Hence $\{\overline{x} : \Spec(k) \to X : f(\overline{x}) = \overline{y}\}$ is equal to the set of points of $X_{\overline{y}}$. Thus the computation of the stalk follows from the more general Lemma \ref{lemma-stalk-f-shriek-separated}.
Please read the proof of \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-shriek-into-star-separated-etale} before reading the proof of this lemma. Let $V$ be an object of $X_\etale$. Recall that $$ j_{p!}\mathcal{F}(V) = \bigoplus\nolimits_{\varphi : V \to U} \mathcal{F}(V \xrightarrow{\varphi} U) $$ Given $\varphi$ we obtain an open subscheme $Z_\varphi \subset U_V = U \times_X V$, namely, the image of the graph of $\varphi$. Via $\varphi$ we obtain an isomorphism $V \to Z_\varphi$ over $U$ and we can think of an element $$ s_\varphi \in \mathcal{F}(V \xrightarrow{\varphi} U) = \mathcal{F}(Z_\varphi) = H_{Z_\varphi}(\mathcal{F}) $$ as a section of $\mathcal{F}$ over $Z_{\varphi}$. Since $Z_\varphi \subset U_V$ is open, we actually have $H_{Z_\varphi}(\mathcal{F}) = \mathcal{F}(Z_\varphi)$ and we can think of $s_\varphi$ as an element of $H_{Z_\varphi}(\mathcal{F})$. Having said this, our map $j_{p!}\mathcal{F} \to f_{p!}\mathcal{F}$ is defined by the rule $$ \sum\nolimits_{i = 1, \ldots, n} s_{\varphi_i} \longmapsto \sum\nolimits_{i = 1, \ldots, n} (Z_{\varphi_i}, s_{\varphi_i}) $$ with right hand side a sum as in (\ref{equation-formal-sum}). We omit the verification that this is compatible with restriction mappings and functorial in $\mathcal{F}$. \medskip\noindent To finish the proof, we claim that given a geometric point $\overline{y} : \Spec(k) \to Y$ there is a commutative diagram $$ \xymatrix{ (j_{p!}\mathcal{F})_{\overline{y}} \ar[r] \ar[d] & \bigoplus_{j(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}} \ar@{=}[d] \\ (f_{p!}\mathcal{F})_{\overline{y}} \ar[r] & \bigoplus_{f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}} } $$ where the top horizontal arrow is constructed in the proof of \'Etale Cohomology, Proposition \ref{etale-cohomology-proposition-describe-jshriek}, the bottom horizontal arrow is constructed in the proof of Lemma \ref{lemma-finite-support-stalk}, the right vertical arrow is the obvious equality, and the left veritical arrow is the map defined in the previous paragraph on stalks. The claim follows in a straightforward manner from the explicit description of all of the arrows involved here and in the references given. Since the horizontal arrows are isomorphisms we conclude so is the left vertical arrow. Hence we find that our map induces an isomorphism on sheafifications by \'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-exactness-stalks}.
Here as usual we set $X_{i_0 \ldots i_p} = X_{i_0} \cap \ldots \cap X_{i_p}$ and we denote $f_{i_0 \ldots i_p}$ the restriction of $f$ to $X_{i_0 \ldots i_p}$. The maps in the complex are the maps constructed in Remark \ref{remark-covariance-lqf-f-shriek} with sign rules as in the {\v C}ech complex. Exactness follows easily from the description of stalks in Lemma \ref{lemma-lqf-f-shriek-stalk}. Details omitted.
With conventions as in Remark \ref{remark-construct-map-presheaves-downstairs} we will explicitly construct a map $$ c : f_{p!}\mathcal{F} \longrightarrow g_*f'_{p!}(g')^{-1}\mathcal{F} $$ of abelian presheaves on $Y_\etale$. By the discussion in Remark \ref{remark-construct-map-presheaves-downstairs} this will determine a canonical map $g^{-1}f_!\mathcal{F} \to f'_!(g')^{-1}\mathcal{F}$. Finally, we will show this map induces isomorphisms on stalks and conclude by \'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-exactness-stalks}. \medskip\noindent Construction of the map $c$. Let $V \in Y_\etale$ and consider a section $s = \sum_{i = 1, \ldots, n} (Z_i, s_i)$ as in (\ref{equation-formal-sum}) defining an element of $f_{p!}\mathcal{F}(V)$. The value of $g_*f'_{p!}(g')^{-1}\mathcal{F}$ at $V$ is $f'_{p!}(g')^{-1}\mathcal{F}(V')$ where $V' = V \times_Y Y'$. Denote $Z'_i \subset X'_{V'}$ the base change of $Z_i$ to $V'$. By (\ref{item-pullback}) there is a pullback map $H_{Z_i}(\mathcal{F}) \to H_{Z'_i}((g')^{-1}\mathcal{F})$. Denoting $s'_i \in H_{Z'_i}((g')^{-1}\mathcal{F})$ the image of $s_i$ under pullback, we set $c(s) = \sum_{i = 1, \ldots, n} (Z'_i, s'_i)$ as in (\ref{equation-formal-sum}) defining an element of $f'_{p!}(g')^{-1}\mathcal{F}(V')$. We omit the verification that this construction is compatible the relations (\ref{item-sum}) and (\ref{item-sub}) and compatible with restriction mappings. The construction is clearly functorial in $\mathcal{F}$. \medskip\noindent Let $\overline{y}' : \Spec(k) \to Y'$ be a geometric point with image $\overline{y} = g \circ \overline{y}'$ in $Y$. Observe that $X'_{\overline{y}'} = X_{\overline{y}}$ by transitivity of fibre products. Hence $g'$ produces a bijection $\{f'(\overline{x}') = \overline{y}'\} \to \{f(\overline{x}) = \overline{y}\}$ and if $\overline{x}'$ maps to $\overline{x}$, then $((g')^{-1}\mathcal{F})_{\overline{x}'} = \mathcal{F}_{\overline{x}}$ by \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-stalk-pullback}. Now we claim that the diagram $$ \xymatrix{ (g^{-1}f_!\mathcal{F})_{\overline{y}'} \ar@{=}[r] \ar[d] & (f_!\mathcal{F})_{\overline{y}} \ar[r] \ar[ld] & \bigoplus\nolimits_{f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}} \ar[d] \\ (f'_!(g')^{-1}\mathcal{F})_{\overline{y}'} \ar[rr] & & \bigoplus\nolimits_{f'(\overline{x}') = \overline{y}'} (g')^{-1}\mathcal{F}_{\overline{x}'} } $$ commutes where the horizontal arrows are given in the proof of Lemma \ref{lemma-finite-support-stalk} and where the right vertical arrow is an equality by what we just said above. The southwest arrow is described in Remark \ref{remark-construct-map-presheaves-downstairs} as the pullback map, i.e., simply given by our construction $c$ above. Then the simple description of the image of a sum $\sum (Z_i, z_i)$ in the stalk at $\overline{x}$ given in the proof of Lemma \ref{lemma-finite-support-stalk} immediately shows the diagram commutes. This finishes the proof of the lemma.