Improve handling LC/X versus X

sites-cohomology.tex

@@ -4947,56 +4947,63 @@ \section{Cohomology on Hausdorff and locally quasi-compact spaces}
 
 \begin{lemma}
 \label{lemma-describe-pullback-pi}
-Let $X$ be an object of $\textit{LC}_{qc}$. Let $\mathcal{F}$ be a
-sheaf on $X_{Zar}$. Then the sheaf $\pi^{-1}\mathcal{F}$ on
-$\textit{LC}_{Zar}/X$ is given by the rule
+Let $X$ be an object of $\textit{LC}$. Let $\mathcal{F}$ be a
+sheaf on $X$. The rule
 $$
-\pi^{-1}\mathcal{F}(Y) = \Gamma(Y_{Zar}, f^{-1}\mathcal{F})
+\textit{LC} \longrightarrow \textit{Sets},\quad
+(f : Y \to X) \longmapsto \Gamma(Y, f^{-1}\mathcal{F})
 $$
-for $f : Y \to X$ in $\textit{LC}_{qc}$. Moreover $\pi^{-1}\mathcal{F}$
-is a sheaf for the qc topology, i.e., the sheaf
-$\epsilon^{-1}\pi^{-1}\mathcal{F}$ on $\textit{LC}_{qc}$
-is given by the same formula.
+is a sheaf in the qc topology, i.e., this rule defines a sheaf on
+$\textit{LC}_{qc}$ and a fortiori on $\textit{LC}_{Zar}$.
+If we view $\mathcal{F}$ as a sheaf on $X_{Zar}$, then the sheaf
+above is equal to $\epsilon^{-1}\pi^{-1}\mathcal{F}$ on
+$\textit{LC}_{qc}$ and $\pi^{-1}\mathcal{F}$ on $\textit{LC}_{Zar}$.
 \end{lemma}
 
 \begin{proof}
-Of course the pullback $f^{-1}$ on the right hand side indicates
-usual pullback of sheaves on topological spaces
-(Sites, Example \ref{sites-example-continuous-map}).
-The equality of the lemma follows directly from the definitions.
+Of course the pullback $f^{-1}$ in the formula denotes usual
+pullback of sheaves on topological spaces. Let us denote
+$\mathcal{F}^a$ the presheaf on the category $\textit{LC}$
+defined by the rule given in the lemma. It is immediate
+from the defitions that $\mathcal{F}^a$ is a sheaf for the Zar
+topology.
 
 \medskip\noindent
-Let $\mathcal{V} = \{g_i : Y_i \to Y\}_{i \in I}$ be a covering of
-$\textit{LC}_{qc}/X$. It suffices to show that
-$\pi^{-1}\mathcal{F}(Y) \to H^0(\mathcal{V}, \pi^{-1}\mathcal{F})$
+Let $Y \to X$ be a morphism in $\textit{LC}$.
+$\mathcal{V} = \{g_i : Y_i \to Y\}_{i \in I}$ be a qc covering.
+To prove $\mathcal{F}^a$ is a sheaf for the qc topology it
+suffices to show that
+$\mathcal{F}^a(Y) \to H^0(\mathcal{V}, \mathcal{F}^a)$
 is an isomorphism, see Sites, Section \ref{sites-section-sheafification}.
 We first point out that the map is injective as a qc covering
 is surjective and we can detect equality of sections at stalks
 (use Sheaves, Lemmas \ref{sheaves-lemma-sheaf-subset-stalks} and
-\ref{sheaves-lemma-stalk-pullback-presheaf}). Thus we see that
-$\pi^{-1}\mathcal{F}$ is a separated presheaf on $\textit{LC}_{qc}$
+\ref{sheaves-lemma-stalk-pullback-presheaf}). Thus
+$\mathcal{F}^a$ is a separated presheaf on $\textit{LC}_{qc}$
 hence it suffices to show that any element
-$(s_i) \in H^0(\mathcal{V}, \pi^{-1}\mathcal{F})$
-maps to an element in the image of $\pi^{-1}\mathcal{F}(Y)$
+$(s_i) \in H^0(\mathcal{V}, \mathcal{F}^a)$
+maps to an element in the image of $\mathcal{F}^a(Y)$
 after replacing $\mathcal{V}$ by a refinement
 (Sites, Theorem \ref{sites-theorem-plus}).
 
 \medskip\noindent
-Observe that $\pi^{-1}\mathcal{F}|_{Y_{i, Zar}}$ is the pullback
-of $f^{-1}\mathcal{F} = \pi^{-1}\mathcal{F}|_{Y_{Zar}}$ under
+Identifying sheaves on $Y_{i, Zar}$ and sheaves on $Y_i$ we find that
+$\mathcal{F}^a|_{Y_{i, Zar}}$ is the pullback of $f^{-1}\mathcal{F}$ under
 the continuous map $g_i : Y_i \to Y$. Thus we can choose an open covering
 $Y_i = \bigcup V_{ij}$ such that for each $j$ there is an open
-$W_{ij} \subset Y$ and a section $t_{ij} \in \pi^{-1}\mathcal{F}(W_{ij})$
-such that $s|_{U_{ij}}$ is the pullback of $t_{ij}$. In other words,
+$W_{ij} \subset Y$ and a section $t_{ij} \in \mathcal{F}^a(W_{ij})$
+such that $V_{ij}$ maps into $W_{ij}$ and such that
+$s|_{V_{ij}}$ is the pullback of $t_{ij}$. In other words,
 after refining the covering $\{Y_i \to Y\}$ we may assume there
 are opens $W_i \subset Y$ such that $Y_i \to Y$ factors through $W_i$
-and sections $t_i$ of $\pi^{-1}\mathcal{F}$ over $W_i$ which restrict
+and sections $t_i$ of $\mathcal{F}^a$ over $W_i$ which restrict
 to the given sections $s_i$. Moreover, if $y \in Y$ is in the image
 of both $Y_i \to Y$ and $Y_j \to Y$, then the images $t_{i, y}$
 and $t_{j, y}$ in the stalk $f^{-1}\mathcal{F}_y$ agree
 (because $s_i$ and $s_j$ agree over $Y_i \times_Y Y_j$).
 Thus for $y \in Y$ there is a well defined element $t_y$ of
-$f^{-1}\mathcal{F}_y$ agreeing with $t_{i, y}$ whenever $y \in Y_i$.
+$f^{-1}\mathcal{F}_y$ agreeing with $t_{i, y}$ whenever $y$
+is in the image of $Y_i \to Y$.
 We will show that the element $(t_y)$ comes from a global section
 of $f^{-1}\mathcal{F}$ over $Y$ which will finish the proof of the lemma.
 
@@ -5006,34 +5013,58 @@ \section{Cohomology on Hausdorff and locally quasi-compact spaces}
 Pick $i_1, \ldots, i_n \in I$ and
 quasi-compact subsets $E_j \subset Y_{i_j}$ such that
 $\bigcup g_{i_j}(E_j)$ is a neighbourhood of $y_0$.
-Then we can find an open neighbourhood $V \subset Y$ of $y_0$
-contained in $W_{i_1} \cap \ldots \cap W_{i_n}$ such that
-the sections $t_{i_j}|_V$, $j = 1, \ldots, n$ agree.
-Hence we see that $(t_y)_{y \in V}$ comes from this section
-and the proof is finished.
+Let $V \subset Y$ be an open neighbourhood of $y_0$ contained
+in $\bigcup g_{i_j}(E_j)$ and contained in $W_{i_1} \cap \ldots \cap W_{i_n}$.
+Since $t_{i_1, y_0} = \ldots = t_{i_n, y_0}$, after shrinking $V$
+we may assume the sections $t_{i_j}|_V$, $j = 1, \ldots, n$ of
+$f^{-1}\mathcal{F}$ agree. As $V \subset \bigcup g_{i_j}(E_j)$
+we see that $(t_y)_{y \in V}$ comes from this section.
+
+\medskip\noindent
+We still have to show that $\mathcal{F}^a$ is equal to
+$\epsilon^{-1}\pi^{-1}\mathcal{F}$ on $\textit{LC}_{qc}$,
+resp.\ $\pi^{-1}\mathcal{F}$ on $\textit{LC}_{Zar}$.
+In both cases the pullback is defined by taking the presheaf
+$$
+(f : Y \to X)
+\longmapsto
+\colim_{f(Y) \subset U \subset X} \mathcal{F}(U)
+$$
+and then sheafifying. Sheafifying in the Zar topology
+exactly produces our sheaf $\mathcal{F}^a$ and the fact
+that $\mathcal{F}^a$ is a qc sheaf, shows that it works as well
+in the qc topology.
 \end{proof}
 
 \begin{lemma}
 \label{lemma-compare-cohomology-LC}
-Let $X$ be an object of $\textit{LC}_{qc}$. Let $\mathcal{F}$ be an
-abelian sheaf on $X_{Zar}$. Then we have
+Let $X$ be an object of $\textit{LC}$. Let $\mathcal{F}$ be an
+abelian sheaf on $X$. Then
 $$
-H^q(X_{Zar}, \mathcal{F}) =
-H^q(\textit{LC}_{qc}/X, \epsilon^{-1}\pi^{-1}\mathcal{F})
+H^q(X, \mathcal{F}) =
+H^q(\textit{LC}_{Zar}/X, \mathcal{F}^a) =
+H^q(\textit{LC}_{qc}/X, \mathcal{F}^a)
 $$
+where $\mathcal{F}^a = \epsilon^{-1}\pi^{-1}\mathcal{F} = \pi^{-1}\mathcal{F}$
+is as in Lemma \ref{lemma-describe-pullback-pi}.
 In particular, if $A$ is an abelian group, then we have
 $H^q(X, \underline{A}) = H^q(\textit{LC}_{qc}/X, \underline{A})$.
 \end{lemma}
 
 \begin{proof}
-The statement is more precisely that the canonical map
+The equality $H^q(X, \mathcal{F}) = H^q(\textit{LC}_{Zar}/X, \mathcal{F}^a)$
+is a general fact coming from the trivial observation that
+coverings of $X$ in $\textit{LC}_{Zar}$ are the same thing as open
+coverings of $X$. The reader who wishes to see a detailed proof
+should apply Lemma \ref{lemma-cohomology-bigger-site} to the functor
+$X_{Zar} \to \textit{LC}_{Zar}$. In the rest of the proof we
+show that
 $$
-H^q(X_{Zar}, \mathcal{F}) \longrightarrow
-H^q(\textit{LC}_{qc}/X, \epsilon^{-1}\pi^{-1}\mathcal{F})
+H^q(X, \mathcal{F}) = H^q(\textit{LC}_{qc}/X, \mathcal{F}^a)
 $$
-is an isomorphism for all $q$.
-The result holds for $q = 0$ by Lemma \ref{lemma-describe-pullback-pi}.
-We argue by induction on $q$. Pick $q_0 > 0$. We will assume the result
+We argue by induction on $q$.
+The base case $q = 0$ is Lemma \ref{lemma-describe-pullback-pi}.
+Pick $q_0 > 0$. We will assume the result
 holds for $q < q_0$ and prove it for $q_0$.
 
 \medskip\noindent
@@ -5047,25 +5078,25 @@ \section{Cohomology on Hausdorff and locally quasi-compact spaces}
 E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}))
 \longrightarrow
 E_2^{p, q} =
-\check{H}^p(\mathcal{U}, \underline{H}^q(\epsilon^{-1}\pi^{-1}\mathcal{F}))
+\check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}^a))
 $$
 between the spectral sequences of
 Cohomology, Lemma \ref{cohomology-lemma-cech-spectral-sequence} and
-Lemma \ref{lemma-cech-spectral-sequence}.
-Since the maps
+Lemma \ref{lemma-cech-spectral-sequence}. The maps
 $\underline{H}^q(\mathcal{F})(U_{i_0 \ldots i_p}) \to
-\underline{H}^q(\epsilon^{-1}\pi^{-1}\mathcal{F}))(U_{i_0 \ldots i_p})$
-are isomorphisms for $q < q_0$ we see that
+\underline{H}^q(\mathcal{F}^a)(U_{i_0 \ldots i_p})$
+are isomorphisms for $q < q_0$ by the induction hypothesis
+applied to $\mathcal{F}|_{U_{i_0 \ldots i_p}}$. Thus
 $$
 \text{Ker}(H^{q_0}(X, \mathcal{F}) \to \prod H^{q_0}(U_i, \mathcal{F}))
 $$
 maps isomorphically to the corresponding subgroup of
-$H^{q_0}(\textit{LC}_{qc}/X, \epsilon^{-1}\pi^{-1}\mathcal{F})$.
+$H^{q_0}(\textit{LC}_{qc}/X, \mathcal{F}^a)$.
 In this way we conclude that our map is injective for $q_0$.
 
 \medskip\noindent
 Surjective.
-Let $\xi \in H^{q_0}(\textit{LC}_{qc}/X, \epsilon^{-1}\pi^{-1}\mathcal{F})$.
+Let $\xi \in H^{q_0}(\textit{LC}_{qc}/X, \mathcal{F}^a)$.
 If for every $x \in X$ we can find a neighbourhood $x \in U \subset X$
 such that $\xi|_U = 0$, then we can use the {\v C}ech complex argument
 of the previous paragraph to conclude that $\xi$ is in the image of
@@ -5077,16 +5108,17 @@ \section{Cohomology on Hausdorff and locally quasi-compact spaces}
 We may replace $X$ by $\bigcup f_{i_j}(E_j)$ and set $Y = \coprod E_{i_j}$.
 Then $Y \to X$ is a surjective continuous map of Hausdorff and
 quasi-compact topological spaces,
-$\xi \in H^{q_0}(\textit{LC}_{qc}/X, \epsilon^{-1}\pi^{-1}\mathcal{F})$,
+$\xi \in H^{q_0}(\textit{LC}_{qc}/X, \mathcal{F}^a)$,
 and $\xi|_Y = 0$. Set $Y_p = Y \times_X \ldots \times_X Y$ ($p + 1$-factors)
 and denote $\mathcal{F}_p$ the pullback of $\mathcal{F}$ to $Y_p$.
-Then the spectral sequence
+Consider the spectral sequence
 $$
 E_1^{p, q} =
-\check{C}^p(\{Y \to X\}, \underline{H}^q(\epsilon^{-1}\pi^{-1}\mathcal{F}))
+\check{C}^p(\{Y \to X\}, \underline{H}^q(\mathcal{F}^a))
 $$
-of Lemma \ref{lemma-cech-spectral-sequence} has
-rows for $q < q_0$ which are (by induction) the complexes
+of Lemma \ref{lemma-cech-spectral-sequence}. By induction hypothesis
+applied to the pullback of $\mathcal{F}$ to $Y_p$ the $q$th
+row of $E_1^{p, q}$ for $q < q_0$ is the complex
 $$
 H^q(Y_0, \mathcal{F}_0) \to
 H^q(Y_1, \mathcal{F}_1) \to
@@ -5105,9 +5137,9 @@ \section{Cohomology on Hausdorff and locally quasi-compact spaces}
 H^q(Y_1 \times_X U, \mathcal{F}_1) \to
 H^q(Y_2 \times_X U, \mathcal{F}_2) \to \ldots\right)
 $$
-are exact (some details omitted). By the proper base change
+are exact (small detail omitted). By the proper base change
 theorem in topology
-(for example Cohomology, Lemma \ref{cohomology-lemma-proper-base-change})
+(Cohomology, Lemma \ref{cohomology-lemma-proper-base-change})
 the colimit is equal to
 $$
 H^q(Y_x, \underline{\mathcal{F}_x}) \to
@@ -5139,6 +5171,108 @@ \section{Cohomology on Hausdorff and locally quasi-compact spaces}
 has the required exactness properties this finishes the proof of the lemma.
 \end{proof}
 
+\begin{lemma}
+\label{lemma-cohomological-descent-LC}
+Let $X$ be an object of $\textit{LC}$. For $K \in D^+(X)$ the maps
+$$
+K \to R\pi_*\pi^{-1}K
+\quad\text{and}\quad
+K \to R(\pi \circ \epsilon)_*(\pi \circ \epsilon)^{-1}K
+$$
+are isomorphisms where
+$\pi : \Sh(\textit{LC}_{Zar}/X) \to \Sh(X)$ and
+$\pi \circ \epsilon : \Sh(\textit{LC}_{qc}/X) \to \Sh(X)$
+are the morphisms of topoi described above.
+\end{lemma}
+
+\begin{proof}
+This is a consequence of Lemma \ref{lemma-compare-cohomology-LC}
+and simple properties of cohomology; we suggest the reader skip the proof.
+Suppose that $\mathcal{F}$ is an abelian sheaf on $X$.
+Let $\mathcal{F}^a = \epsilon^{-1}\pi^{-1}\mathcal{F} = \pi^{-1}\mathcal{F}$
+be as in Lemma \ref{lemma-describe-pullback-pi}. Recall that
+$$
+R^q\pi_*\mathcal{F}^a
+\quad\text{resp. }\quad
+R^q(\pi \circ \epsilon)_*\mathcal{F}^a
+$$
+is the sheaf associated to the presheaf
+$$
+U \longmapsto H^q(\textit{LC}_{Zar}/U)
+\quad\text{resp. }\quad
+H^q(\textit{LC}_{qc}/U)
+$$
+See Lemma \ref{lemma-higher-direct-images}. By
+Lemma \ref{lemma-compare-cohomology-LC} this is equal to
+$U \mapsto H^q(U, \mathcal{F})$, i.e., the presheaf
+$\underline{H}^q(\mathcal{F}$. Since this presheaf sheafifies
+to zero for $q > 0$ and to $\mathcal{F}$ for $q = 0$ we find
+that $R^q\pi_*\mathcal{F}^a = 0$,
+resp.\ $R^q(\pi \circ \epsilon)_*\mathcal{F}^a = 0$
+for $q > 0$ and that $\pi_*\mathcal{F}^a = \mathcal{F}$,
+resp.\ $(\pi \circ \epsilon)_*\mathcal{F}^a = \mathcal{F}$.
+Represent $K$ by a bounded below complex $\mathcal{F}^\bullet$.
+By Leray's acyclicity lemma
+(Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity})
+applied to $\pi^{-1}\mathcal{F}^\bullet$,
+resp.\ $(\pi \circ \epsilon)^{-1}\mathcal{F}^\bullet$
+and $\pi_*$, resp.\ $(\pi \circ \epsilon)_*$ we conclude.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-push-pull-LC}
+Let $f : X \to Y$ be a morphism of $\textit{LC}$.
+Then there is a commutative diagram
+$$
+\xymatrix{
+\Sh(\textit{LC}_{qc}/X) \ar[r]_j \ar[d]_a &
+\Sh(\textit{LC}_{qc}/Y) \ar[d]^b \\
+\Sh(X) \ar[r]^f &
+\Sh(Y)
+}
+$$
+of topoi. If $f$ is proper, then $b^{-1} \circ f_* = j_* \circ a^{-1}$.
+\end{lemma}
+
+\begin{proof}
+The morphism of topoi $j$ is the one from
+Sites, Lemma \ref{sites-lemma-relocalize}
+which in our case comes from the continuous functor
+$Z/Y \mapsto Z \times_Y X/X$, see
+Sites, Lemma \ref{sites-lemma-relocalize-given-fibre-products}.
+The diagram commutes simply because the corresponding
+continuous functors compose correctly
+(see Sites, Lemma \ref{sites-lemma-composition-morphisms-sites}).
+Let $\mathcal{F}$ be a sheaf on $X$. Let $g : Z \to Y$ be an object of
+$\textit{LC}/Y$. Consider the fibre product
+$$
+\xymatrix{
+Z' \ar[r]_{f'} \ar[d]_{g'} & Z \ar[d]^g \\
+X \ar[r]^f & Y
+}
+$$
+Then we have
+$$
+(j_*a^{-1}\mathcal{F})(Z/Y) =
+(a^{-1}\mathcal{F})(Z'/X)  =
+\Gamma(Z', (g')^{-1}\mathcal{F})  =
+\Gamma(Z, f'_*(g')^{-1}\mathcal{F})
+$$
+the second equality by Lemma \ref{lemma-describe-pullback-pi}.
+On the other hand
+$$
+(b^{-1}f_*\mathcal{F})(Z/Y) = \Gamma(Z, g^{-1}f_*\mathcal{F})
+$$
+again by Lemma \ref{lemma-describe-pullback-pi}.
+Hence by proper base change for sheaves of sets
+(Cohomology, Lemma \ref{cohomology-lemma-proper-base-change-sheaves-of-sets})
+we conclude the two sets are canonically isomorphic.
+The isomorphism is compatible with restriction mappings
+and defines an isomorphism $b^{-1}f_*\mathcal{F} = j_*a^{-1}\mathcal{F}$.
+Thus an isomorphism of functors
+$b^{-1} \circ f_* = j_* \circ a^{-1}$.
+\end{proof}
+
 \begin{lemma}
 \label{lemma-proper-surjective-is-qc-covering}
 Let $f : X \to Y$ be a morphism of $\textit{LC}$.