Introduction to Verdier Duality

Xinyu Zhou

Abstract.

We give a brief introduction to the Verdier duality. The main references are [1], [2], and [3].

Notations

In this note, $X$ and $Y$ are topological manifolds or more generally locally compact Hausdorff topological spaces of finite cohomological dimensions. We will fix a base field, which will be denoted as $k$ or $A$ . The corresponding constant sheaf on $X$ is denoted as $\underline{A}_{X}$ . A sheaf $\operatorname{\mathcal{F}}$ on $X$ is understood as a sheaf of $\underline{A}_{X}$ -modules. Therefore, one can also regard $X$ as a ringed space $(X,\operatorname{\mathcal{O}}_{X})$ , where $\operatorname{\mathcal{O}}_{X}:=\underline{A}_{X}$ ; so $\operatorname{\mathcal{F}}$ is an $\operatorname{\mathcal{O}}_{X}$ -module. We will work with the derived category $D^{b}(X):=D^{b}(\mathrm{Mod}_{\operatorname{\mathcal{O}}_{X}})$ of bounded complexes of $\operatorname{\mathcal{O}}_{X}$ -modules, although many constructions would work for the derived category $D^{+}(X)$ of bounded-below complexes. If $\operatorname{\mathcal{F}}$ and $\operatorname{\mathcal{G}}$ are two sheaves on $X$ , the sheaf-hom will be denoted as $\underline{\operatorname{\mathrm{Hom}}}(\operatorname{\mathcal{F}},% \operatorname{\mathcal{G}})$ .

Let $\operatorname{\mathcal{F}}$ be a sheaf on $X$ . For a subset $Z$ of $X$ (not necessarily open or closed), write $i:Z\to X$ for the inclusion map and consider the subspace topology on $Z$ . We endow $Z$ with a structure sheaf $\operatorname{\mathcal{O}}_{Z}:=i^{-1}(\operatorname{\mathcal{O}}_{X})$ . In the case $\operatorname{\mathcal{O}}_{X}=\underline{A}_{X}$ , the structure sheaf $\operatorname{\mathcal{O}}_{Z}$ is the constant sheaf $\underline{A}_{Z}$ . The pullback of $\operatorname{\mathcal{F}}$ is defined as $i^{\ast}\operatorname{\mathcal{F}}:=i^{-1}\operatorname{\mathcal{F}}\otimes_{i% ^{-1}\operatorname{\mathcal{O}}_{X}}\operatorname{\mathcal{O}}_{Z}$ . The group $\Gamma(Z,\operatorname{\mathcal{F}})$ is understood as $\Gamma(Z,i^{\ast}\operatorname{\mathcal{F}})$ . Also notice $i^{-1}=i^{\ast}$ .

1. Motivation

Let $X$ be an $n$ -dimensional connected orientable topological manifold, and let $k$ be a field. For any $i$ , there is a natural pairing of singular cohomology (with compact support)

H^{i}_{c}(X;k)\times H^{n-i}(X;k)\to H^{n}_{c}(X;k)\xrightarrow{\sim}k

induced by the cup product and an orientation map $H^{n}_{c}(M;k)\xrightarrow{\sim}k$ . The Poincaré duality states the following.

Theorem 1.1 (Poincaé duality).

The pairing above is non-degenerate and thus induces an isomorphism

H^{i}_{c}(X;k)^{\vee}\cong H^{n-i}(X;k)

where $H^{i}_{c}(X;k)^{\vee}$ is the dual as a $k$ -vector space.

Example 1.2.

Let $X=\operatorname{\mathbb{C}}\!\operatorname{\mathbf{P}}^{n}$ . Recall

H^{i}(\operatorname{\mathbb{C}}\!\operatorname{\mathbf{P}}^{n},k)=\begin{cases% *}$k$&$0\leq i\leq 2n$ \text{ even}\\ 0&\text{otherwise}\end{cases*}

Example 1.3.

Let $X=\Sigma_{g}$ be the compact orientable surface of genus $g$ . Then

H^{i}(\Sigma_{g},k)=\begin{cases}k&i=0,2\\ k^{2g}&i=1\\ 0&\text{otherwise}\end{cases}

Example 1.4.

Let $X=\operatorname{\mathbb{R}}\!\operatorname{\mathbf{P}}^{n}$ be the $n$ -dimensional real projective space. $\operatorname{\mathbb{R}}\!\operatorname{\mathbf{P}}^{n}$ is orientable if and only if $n$ is odd. If $n$ is odd, its cohomology is

H^{i}(\operatorname{\mathbb{R}}\!\operatorname{\mathbf{P}}^{n},\operatorname{% \mathbb{Z}})=\begin{cases*}\operatorname{\mathbb{Z}}&$i=0,n$\\ \operatorname{\mathbb{Z}}/2\operatorname{\mathbb{Z}}&$0<i<n$\text{ even}\\ 0&\text{otherwise}\end{cases*}

If $n$ is even, its cohomology is

H^{i}(\operatorname{\mathbb{R}}\!\operatorname{\mathbf{P}}^{n},\operatorname{% \mathbb{Z}})=\begin{cases*}\operatorname{\mathbb{Z}}&$i=0$\\ \operatorname{\mathbb{Z}}/2\operatorname{\mathbb{Z}}&$0<i\leq n$ \text{ even}% \\ 0&\text{otherwise}\end{cases*}

Observe that in the case that $\operatorname{\mathbb{R}}\!\operatorname{\mathbf{P}}^{n}$ is not orientable, the symmetry of the cohomology groups is broken.

It is natural to envision a generalization of the Poincaré duality to more general spaces. But a naïve extension would fail as shown by the next example.

Example 1.5.

Let $X$ be the figure-eight. The expected dimension of $X$ should be $1$ . One has $H^{0}(X,\operatorname{\mathbb{R}})=\operatorname{\mathbb{R}}$ and $H^{1}(X,\operatorname{\mathbb{R}})=\operatorname{\mathbb{R}}^{2}$ . So the cohomology groups don’t satisfy the Poincaré duality.

The problem is solved by the Verdier duality. But before going into that, let us first reinterpret the Poincaré duality using sheaf cohomology.

Let $X$ be an $n$ -dimensional topological manifold (not necessarily orientable). We also fix a coefficient field $A$ ¹¹1Many constructions would work for $A$ being a (commutative) ring. But for simplicity, we only assume $A$ is a field.. Write $\underline{A}_{X}$ for the constant sheaf on $X$ with values in $A$ . The orientation sheaf $\operatorname{\mathrm{Or}}_{X}$ of $X$ is the sheaf associated to the presheaf

U\mapsto H^{n}_{c}(U,A)^{\vee}

The orientation sheaf $\operatorname{\mathrm{Or}}_{X}$ can be equivalently defined as the sheaf associated to the presheaf

U\mapsto H_{n}(X,X-U;A)

Proposition 1.6.

The two definitions are equivalent.

Proof.

Denote by $\operatorname{\mathrm{Or}}_{1}$ and $\operatorname{\mathrm{Or}}_{2}$ the two sheaves, and denote by $\operatorname{\mathrm{Or}}_{X}^{\mathrm{pre}}$ and $\operatorname{\mathrm{Or}}_{X}^{\mathrm{pre}}$ the corresponding presheaves. For any open subset $U\subseteq X$ , there is a natural map

(1)		$\displaystyle H_{n}(X,X-U;A)$	$\displaystyle\to H^{n}_{c}(U;A)^{\vee}$
(2)		$\displaystyle[\xi]$	$\displaystyle\mapsto\left(\int_{\xi}\cdot:H^{n}_{c}(U,A)\to A\right)$

If $U$ is small enough, it is homeomorphic to an open subset of $\operatorname{\mathbb{R}}^{n}$ . In this case, $H_{n}(X,X-U;A)$ is naturally identified with $H_{n}(S^{n};A)\cong A$ , and $H^{n}_{c}(U;A)$ is identified with $H^{n}(S^{n};A)\cong A$ . The natural map 1 becomes an isomorphism. This should the natural map $\operatorname{\mathrm{Or}}_{2}\to\operatorname{\mathrm{Or}}_{1}$ is an isomorphism. Furthermore, this makes clear that the orientation sheaf is locally constant. ∎

In the case that $X$ is orientable, there are non-canonical isomorphisms $\operatorname{\mathrm{Or}}_{X}\cong\underline{A}_{X}$ , and each such isomorphism corresponds to a choice of orientation on $X$ . The Poincaré duality can be restated as: there is a non-degenerate pairing

H^{i}_{c}(X,\underline{A}_{X})\times H^{n-i}(X,\operatorname{\mathrm{Or}}_{X})% \to H^{n}_{c}(X,\operatorname{\mathrm{Or}}_{X})\xrightarrow{\sim}A

for any $i$ .

2. $f_{!}$ and $f^{!}$

Let $f:X\to Y$ be a continuous map between locally compact topological spaces. Let $\operatorname{\mathcal{F}}$ be a sheaf on $X$ . Recall the support of a section $s\in\Gamma(X,\operatorname{\mathcal{F}})$ is the closed subset

\operatorname{Supp}s:=\overline{\{x\in X\mid s(x)\neq 0\}}

where $s(x)$ is the image of $s$ in the stalk $\operatorname{\mathcal{F}}_{x}$ at the point $x$ . Using this notion, we can define:

Definition 2.1.

Define a subgroup $\Gamma_{c}(X,\operatorname{\mathcal{F}})$ of $\Gamma(X,\operatorname{\mathcal{F}})$ by

\Gamma_{c}(X,\operatorname{\mathcal{F}}):=\{s\in\Gamma(X,\operatorname{% \mathcal{F}})\mid\operatorname{Supp}s\text{ is compact}\}.

The functor $\Gamma_{c}(X,\cdot)$ is left-exact and has right derived functor $R\Gamma_{c}(X,\cdot)$ . The $i$ -th derived functor is called the (hyper)cohomology in degree $i$ with compact support and is often denoted as

\mathbb{H}^{i}(X,\operatorname{\mathcal{F}}^{\bullet}):=h^{i}(R\Gamma_{c}(X,% \operatorname{\mathcal{F}}^{\bullet}))

or as $H_{c}^{i}(X,\operatorname{\mathcal{F}})$ if $\operatorname{\mathcal{F}}^{\bullet}=\operatorname{\mathcal{F}}[0]$ is concentrated in degree 0.

Recall that a map between locally compact topological spaces is called proper if the inverse image of a compact set is compact.

Definition 2.2.

For any sheaf $\operatorname{\mathcal{F}}\in\mathrm{Sh}(X)$ , the pushforward with proper support $f_{!}\operatorname{\mathcal{F}}\in\mathrm{Sh}(Y)$ is defined by assigning to every open subset $U\subseteq Y$ an $A$ -vector space

\Gamma(U,f_{!}\operatorname{\mathcal{F}}):=\{s\in\Gamma(f^{-1}(U),% \operatorname{\mathcal{F}})\mid f|_{\operatorname{Supp}s}:\operatorname{Supp}s% \to U\text{ is proper}\}

The assignment $\operatorname{\mathcal{F}}\mapsto f_{!}\operatorname{\mathcal{F}}$ defines a left-exact functor

f_{!}:\mathrm{Sh}(X)\to\mathrm{Sh}(Y)

and therefore also gives a derived functor

Rf_{!}:\mathrm{D}^{b}(X)\to\mathrm{D}^{b}(Y)

The functor $f_{!}$ can be thought as the relative version of $\Gamma_{c}(X,\cdot)$ as illustrated by the following example.

Example 2.3.

If $Y$ is a point, then $f_{!}(\operatorname{\mathcal{F}})=\Gamma_{c}(X,\operatorname{\mathcal{F}})$ and $R^{i}f_{!}(\operatorname{\mathcal{F}})=H^{i}_{c}(X,\operatorname{\mathcal{F}})$ .

Example 2.4.

If $f$ is proper (e.g. a closed inclusion), then $f_{!}=f_{\ast}$ .

We want to define a functor $f^{!}:\mathrm{Sh}(Y)\to\mathrm{Sh}(X)$ right adjoint to $f_{!}$ . That is, $f^{!}$ should satisfy

\underline{\operatorname{\mathrm{Hom}}}(f_{!}\operatorname{\mathcal{F}},% \operatorname{\mathcal{G}})\cong f_{\ast}\underline{\operatorname{\mathrm{Hom}% }}(\operatorname{\mathcal{F}},f^{!}\operatorname{\mathcal{G}})

for any $\operatorname{\mathcal{F}}\in\mathrm{Sh}(X)$ and $\operatorname{\mathcal{G}}\in\mathrm{Sh}(Y)$ . However, the following example shows such $f^{!}$ does not always exist.

Example 2.5.

Let $j:U\to X$ be the inclusion of an open subset $U\subseteq X$ . For $\operatorname{\mathcal{F}}=\underline{A}_{U}$ and $\operatorname{\mathcal{G}}\in\mathrm{Sh}(Y)$ , we have

	$\displaystyle\Gamma(U,j^{!}\operatorname{\mathcal{G}})$	$\displaystyle=\operatorname{\mathrm{Hom}}(\underline{A}_{U},j^{!}\operatorname% {\mathcal{G}})$
		$\displaystyle=\Gamma(U,\underline{\operatorname{\mathrm{Hom}}}(\underline{A}_{% U},j^{!}\operatorname{\mathcal{G}}))$
		$\displaystyle=\Gamma(U,j_{\ast}\underline{\operatorname{\mathrm{Hom}}}(% \underline{A}_{U},j^{!}\operatorname{\mathcal{G}}))$
		$\displaystyle=\Gamma(U,\underline{\operatorname{\mathrm{Hom}}}(j_{!}\underline% {A}_{U},\operatorname{\mathcal{G}}))$
		$\displaystyle=\operatorname{\mathrm{Hom}}(j^{\ast}j_{!}\underline{A}_{U},j^{% \ast}\operatorname{\mathcal{G}})$
		$\displaystyle=\operatorname{\mathrm{Hom}}(\underline{A}_{U},j^{\ast}% \operatorname{\mathcal{G}})$
		$\displaystyle=\Gamma(U,j^{\ast}\operatorname{\mathcal{G}})$

Performing the same calculation for every open $V\subset U$ , one obtains

j^{!}\operatorname{\mathcal{G}}\cong j^{\ast}\operatorname{\mathcal{G}}

for open inclusions $j$ .
Now for a continuous map $f:X\to Y$ , by a similar calculation, one gets

\Gamma(U,f^{!}\operatorname{\mathcal{G}})=\operatorname{\mathrm{Hom}}(f_{!}(j_% {!}\underline{A}_{U}),\operatorname{\mathcal{G}})

This implies that, if $f^{!}$ existed, then $f^{!}\operatorname{\mathcal{G}}$ should be given by the assignment

U\mapsto\operatorname{\mathrm{Hom}}(f_{!}(j_{!}\underline{A}_{U}),% \operatorname{\mathcal{G}})

But this is in general not a sheaf.

However, we claim that $f^{!}$ can be defined on the derived level. That is, we can define a functor

Rf^{!}=f^{!}:\mathrm{D}^{b}(Y)\to\mathrm{D}^{b}(X)

which satisfies the derived version of adjointness condition, which we will state later.

To define $f^{!}$ on the derived level, we need the notions of c-soft and flat sheaves.

Definition 2.6.

A sheaf $\operatorname{\mathcal{F}}\in\mathrm{Sh}(X)$ is c-soft if the map $\Gamma(X,\operatorname{\mathcal{F}})\to\Gamma(K,\operatorname{\mathcal{F}})$ is surjective for any compact subset $K\subset X$ .

The following characterization of c-softness is useful.

Lemma 2.7.

Let $\operatorname{\mathcal{F}}$ be a sheaf on $X$ . The following are equivalent:

(1)

$\operatorname{\mathcal{F}}$ is c-soft;
(2)

$H^{1}_{c}(U,\operatorname{\mathcal{F}}|_{U})=0$ for all open subsets $U\subset X$ ;
(3)

$H^{i}_{c}(U,\operatorname{\mathcal{F}}|_{U})=0$ for all $i\geq 1$ and all open subsets $U\subset X$ .

Corollary 2.8.

text

(1)

Any injective or flasuqe sheaf is c-soft .
(2)

If a sheaf on the open subset $U\subset X$ is c-soft, then its extension by zero is c-soft on $X$ .
(3)

The restriction of a c-soft sheaf to a closed subspace is c-soft.
(4)

The restriction of a c-soft sheaf to an open subspace is c-soft.

Proof.

Easy. Omitted. ∎

Proposition 2.9.

Assume $X$ has cohomological dimension $n$ . Consider an exact sequence of sheaves on $X$

0\to A\to F_{0}\to\cdots\to F_{k-1}\to B\to 0

where $F_{i}$ are c-soft.

(1)

If $k\geq n$ , then $B$ is c-soft.
(2)

If $k\geq 0$ and $A$ is also c-soft, then $B$ is c-soft.

Proof.

Since each $F_{i}$ is $\Gamma_{c}$ -acyclic, it is easy to see

H^{i+k}_{c}(U,A)\cong H^{i}_{c}(U,B)

for any open subset $U\subseteq X$ and $i\geq 1$ . If $k\geq n$ , then $H^{i+k}_{c}(U,A)=0$ . Hence $H^{i}_{c}(U,B)=0$ , and consequently, $B$ is c-soft. If $A$ is c-soft, then already $H^{i}_{c}(U,A)=0$ for any $i\geq 1$ . Therefore, $B$ is c-soft. ∎

Definition 2.10.

A sheaf $\mathcal{K}\in\mathrm{Sh}(X)$ is flat if for any injective map $\operatorname{\mathcal{F}}\hookrightarrow\operatorname{\mathcal{G}}$ of sheaves on $X$ , the map $\operatorname{\mathcal{F}}\otimes\mathcal{K}\to\operatorname{\mathcal{G}}% \otimes\mathcal{K}$ is again injective. A complex $\mathcal{K}^{\bullet}$ of sheaves on $X$ is flat if each term $\mathcal{K}^{i}$ is flat.

Proposition 2.11.

Assume $X$ has cohomological dimension $n$ . Consider an exact sequence of sheaves on $X$

0\to A\to F_{0}\to\cdots\to F_{k-1}\to B\to 0

where $F_{i}$ are flat.

(1)

If $k\geq n$ , then $B$ is flat.
(2)

If $k\geq 0$ and $A$ is also flat, then $B$ is flat.

Definition 2.12 (Cohomological dimension).

The cohomological dimension $\dim X$ of a locally compact Hausdorff space $X$ is the smallest integer $n$ for which

H^{i}_{c}(X,\operatorname{\mathcal{F}})=0

for all $i>n$ and all sheaves $\operatorname{\mathcal{F}}$ on $X$ .

Example 2.13.

A stratified pseudomanifold as introduced in [1] Section 4.1.2 has a finite cohomological dimension. In particular, any topological manifold has a finite cohomological dimension.

For now on, assume $X$ and $Y$ have finite cohomological dimension. Assume $\dim X=n$ . Then any sheaf $\operatorname{\mathcal{F}}$ on $X$ admits a resolution by c-soft flat sheaves

0\to\operatorname{\mathcal{F}}\to K_{0}\to\cdots\to K_{n}\to 0

This is obtained by first considering the Godement resolution

0\to\operatorname{\mathcal{F}}\to G_{0}\to G_{1}\to\cdots

Let us first review how the Godement resolution is defined. For a sheaf $\operatorname{\mathcal{F}}$ on $X$ , we set $\mathrm{Gode}(\operatorname{\mathcal{F}})$ to be the sheaf

U\mapsto\prod_{x\in U}\operatorname{\mathcal{F}}_{x}.

There is a natural map $\operatorname{\mathcal{F}}\to\mathrm{Gode}(\operatorname{\mathcal{F}})$ given by on any open subset $U$ the maps $\operatorname{\mathcal{F}}(U)to\operatorname{\mathcal{F}}_{x}$ to stalks at $x\in U$ . It is easy to see the natural map $\operatorname{\mathcal{F}}\to\mathrm{Gode}(\operatorname{\mathcal{F}})$ is injective. Its cokernel is denoted as $\operatorname{\mathcal{G}}$ .

0\to\operatorname{\mathcal{F}}\to\mathrm{Gode}(\operatorname{\mathcal{F}})\to% \operatorname{\mathcal{G}}\to 0

We can now apply $\mathrm{Gode}(\cdot)$ to $\operatorname{\mathcal{G}}$ and obtain a sheaf morphism $\mathrm{Gode}(\operatorname{\mathcal{F}})\to\mathrm{Gode}(\operatorname{% \mathcal{G}})$ by composing $\mathrm{Gode}(\operatorname{\mathcal{F}})\to\operatorname{\mathcal{G}}$ and $\operatorname{\mathcal{G}}\to\mathrm{Gode}(\operatorname{\mathcal{G}})$ . Now write $G_{0}:=\mathrm{Gode}(\operatorname{\mathcal{F}})$ and $G_{1}:=\mathrm{Gode}(\operatorname{\mathcal{G}})$ . Inductively, we define $G_{i}:=\mathrm{Gode}(\operatorname{\mathrm{coker}}(G_{i-2}\to G_{i-1}))$ . We therefore get a complex of sheaves

0\to\operatorname{\mathcal{F}}\to\operatorname{\mathcal{G}}_{0}\to% \operatorname{\mathcal{G}}_{1}\to\cdots

Furthermore, one verifies easily that each term $G_{i}$ is flasque.

We now can obtain a desired c-soft flat resolution of $\operatorname{\mathcal{F}}$ by truncating the Godement resolution. Denote by $d_{i}:G_{i}\to G_{i+1}$ the differential map. Set

K_{i}:=\begin{cases}G_{i}&0\leq i<n\\ \operatorname{\mathrm{im}}d_{n-1}&i=n\end{cases}

Proposition 2.14.

All $K_{i}$ are c-soft and flat.

Proof.

If $0\leq i<n$ , $K_{i}$ is c-soft since it is flasque. It is also soft by construction. If $i=n$ , the statement follows from Proposition 2.9 and Proposition 2.11. ∎

If $\operatorname{\mathcal{F}}^{\bullet}$ is a bounded complex of sheaves on $X$ , then similar to the Cartan-Eilenberg resolution, we can construct a quasi-isomorphism

\operatorname{\mathcal{F}}^{\bullet}\xrightarrow{\text{qis}}K^{\bullet}

where $K^{\bullet}$ is a bounded complex of c-soft flat sheaves on $X$ .

Fix a c-soft sheaf $K$ on $X$ . For any sheaf $\operatorname{\mathcal{F}}$ on $U$ and for any open inclusion $i:U\hookrightarrow X$ , we will write $\operatorname{\mathcal{F}}^{X}:=i_{!}\operatorname{\mathcal{F}}$ to avoid the need of a symbol for the open inclusion. One can verify that the assignment

U\mapsto\operatorname{\mathrm{Hom}}(f_{!}(K|_{U})^{X},\operatorname{\mathcal{G% }})

is a sheaf on $X$ ; we denote this sheaf by $f^{!}_{K}\operatorname{\mathcal{G}}$ . For a fixed bounded c-soft flat resolution $\mathcal{K}^{\bullet}$ of $\underline{A}_{X}$ and a bounded complex $\operatorname{\mathcal{G}}^{\bullet}$ of sheaves on $X$ , define $f^{!}_{\mathcal{K}^{\bullet}}\operatorname{\mathcal{G}}^{\bullet}$ to be the simple complex associated to the double complex $\left(f^{!}_{K^{i}}\operatorname{\mathcal{G}}^{j}\right)_{i,j\in\operatorname{% \mathbb{Z}}}$ . In other words, $f^{!}_{\mathcal{K}^{\bullet}}\operatorname{\mathcal{G}}^{\bullet}$ has associated presheaf

U\mapsto\operatorname{\mathrm{Hom}}^{\bullet}(f_{!}(\mathcal{K}^{\bullet}|_{U}% )^{X},\operatorname{\mathcal{G}}^{\bullet})

Proposition 2.15.

If $\mathcal{K}^{\bullet}$ and $\mathcal{L}^{\bullet}$ are two bounded c-soft flat resolutions of $\underline{A}_{X}$ with a quasi-isomorphism $\tau:\mathcal{K}^{\bullet}\xrightarrow{\text{qis}}\mathcal{L}^{\bullet}$ extending the identity map on $\underline{A}_{X}$ , then $\tau$ induces a quasi-isomorphism $f^{!}_{\mathcal{L}^{\bullet}}\operatorname{\mathcal{G}}\to f^{!}_{\mathcal{K}^% {\bullet}}\operatorname{\mathcal{G}}$ . In particular, we obtain a well-defined functor

f^{!}:\mathrm{D}^{b}(Y)\to\mathrm{D}^{b}(X)

independent of the choice of c-soft flat resolutions of $\underline{A}_{X}$ .

3. Dualizing sheaf and Verdier duality

Definition 3.1.

Assume $Y=\mathrm{pt}$ is a point. The dualizing complex of $X$ is defined to be

\mathbb{D}_{X}^{\bullet}=f^{!}\underline{A}_{\mathrm{pt}}

The Verdier dual of a bounded complex $\operatorname{\mathcal{F}}^{\bullet}\in\mathrm{D}^{b}(X)$ is defined to be

\operatorname{\mathcal{D}}_{X}\operatorname{\mathcal{F}}^{\bullet}:=R% \operatorname{\mathrm{Hom}}^{\bullet}(\operatorname{\mathcal{F}}^{\bullet},% \mathbb{D}_{X}^{\bullet})

The next important theorem shows the connection between the dualizing complex and the orientation sheaf on a manifold.

Theorem 3.2.

Let $X$ be an $n$ -dimensional topological manifold with orientation sheaf $\operatorname{\mathrm{Or}}_{X}$ . There is a canonical isomorphism in $\mathrm{D}^{b}(X)$ :

\mathbb{D}_{X}^{\bullet}\cong\operatorname{\mathrm{Or}}_{X}[n].

Now we have the statement of the (local) Verdier duality.

Theorem 3.3 (Verdier duality).

For $\operatorname{\mathcal{F}}^{\bullet}\in D^{b}(X)$ and $\operatorname{\mathcal{G}}^{\bullet}\in D^{b}(Y)$ , there is a canonical isomorphism

R\underline{\operatorname{\mathrm{Hom}}}^{\bullet}(Rf_{!}\operatorname{% \mathcal{F}},\operatorname{\mathcal{G}}^{\bullet})\cong Rf_{\ast}R\underline{% \operatorname{\mathrm{Hom}}}^{\bullet}(\operatorname{\mathcal{F}}^{\bullet},Rf% ^{!}\operatorname{\mathcal{G}}^{\bullet})

Notice this is a kind of adjointness property, which we promised to state in the previous section. Applying the functor $h^{0}\Gamma(Y,\cdot)$ , one obtains the global version of the Verdier duality.

Corollary 3.4 (Global Verdier duality).

With notations as above, there is an isomorphism

\operatorname{\mathrm{Hom}}_{D^{b}(Y)}(Rf_{!}\operatorname{\mathcal{F}},% \operatorname{\mathcal{G}}^{\bullet})\cong\operatorname{\mathrm{Hom}}_{D^{b}(X% )}(\operatorname{\mathcal{F}}^{\bullet},Rf^{!}\operatorname{\mathcal{G}}^{% \bullet}).

4. Examples

We now consider some example of the Verdier duality. The primary one is of course the Poincaré duality.

Proposition 4.1 (Poincaré duality).

Let $X$ be an $n$ -dimensional orientable topological manifold. Then there is a canonical isomorphism of $A$ -vector spaces

H^{n-i}(X;A)\cong H^{i}_{c}(X;A)^{\vee}

for any $i$ .

Proof.

Since $Y=\mathrm{pt}$ is a point, the Verdier duality for $f:X\to\mathrm{pt}$ simplifies to

(3)

\operatorname{\mathrm{Hom}}(R\Gamma(\mathrm{pt},Rf_{!}\underline{A}_{X}),% \underline{A}_{\mathrm{pt}})\cong Rf_{\ast}R\underline{\operatorname{\mathrm{% Hom}}}^{\bullet}(\underline{A}_{X},\underline{A}_{X}[n])

in $D^{b}(\mathrm{pt})\cong D^{b}(A)$ . Choose an injective resolution $\underline{A}_{X}\to I^{\bullet}$ . For $\underline{\operatorname{\mathrm{Hom}}}(\underline{A}_{X},I^{\bullet}[n])=% \underline{\operatorname{\mathrm{Hom}}}(\underline{A}_{X},I^{\bullet})[n]$ , we choose an injective resolution

\underline{\operatorname{\mathrm{Hom}}}(\underline{A}_{X},I^{\bullet})[n]% \xrightarrow{\text{qis}}J^{\bullet}.

Now we apply the functor $h^{-i}\Gamma(\mathrm{pt},\cdot)$ on $J^{\bullet}$ ; this gives

h^{-i}\Gamma(\mathrm{pt},J^{\bullet})=\mathbb{H}^{-i}(X,\underline{% \operatorname{\mathrm{Hom}}}(\underline{A}_{X},I^{\bullet})[n])=H^{n-i}(X,A)

Similarly, applying $h^{-i}\Gamma(\mathrm{pt},\cdot)$ to the left-hand side gives $H^{i}_{c}(X,A)^{\vee}$ . Therefore, we conclude that there is an isomorphism

H^{i}_{c}(X,A)^{\vee}\cong H^{n-i}(X,A)

∎

References

[1] Banagl, M. (2007). Topological invariants of stratified spaces. Springer Berlin, Heidelberg.
[2] Borel, A. (1984). Intersection Cohomology (Vol. 50). Birkhäuser Boston.
[3] Maxim, L. G. (2019). Intersection Homology & Perverse Sheaves: With Applications to Singularities (Graduate Texts in Mathematics vol. 281). Springer Cham.

Introduction to Verdier Duality

Abstract.

Notations

1. Motivation

Theorem 1.1 (Poincaé duality).

Example 1.2.

Example 1.3.

Example 1.4.

Example 1.5.

Proposition 1.6.

Proof.

2. f! and f!

Definition 2.1.

Definition 2.2.

Example 2.3.

Example 2.4.

Example 2.5.

Definition 2.6.

Lemma 2.7.

Corollary 2.8.

Proof.

Proposition 2.9.

Proof.

Definition 2.10.

Proposition 2.11.

Definition 2.12 (Cohomological dimension).

Example 2.13.

Proposition 2.14.

Proof.

Proposition 2.15.

3. Dualizing sheaf and Verdier duality

Definition 3.1.

Theorem 3.2.

Theorem 3.3 (Verdier duality).

Corollary 3.4 (Global Verdier duality).

4. Examples

Proposition 4.1 (Poincaré duality).

Proof.

References

2. $f_{!}$ and $f^{!}$