Verdier duality

Verdier duality states that (subject to suitable finiteness conditions discussed below) certain derived image functors for sheaves are actually adjoint functors. There are two versions.

Global Verdier duality states that for a continuous map ${\displaystyle f\colon X\to Y}$ of locally compact Hausdorff spaces, the derived functor of the direct image with compact (or proper) supports ${\displaystyle Rf_{!}}$ has a right adjoint ${\displaystyle f^{!}}$ in the derived category of sheaves, in other words, for (complexes of) sheaves (of abelian groups) ${\displaystyle {\mathcal {F}}}$ on ${\displaystyle X}$ and ${\displaystyle {\mathcal {G}}}$ on ${\displaystyle Y}$ we have

${\displaystyle RHom(Rf_{!}{\mathcal {F}},{\mathcal {G}})\cong RHom({\mathcal {F}},f^{!}{\mathcal {G}}).}$

Local Verdier duality states that

${\displaystyle R\,{\mathcal {H}}om(Rf_{!}{\mathcal {F}},{\mathcal {G}})\cong Rf_{\ast }R\,{\mathcal {H}}om({\mathcal {F}},f^{!}{\mathcal {G}})}$

in the derived category of sheaves on Y. It is important to note that the distinction between the global and local versions is that the former relates morphisms between complexes of sheaves in the derived categories, whereas the latter relates internal Hom-complexes and so can be evaluated locally. Taking global sections of both sides in the local statement gives the global Verdier duality.

These results hold subject to the compactly supported direct image functor ${\displaystyle f_{!}}$ having finite cohomological dimension. This is the case if the there is a bound ${\displaystyle d\in \mathbf {N} }$ such that the compactly supported cohomology ${\displaystyle H_{c}^{r}(X_{y},\mathbf {Z} )}$ vanishes for all fibres ${\displaystyle X_{y}=f^{-1}(y)}$ (where ${\displaystyle y\in Y}$) and ${\displaystyle r>d}$. This holds if all the fibres ${\displaystyle X_{y}}$ are at most ${\displaystyle d}$-dimensional manifolds or more generally at most ${\displaystyle d}$-dimensional CW-complexes.

The discussion above is about derived categories of sheaves of abelian groups. It is instead possible to consider a ring ${\displaystyle A}$ and (derived categories of) sheaves of ${\displaystyle A}$-modules; the case above corresponds to ${\displaystyle A=\mathbf {Z} }$.

The dualizing complex ${\displaystyle D_{X}}$ on ${\displaystyle X}$ is defined to be

${\displaystyle \omega _{X}=p^{!}(k),}$

where p is the map from ${\displaystyle X}$ to a point. Part of what makes Verdier duality interesting in the singular setting is that when ${\displaystyle X}$ is not a manifold (a graph or singular algebraic variety for example) then the dualizing complex is not quasi-isomorphic to a sheaf concentrated in a single degree. From this perspective the derived category is necessary in the study of singular spaces.

If ${\displaystyle X}$ is a finite-dimensional locally compact space, and ${\displaystyle D^{b}(X)}$ the bounded derived category of sheaves of abelian groups over ${\displaystyle X}$, then the Verdier dual is a contravariant functor

${\displaystyle D\colon D^{b}(X)\to D^{b}(X)}$

defined by

${\displaystyle D({\mathcal {F}})=R\,{\mathcal {H}}om({\mathcal {F}},\omega _{X}).}$

It has the following properties:

Poincaré duality can be derived as a special case of Verdier duality. Here one explicitly calculates cohomology of a space using the machinery of sheaf cohomology.

Suppose X is a compact orientable n-dimensional manifold, k is a field and ${\displaystyle k_{X}}$ is the constant sheaf on X with coefficients in k. Let ${\displaystyle f=p}$ be the constant map to a point. Global Verdier duality then states

${\displaystyle [Rp_{!}k_{X},k]\cong [k_{X},p^{!}k].}$

To understand how Poincaré duality is obtained from this statement, it is perhaps easiest to understand both sides piece by piece. Let

${\displaystyle k_{X}\to (I_{X}^{\bullet }=I_{X}^{0}\to I_{X}^{1}\to \cdots )}$

be an injective resolution of the constant sheaf. Then by standard facts on right derived functors

${\displaystyle Rp_{!}k_{X}=p_{!}I_{X}^{\bullet }=\Gamma _{c}(X;I_{X}^{\bullet })}$

is a complex whose cohomology is the compactly supported cohomology of X. Since morphisms between complexes of sheaves (or vector spaces) themselves form a complex we find that

${\displaystyle \mathrm {Hom} ^{\bullet }(\Gamma _{c}(X;I_{X}^{\bullet }),k)=\cdots \to \Gamma _{c}(X;I_{X}^{2})^{\vee }\to \Gamma _{c}(X;I_{X}^{1})^{\vee }\to \Gamma _{c}(X;I_{X}^{0})^{\vee }\to 0}$

where the last non-zero term is in degree 0 and the ones to the left are in negative degree. Morphisms in the derived category are obtained from the homotopy category of chain complexes of sheaves by taking the zeroth cohomology of the complex, i.e.

${\displaystyle [Rp_{!}k_{X},k]\cong H^{0}(\mathrm {Hom} ^{\bullet }(\Gamma _{c}(X;I_{X}^{\bullet }),k))=H_{c}^{0}(X;k_{X})^{\vee }.}$

For the other side of the Verdier duality statement above, we have to take for granted the fact that when X is a compact orientable n-dimensional manifold

${\displaystyle p^{!}k=k_{X}[n],}$

which is the dualizing complex for a manifold. Now we can re-express the right hand side as

${\displaystyle [k_{X},k_{X}[n]]\cong H^{n}(\mathrm {Hom} ^{\bullet }(k_{X},k_{X}))=H^{n}(X;k_{X}).}$

We finally have obtained the statement that

${\displaystyle H_{c}^{0}(X;k_{X})^{\vee }\cong H^{n}(X;k_{X}).}$

By repeating this argument with the sheaf k_X replaced with the same sheaf placed in degree i we get the classical Poincaré duality

${\displaystyle H_{c}^{i}(X;k_{X})^{\vee }\cong H^{n-i}(X;k_{X}).}$

• Poincaré duality
• Six operations
• Coherent duality
• Derived category

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• Grothendieck, Alexandre (1977), Séminaire de Géométrie Algébrique du Bois Marie - 1965-66 - Cohomologie l-adique et Fonctions L - (SGA 5), Lecture notes in mathematics, vol. 589, Berlin, New York: Springer-Verlag, pp. xii+484, ISBN 978-3-540-08248-4, Exposés I and II contain the corresponding theory in the étale situation
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