VII.tex

\section{The functor ${ }^{f}$ :}

The inclusion $h: W \rightarrow x$ of a locally subspace of $a$ locally compact space gives rise to a very useful functor $h_{-!}: S h(w) \rightarrow S h(X)$, compare II. 6 and III.7. In order to generalize this notion let us observe that $h$ is a proper map if and only if $W$ is a closed subspace of $X$, III. 6 .

Definition 1.1. Let $f: X \rightarrow Y$ be a continuous map between locally compact spaces. For a sheaf $F$ on $X$ and an open subset $V$ of $Y$ put

$$
\Gamma\left(V, f_{!} F\right)=\left\{\begin{array}{l|l}
s \in \Gamma\left(f^{-1}(V), F\right) & \begin{array}{l}
\operatorname{supp}(s) \xrightarrow{f} Y \\
\text { is a proper map }
\end{array}
\end{array}\right\}
$$

We consider $f_{!} F$ as a subpresheaf of $f_{*} F$.

Proposition 1.2. The presheaf $f_{!} F$ is a sheaf on $Y$.

begin{proof} It suffices to prove that the map $f: X \rightarrow Y$ has the following property:

Given a family $\left(V_{i}\right)' \in$ of open subsets of $Y$ with union $V$ and a closed subset $S$ of $f^{-1}(V)$. If the restriction $f: s \cap V_{i} \rightarrow V_{i}$ is proper for all $i$, then the restriction $f: \mathrm{S} \rightarrow V$ is proper.

Let $K$ be a compact subset of $V$. We are going to construct a family $\left(K_{i}\right)_{i \in I}$ of compact subsets of $K_{\text {, with }} K_{i} \subseteq V_{i}$ for each $i \in I$, whose union is $K$ and such that $K_{i}$ is empty except for finitely many $i \epsilon I$.

For the construction of the $K_{i}$ 's we may assume that I is a finite set by Borel-Heine. For $x \in K$ choose a compact neighbourhood $K_{x}$ of $x$ contained in some $\mathrm{v}_{i}, i \in I$. Using Borel-Heine we obtain a finite covering of $K$ by compact subsets of $K$ each of which is contained in some $V_{i}$, $i \in I$. Let $K_{i}$ be the union at those of the components of the covering which are contained in $U_{\dot{1}}$. This gives the desired family $\left(K_{i}\right)_{i \in I}$. To conclude the proof notice that

$$
s \cap f^{-1}(K)=s \cap f^{-1}\left(U K_{i}\right)=U\left(S \cap f^{-1}\left(K_{i}\right)\right)
$$

which shows that $S \cap f^{-1}(K)$ is compact.

end{proof}

Let us consider the left exact functor

1.3

$$
f_{!}: \SH(X) \longrightarrow \Sh(Y)
$$

The i'th derived functor evaluated on the sheaf $F$ will be denoted $R^{i} f_{!} F$.

Theorem 1.4. Let $E: X \rightarrow Y$ denote a continuous map between locally compact spaces. For $y \in Y$ we have a natural isomorphism

$$
\left(R^{i} f_{!} F\right)_{y}=H_{c}^{i}\left(f^{-1}(y), F\right) \quad ; i \in \BbbZ
$$

as $F$ varies through the category of sheaves on $x$.

begin{proof} Let us first treat the case $i=0$. Consider an open neighbourhood $V$ of $y$ in $Y$ and the restriction

$$
\Gamma(V, f, F) \rightarrow \Gamma\left(f^{-1}(y), F\right)
$$

Consider a $s \in \Gamma\left(V, f_{!} F\right)$ which maps to zero. This means that Supp(s) $\cap f^{-1}(y)$ is empty or otherwise expressed $y \notin f(\operatorname{Supp}(s))$. Let $W$ denote the complement of Supp ( $s$ ) in $V$, this is an open neighbourhood of $y$ to which the restriction of $s$ is zero. Thus we have proved that the restriction

$$
\left(f_{!} F\right)_{y} \rightarrow \Gamma_{c}\left(f^{-1}(y), F\right)
$$

is injective. In case $F$ is soft the restriction map is surjective as it follows by remarking that $\Gamma_{c}(X, F)$ is a subgroup of $\Gamma(Y, f, F)$ and that restriction from $\Gamma_{c}(X, F)$ to $\Gamma_{c}\left(f^{-1}(Y), F\right)$ is surjective III.2.6. - In the general case consider an exact sequence $0 \rightarrow F \rightarrow S \rightarrow T$ with $S$ and $T$ soft and use the resulting diagram

\begin{center}
\includegraphics[max width=\textwidth]{2024_03_22_0f4161bdf598cd7e5dd2g-326}
\end{center}

to conclude that restriction is an isomorphism. - Let us now choose an injective resolution $F \rightarrow I^{\bullet}$. We have

$$
\left(R^{i_{f}}{ }_{!}\right)_{Y}=\left(H^{i} E_{!} I^{*}\right)_{Y}=H^{i}\left(f_{!} I^{*}\right)_{Y}=H^{i} \Gamma_{C}\left(f^{-1}(Y), I^{*}\right)
$$

The result follows by noticing that the restriction of $I^{*}$ to $f^{-1}(y)$ is a soft resolution of the restriction of $F$ to $f^{-1}(y)$, III.2.5.

end{proof}

A diagram as below is called cartesian if $(c, h): A \rightarrow X \times B$ induces an isomorphism between $A$ and the subspace $\{(x, b) \in X \times B \mid f(x)=p(b)\}$ of $X \times B$.

Corollary 1.5. Consider a cartesian square of locally compact spaces. For any sheaf $F$ on $X$ $A \rightarrow x$ we have

$$
p^{*} R^{i_{1}}{ }_{!} F \xrightarrow[\rightarrow]{\sim} R^{i_{h}} q^{*} F \quad ; i \in \BbbZ
$$

\begin{center}
\includegraphics[max width=\textwidth]{2024_03_22_0f4161bdf598cd7e5dd2g-326(1)}
\end{center}

begin{proof} The canonical morphism

$$
p^{\star} f_{!} F \longrightarrow h_{!} q^{*} F
$$

Coroilary 1.6. Let $f: X \rightarrow Y$ denote a continuous map between locally compact spaces. A soft sheaf $S$ on $x$ is transformed into a soft sheaf $f_{!} S$ on $Y$. Moreover $R^{i} f_{!} S=0$ for $i>0$.

begin{proof} Let us prove that for any compact subset $K$ of $Y$ and any sheaf $S$ on $x$

1.7

$$
\Gamma(K, f, S)=\Gamma_{C}\left(E^{-1}(K), S\right)
$$

To this end let $p: K \rightarrow Y$ and $g: f^{-1}(\mathrm{~K}) \rightarrow X$ denote the inclusions and $h: f^{-1}(K) \rightarrow K$ the restriction of $f$. According to 1.5 we have $p^{\star} f_{!} S=h_{!} q^{\star} S$. Apply $\Gamma(K,-)$ to this identity to get 1.7 .

In case $S$ is soft we shall prove that $f_{!} S$ is soft, i.e. that the restriction map

$$
\Gamma\left(Y, f_{!} \mathrm{S}\right) \longrightarrow \Gamma\left(K, f_{!} \mathrm{S}\right)
$$

is surjective. Notice that $\Gamma\left(Y, f_{!} S\right)$ contains $\Gamma_{c}(X, S)$ as

a subgroup. Restriction from this group to $\Gamma_{c}\left(f^{-1}(K), S\right)$ is surjective according to III.2.6 and the result follows from 1.7 . The second part follows from 1.4 by localization using the fact that $S$ induces soft sheaves on any of the fibres of $f$, III.2.5.

end{proof}
is an isomorphism as one sees by localization using 1.5 and identification of the fibre of $h$ over $b \in B$ with the fiber of $f$ over $p(b)$. Let us remark that in case $S$ is soft, then $q^{*} S$ is acyclic for $h_{!}$: Notice that $q^{\star} S$ induces soft sheaves on the fibres of $h$ and apply 1.4. For a soft resolution $S^\bullet$ of $F$ we have

$$
p^{\star} H^{i} f_{!} S^{*}=H^{i} p^{\star} f_{!} S^{*}=H^{i} h_{!} q^{\star} S^{*}
$$

from which the result follows using I.7.5.

end{proof}

Let us now consider a commutative ring $k$ and interprete f! as a functor

$$
f_{!}: \Sh(X, k) \longrightarrow \Sh(Y, k)
$$

and similarly for the derived functor

1.8

$$
R f_{!}: D^{+}(X, k) \longrightarrow D^{+}(Y, k)
$$

Given two continuous maps between locally compact spaces $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ then it is easily seen that

1.9

$$
(f \circ g)_{!}=g_{!} \circ f_{!}
$$

and as a consequence of 1.6 and $I .7 .15$

\section{2 The Künneth formula}
Let $k$ denote a noetherian ring and $f: X \rightarrow Y$ a continuous map of locally compact spaces both of finite dimension. We are first going to construct with the notation of XI. 2 a functor

$$
\operatorname{Rf}_{!}: D^{-}(X, k) \longrightarrow D^{-}(Y, k)
$$

on the basis of the following

Lemma 2.2. Let $x$ denote a locally compact space of finite dimension and $k$ a commutative ring. Given a soft sheaf $S$ and a sheaf $F$ on $X$. If either $S$ or $F$ is flat then $S \otimes_{k} F$ is a soft sheaf.

begin{proof} In case $S$ is flat we can use the proof of $v \cdot 1.3$. In case $F$ is flat, that proof needs a small modification: notice that $K=\ker \partial_{n-1}$ is a flat sheaf II.11.1.

end{proof}

Lemma 2.3. Let $x$ denote a locally compact space of finite dimension and $k$ a noetherian ring. For any $F^{*}$ in $K^{-}(X, k)$ there exists a quasi-isomorphism $F^\bullet \rightarrow T^\bullet$ in $K^{-}(X, k)$ where $T^\bullet$ is a complex of soft sheaves (a soft resulution of $F^{*}$ )

begin{proof} Choose according to VI. 1.3 a quasi-isomorphism $k \rightarrow S^{\circ}$ where $S^{*}$ is a bounded complex of soft and flat sheaves.

According to II.11.2 this gives a quasi-isomorphism $F^{\bullet} \rightarrow \mathrm{S}^{\bullet} \otimes F^{\bullet}$. The complex $\mathrm{S}^{\bullet} \otimes F^\bullet$ is a bounded above complex of soft sheaves by Lemma 2.2 .

Construction of $\mathrm{Rf}_{1}: D^{-}(X, k) \rightarrow D^{-}(Y, k)$

Let $S^{-}(x, k)$ denote the homotopy category of bounded above complexes of soft k-sheaves. It follows from 2.2 and XI. 2 that we may identify $D^{-}(X, k)$ with the category obtained by inverting all quasi-isomorphisms in $S^{-}(x, k)$. - The functor f! extends to a functor

$$
f_{!}: S^{-}(X, k) \longrightarrow S^{-}(Y, k)
$$

according to 1.6 . It remains to establish that $f_{!}$transforms quasi-isomorphisms into quasi-isomorphisms. To see this remark that $R^{n} f_{!}=0$ for $n>\operatorname{dim} x$ and use I.7.6.

Let us remark that the functor $f^{*}: \Sh(Y, k) \rightarrow \Sh(X, k)$ being exact extends immediately to a functor

$$
f^{*}: D^{-}(Y, k) \longrightarrow D^{-}(X, k)
$$

With the notations above and the notation of II. 11.10 we have the fundamental

Projection formula 2.4.

$$
\operatorname{Rf}_{!}\left(F^\bullet \stackrel{L}{\otimes} f^{\star} G^{*}\right)=\left(R_{!} F^{*}\right) \stackrel{L}{\otimes} G^{*}
$$

for $F^\bullet$ in $D^{-}(X, k)$ and $G$ in $D^{-}\left(Y^{*}, k\right)$.

begin{proof} Let us first prove that for a sheaf $F$ on $x$ and a flat sheaf $G$ on $Y$ there is a canonical isomorphism

2.5

$$
f_{!}(F) \otimes G \similarrightarrow f_{!}\left(F \otimes f^{*} G\right)
$$

Let us first notice that there is a natural transformation from the left hand side of the formula to the right hand side. Thus it follows from 1.4 that we may assume that $Y$ is a point. Next, let us remark that both sides of the formula are left exact functors in F. Thus we may assume that $F$ is soft.

It is now time to vary the k-module G. The case where $G$ is a free module follows from the fact that $\Gamma_{c}$ preserves direct sums, III.5.1. In general consider an exact sequence

$$
0 \rightarrow L_{2} \rightarrow L_{1} \rightarrow L_{0} \rightarrow G \rightarrow 0
$$

where $L_{1}$ and $L_{0}$ are free modules. Since $G$ is flat we can conclude that $L_{2}$ is flat. There results an exact sequence of sheaves on $X$

$$
0 \rightarrow S \otimes L_{2} \rightarrow S \otimes L_{1} \rightarrow S \otimes L_{0} \rightarrow S \otimes G \rightarrow 0
$$

This is a sequence of soft sheaves on $x$ by 2.2 . It follows from 1.7 .5 that the sequence

$$
0 \rightarrow \Gamma_{c}\left(X, S \otimes L_{2}\right) \rightarrow \Gamma_{c}\left(X, S \otimes L_{1}\right) \rightarrow \Gamma_{c}\left(X, S \otimes L_{0}\right) \rightarrow \Gamma_{c}(X, S \otimes G) \rightarrow 0
$$

is exact, in particular we deduce a commutative exact diagram

\begin{center}
\includegraphics[max width=\textwidth]{2024_03_22_0f4161bdf598cd7e5dd2g-332}
\end{center}

from which we conclude that the vertical arrow to the right is an isomorphism. This proves 2.5 .

Let us notice that in case $F$ is soft, then $F \otimes f^{*} G$ is soft as it follows from 2.2 and the fact that $f *_{G}$ is flat. With this remark in hand it is a simple matter to extend the formula 2.5 to the derived categories.

end{proof}

Base change 2.6. Consider a cartesian square of locally compact spaces of finite dimension. Then

\begin{center}
\includegraphics[max width=\textwidth]{2024_03_22_0f4161bdf598cd7e5dd2g-332(1)}
\end{center}

$$
p^{\star} \mathrm{Rf}_{!} F^\bullet \longrightarrow \mathrm{Rh}_{!} q^{\star} F^{\bullet}
$$

for all $F^{-}$in $D^{-}(X, k)$.

begin{proof} Let us remark that a soft sheaf $S$ on $x$ transforms into a sheaf $q^{\star} S$ on $A$ which is acyclic for $h_{!}$as it follows from 1.5 and 1.6. For a complex $S^{-}$in $K^{-}(x, k)$ choose a soft resolution $q^{*} S^{*} \rightarrow T^{*}$ in $K^{-}(A, k)$. According to 1.6 we have

$$
p^{*} f_{!} S^{\bullet} \similarrightarrow h_{!} q^{*} S^{*} \rightarrow h_{!} T^{*}
$$

The second arrow is a quasi-isomorphism by I.7.7.

With the notation of 2.6 put $c=f q=p h$.

Künneth formula 2.7.

$$
\mathrm{Rc}_{!}\left(h^{\star} E^\bullet \otimes G^{\star} F^{\bullet}\right)=\left(\mathrm{Rp}_{!} E^\bullet\right)^{L}\left(\mathrm{Rf}_{!} F^{*}\right)
$$

for $E^\bullet$ in $D^{-}(B, k)$ and $F^{-}$in $D^{-}(X, k)$.

begin{proof} Use first the projection formula 2.4 and next the base change formula 2.6 to get

$$
R q_{!}\left(h^{\star} E^{*} \otimes q^{\star} F^{*}\right)=R q_{!}\left(h^{*} E^{*}\right) \stackrel{L}{\&}=f *\left(R p_{!} E^{*}\right) \stackrel{L}{L} F^{*}
$$

Apply $\mathrm{Rf}_{!}$to this and use $\mathrm{Rc}_{!}=\mathrm{Rf}_{!} \circ \mathrm{Rq}_{!}$to get

$$
R c_{!}\left(h^{\star} E^{*} \otimes q^{*} F\right)=R F_{!}\left(f^{*}\left(R p_{!} E^{*}\right) \stackrel{L}{\otimes} F^{*}\right)
$$

Apply the projection formula once more to get the result.

end{proof}

In particular for two locally compact spaces $X$ and $Y$ of finite dimension and a noetherian ring $k$

$$
2.8 \quad R \Gamma_{C} \cdot(X \times Y, k)=R \Gamma_{C} \cdot(X, k) \stackrel{L}{\otimes_{k}} R \Gamma_{C} \cdot(Y, k)
$$

\section{Giobal form of Verdier duality}

Let $k$ be a noetherian ring and $f: X \rightarrow Y$ a continuous map between locally compact spaces of finite dimension.

Theorem 3.1. There exists an additive functor

$$
f^{!}: D^{+}(Y, k) \rightarrow D^{+}(X, k)
$$

and a natural isomorphism

$$
\left[\operatorname{Rf}_{!} F^{\bullet}, G^{*}\right] \stackrel{\sim}{\simeq}\left[F^{\bullet}, f^{!} G^{\bullet}\right]
$$

as $F^\bullet$ varies through $D^{+}(X, k)$ and $G^\bullet$ varies through $D^{+}(Y, k)$.

begin{proof} We shall follow the proof of VI.1.1 closely and give the needed modifications. For a soft and flat sheaf $S$ on $X$ and a sheaf $G$ on $Y$ the functor

$$
F \longrightarrow \hom\left(f_{!}(S \otimes F), G\right)
$$

from $\Sh(x, k)$ to the category of $k$-modules transforms kernels into cokernels and direct sums into direct products: To see this notice that the functor $F \mapsto f_{!}(S \otimes F)$ is exact as it follows from 2.2 and 1.6 . The same functor preserves direct sums or more generally direct limits as it follows from 1.4 and III.5.

\begin{enumerate}
  \item With $S$ and $G$ as above, the presehaf on $x$
\end{enumerate}

$U \longmapsto \hom\left(f_{!}\left(S_{U}\right), G\right)$\\
is a sheaf on $\dot{x}$, here $s_{U}=j_{!} j^{*} S=S \otimes j_{!} k$ where $j: U \rightarrow X$ denotes the inclusion: For open subsets $U$ and $V$ of $x$ consider the transform of the exact sequence

\begin{center}
\includegraphics[max width=\textwidth]{2024_03_22_0f4161bdf598cd7e5dd2g-335}
\end{center}

by the functor $F \mapsto H O m\left(f_{!}(S \otimes F), G\right)$.

\begin{enumerate}
  \setcounter{enumi}{1}
  \item Let $f^{!}(5, G)$ denote the above sheaf on $x$. Here $S$ is a k-flat and soft sheaf on X and $G$ is any sheaf on $Y$.

  \item There is a canonical isomoprphism

\end{enumerate}

$\hom\left(f_{!}(F \otimes S), G\right) \similarrightarrow \hom(F, f !(S, G))$

as $F$ varies through the category $\Sh(X, k)$. - Let us first establish the identification

3.3

$$
f_{\star} f^{!}(\mathrm{S}, G)=\hom\left(f_{!} \mathrm{S}, G\right)
$$

To this end consider the inclusion $j: V \rightarrow Y$ of an open subset of $Y$ and notice that

$$
\hom\left(j * f_{!} S, j * G\right)=\hom\left(j_{!} j^{\star} f_{l} S, G\right)=\hom\left(f_{!}\left(S_{f^{-1}(V)}\right), G\right)
$$

as it follows by the base change property 1.5. -

By adjunction we deduce from 3.3 a morphism of sheaves

$$
f \star \hom(f, S, G) \rightarrow f^{!}(S, G)
$$

Let us now depart from the morphism $\quad f_{\star} F \otimes f_{!} S \rightarrow f_{!}(F \otimes S)$\\
and deduce first a morphism, II. 12

$$
\hom\left(f_{!}(F \otimes S), G\right) \rightarrow \hom\left(f_{\star} F, \hom\left(f_{!} S, G\right)\right)
$$

and by adjunction a morphism

$$
\hom\left(f_{!}(F \otimes S), G\right) \rightarrow \hom\left(F, f_{\star} \hom\left(f_{!} S, G\right)\right)
$$

Combine this with the morphism above to get

$$
\hom(f,(F \otimes S), G) \rightarrow \hom\left(F, f^{!}(S, G)\right)
$$

To prove that this is an isomorphism we shall vary $F$. It suffices to check the case $F=j, k$ where $j$ is the inclusion of an open subset of $X$.

4-7) Needs no essential modifications.

end{proof}

Example 3.4. Let $h: W \rightarrow x$ denote the inclusion of $a$ locally closed subspace. With the notation of II. 6 the functor

$$
h^{!}: \Sh(x, k) \rightarrow \Sh(W, k)
$$

is left exact and transforms injectives into injectives. Thus it extends to a functor

$$
h^{!}: D^{+}(x, k) \rightarrow D^{+}(w, k)
$$

which is a right adjoint to the functor $h_{1}$ described in II. 6 . This provides an extension of Theorem 3.1 for this kind of immersions beyond the framwork of locally compact spaces. The case of a closed subspace will be explored in Chapter VIII.

Let $f: X \rightarrow Y$ be a finite covering space of degree $n$, i.e. for each point $y \in Y$ there exists an open neighbourhood $V$ of $y$ such that $f: f^{-1}(V) \rightarrow V$ is isomorphic to the projection $V \times[1, n] \rightarrow V$. For a sheaf $F$ on $x$ and $y \in Y$ we have a canonical isomorphism

\section{Covering spaces}

$4=1$

$$
\left(f_{\star} F\right)_{y} \similarrightarrow \underset{x \in f^{-1}(y)}{\oplus} F_{x}
$$

from which we conclude that the functor

$$
\text { 4.2 } f_{\star}: \Sh(X, k) \rightarrow \Sh(Y, k) \text { is exact }
$$

For a sheaf $G$ on $Y$ we deduce from 4.1 an isomorphism on the stalks at $y \in Y$

$$
\left(f_{*} f^{*} G\right) \similarrightarrow \underset{x \in f^{-1}(y)}{ }{ }^{G} y
$$

Compose this with "summation" to get the trace map

$$
4.3 \quad \operatorname{tr}_{y}:\left(f^{*} f_{\star} G\right)_{y} \rightarrow G_{y} \quad ; y \in Y
$$

There exists a unique morphism of sheaves

4.4

$$
\operatorname{tr}: f_{\star} f^{\star} G \longrightarrow G
$$

whose stalks are those recorded in 4.3: Uniqueness follows from II.2.2.i. By II. 12 the problem is local on Y. Thus it suffices to treat the case where the covering is trivial which\\
is left to the reader. - For a sheaf $F$ on $X$ the trace map induces an isomorphism

4.5

$$
\hom(F, f * G) \xrightarrow[\rightarrow]{\sim} \hom(f, F, G)
$$

begin{proof} The construction above yields more generally a morphism of sheaves on $Y$

4.6

$$
f_{\star} \hom\left(F, f^{*} G\right) \rightarrow \hom\left(f_{\star} F, G\right)
$$

We shali in fact prove that the morphism 4.6 is an isomorphism: The new problem is local on Y. Thus it suffices to treat the case of a trivial covering which is left to the reader.

end{proof}

Corollary 4.7. The pull back functor

$$
f^{\star}: \Sh(Y, k) \rightarrow \Sh(X, k)
$$

transforms injectives into injectives.

begin{proof} Follows formally from the presence of a left adjoint which is exact by 4.2 .

From our discussion follows that in the case where $x$

and $Y$ are locally compact we have $E^{!}=E^{*}$ in this particular case of Verdier duality 2.1.

For a sheaf $G$ on $Y$ let res: $G \rightarrow f_{\star} f^{*} G$ denote the standard adjunction morphism, II.4.9. By localization on $Y$ follows that

4.8

$$
\text { trores }=n
$$

Let us consider an injective resolution $\underset{\sim}{k} \rightarrow G^{\circ}$. Apply the functor $H \cdot \Gamma(Y,-)$ to the morphisms

4.9

$$
G \cdot \xrightarrow{\text { res }} f_{\star} f^{\star} G^{*} \xrightarrow{\text { tr }} G^{*}
$$

to obtain morphisms on cohomology $f^{*}=$ res

4.10

$$
H^{*}(Y, k) \xrightarrow{\text { res }} H^{*}(x, k) \xrightarrow{\text { tr }} H^\bullet(Y, k)
$$

which satisfies the relation 4.8 . Notice that

$4 \cdot 11$

$\operatorname{tr}\left(f^{\star} \eta \cup \xi\right)=\eta \cup \operatorname{tr}(\xi)$

; $\eta \in H^{\bullet}(Y, k), \xi \in H^{*}(X, k)$

as it follows from the construction above.

\section{5 Local form of Veraier duality}
In this section we shall give a local version of Verdier duality. To do so we shall use the general theory of derived categories as exposed in Chapter XI.

Given a topological space and a commutative ring k. For $F^{*}$ in $D^{-}(X, k)$ and $F^{*}$ in $D^{+}(X, k)$ we shall define $R \hom^{*}(E, F)$ in $D^{+}(X, k)$ : choose an injective resolution $F^{+} \rightarrow J^{+}$and put 5.1 $\quad \operatorname{RHOm}\left(E^{\bullet}, F^{*}\right)=\hom\left(E^{\bullet}, J^{*}\right)$.

This takes a particularly simple form if we represent the derived categories as follows $D^{-}(x, k)$ : The category obtained from the homotopy category of bounded above complexes of flat sheaves by inverting all quasi-isomorphisms. $D^{+}(x, k)$ : The homotopy category of bounded below complexes of injective sheaves. In particular, with $E^{*}$ and $F^{-}$such represented, the complex $\mathrm{Hom}^{\circ}\left(E^{\bullet}, F^{\bullet}\right)$ is automatically a bounded below complex of injective sheaves, II. 12.3 .

Theorem 5.2. Let $k$ be a noetherian ring and $f: X \rightarrow Y$ a continuous map of locally compact spaces of finite dimension. There is a natural isomorphism in $D^{+}(X, k)$

\begin{center}
\includegraphics[max width=\textwidth]{2024_03_22_0f4161bdf598cd7e5dd2g-340}
\end{center}

as $E^{*}$ varies through $D^{-}(X, k)$ and $F^{*}$ through $D^{+}(x, k)$.

begin{proof} Let $S$ be a soft and flat sheaf on $x$. The isomorphism 3.2 extends easily to an isomorphism

5.3

$$
\hom(f,(E \otimes S), F)=f_{\star} \hom\left(E, f^{1}(S, F)\right)
$$

as $E$ varies through $\SH(X, k)$ and $F$ through $\SH(Y, k)$. Let us choose a fixed bounded flat and soft resolution $\underset{\sim}{k} \rightarrow s^{\circ}$ on $X_{\text {; VI }}, 1.3$. For a bounded above complex $E^\bullet$ of flat sheaves on $X$ and a bounded below complex $F^{-}$of injective sheaves on $Y$ we deduce from 5.3 an isomorphism.

5.4

$$
\hom\left(f_{!}\left(E^{*} \otimes S^{*}\right), F^{*}\right) \similarrightarrow f_{\star} \hom^{*}\left(E^{*}, f^{!}\left(S^{*}, F^{*}\right)\right)
$$

We can conclude the proof by the following four references: The morphism $E^{*} \rightarrow E^{*} \otimes \mathrm{S}^{*}$ is a soft resolution of $E^{*}$ by 2.2 . The complex $f^{!}\left(\mathrm{S}^{\bullet}, F^{\circ}\right)$ is a bounded below complex of injective sheaves on $X$ as it follows from the proof of 3.1. The complex Hor ${ }^{*}\left(E^{*}, f^{!}\left(S^{*}, F^{*}\right)\right)$ is a bounded below complex of injective sheaves as it follows from II.12.3. Any bounded above complex admits a flat resolution by II.11.8.

end{proof}

Example 5.5. Consider a finite covering $\mathrm{f:} X \rightarrow Y$. Verdier duality is simply represented by the isomorphism 4.6 .