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 $f\backslash colon\; X\; \backslash to\; Y$ of locally compact Hausdorff spaces, the derived functor of the direct image with compact (or proper) supports $Rf\_!$ has a right adjoint $f^!$ in the derived category of

sheaves, in other words, for (complexes of) sheaves (of abelian groups) $\backslash mathcal\; F$ on $X$ and $\backslash mathcal\; G$ on $Y$ we have

:$RHom(Rf\_!\backslash mathcal\{F\},\backslash mathcal\{G\})\; \backslash cong\; RHom(\backslash mathcal\{F\},f^!\backslash mathcal\{G\}).$

Local Verdier duality states that

:$R\backslash ,\backslash mathcal\{H\}om(Rf\_!\backslash mathcal\{F\},\backslash mathcal\{G\})\; \backslash cong\; Rf\_\{\backslash ast\}R\backslash ,\backslash mathcal\{H\}om(\backslash mathcal\{F\},f^!\backslash mathcal\{G\})$

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 $f\_\{!\}$ having finite cohomological dimension.

This is the case if there is a bound $d\backslash in\backslash mathbf\{N\}$ such that the compactly supported cohomology

$H\_c^\{r\}(X\_y,\backslash mathbf\{Z\})$ vanishes for all fibres $X\_y\; =\; f^\{-1\}(y)$ (where $y\backslash in\; Y$)

and $r>d$. This holds if all the fibres $X\_y$ are at most $d$-dimensional manifolds or more generally at most $d$-dimensional CW-complexes.

The discussion above is about derived categories of sheaves of abelian groups. It is instead possible to consider a ring

$A$ and (derived categories of) sheaves of $A$-modules; the case above corresponds to

$A=\backslash mathbf\{Z\}$.

The dualizing complex $D\_X$ on $X$ is defined to be

:$\backslash omega\_X\; =\; p^!(k)\; ,$

where p is the map from $X$ to a point. Part of what makes Verdier duality interesting in the singular setting is that when $X$ 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 $X$ is a finite-dimensional locally compact space, and $D^b(X)$ the bounded derived category of sheaves of abelian groups over $X$, then the Verdier dual is a contravariant functor

:$D\; \backslash colon\; D^b(X)\backslash to\; D^b(X)$

defined by

:$D(\backslash mathcal\{F\})\; =\; R\backslash ,\backslash mathcal\{H\}om(\backslash mathcal\{F\},\; \backslash omega\_X)\; .$

It has the following properties:

{{unordered list

|1= $D^2(\backslash mathcal\{F\})\backslash cong\; \backslash mathcal\{F\}$ for sheaves with constructible cohomology.

|2= (Intertwining of functors $f\_*$ and $f\_!$). If $f$ is a continuous map from $X$ to $Y$, then there is an isomorphism

:$D(Rf\_\{!\}(\backslash mathcal\{F\}))\; \backslash cong\; Rf\_\backslash ast\; D(\backslash mathcal\{F\})$.

}}

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 $k\_X$ is the constant sheaf on X with coefficients in k. Let $f=p$ be the constant map to a point. Global Verdier duality then states

:$[Rp\_!k\_X,k]\; \backslash cong\; [k\_X,p^!k]\; .$

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

:$k\_X\backslash to\; (I^\{\backslash bullet\}\_X\; =\; I^0\_X\; \backslash to\; I^1\_X\; \backslash to\; \backslash cdots)$

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

:$Rp\_!k\_X=p\_!I^\{\backslash bullet\}\_X=\backslash Gamma\_c(X;I^\{\backslash bullet\}\_X)$

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

:$\backslash mathrm\{Hom\}^\{\backslash bullet\}(\backslash Gamma\_c(X;I^\{\backslash bullet\}\_X),k)=\; \backslash cdots\; \backslash to\; \backslash Gamma\_c(X;I^2\_X)^\{\backslash vee\}\backslash to\; \backslash Gamma\_c(X;I^1\_X)^\{\backslash vee\}\backslash to\; \backslash Gamma\_c(X;I^0\_X)^\{\backslash vee\}\backslash to\; 0$

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.

:$[Rp\_!k\_X,k]\backslash cong\; H^0(\backslash mathrm\{Hom\}^\{\backslash bullet\}(\backslash Gamma\_c(X;I^\{\backslash bullet\}\_X),k))=H^0\_c(X;k\_X)^\{\backslash vee\}.$

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

:$p^!k=k\_X[n],$

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

:$[k\_X,k\_X[n]]\backslash cong\; H^n(\backslash mathrm\{Hom\}^\{\backslash bullet\}(k\_X,k\_X))=H^n(X;k\_X).$

We finally have obtained the statement that

:$H^0\_c(X;k\_X)^\{\backslash vee\}\backslash cong\; H^n(X;k\_X).$

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

:$H^i\_c(X;k\_X)^\{\backslash vee\}\backslash cong\; H^\{n-i\}(X;k\_X).$

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

• {{Citation | last1=Borel | first1=Armand | author1-link=Armand Borel | title=Intersection cohomology | publisher=Birkhäuser | location=Basel, Boston, Berlin | series=Progress in Mathematics | isbn=978-0-8176-3274-8 | year=1984}}
• {{Citation | last1=Gelfand | first1=Sergei I. | last2=Manin | first2=Yuri Ivanovich | author2-link=Yuri Ivanovich Manin | title=Homological algebra | isbn=978-3-540-65378-3 | year=1999 | publisher=Springer | location=Berlin}}
• {{Citation | last1=Grothendieck | first1=Alexandre | author1-link=Alexandre Grothendieck | title=Séminaire de Géométrie Algébrique du Bois Marie - 1965-66 - Cohomologie l-adique et Fonctions L - (SGA 5) | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture notes in mathematics | isbn=978-3-540-08248-4 | year=1977 | volume=589 | pages=xii+484}}, Exposés I and II contain the corresponding theory in the étale situation
• {{Citation | last1=Iversen | first1=Birger | title=Cohomology of sheaves | publisher=Springer-Verlag | location=Berlin, New York | series=Universitext | isbn=978-3-540-16389-3 |mr=842190 | year=1986 | doi=10.1007/978-3-642-82783-9}}
• {{Citation | last1=Kashiwara | first1=Masaki | last2=Schapira | first2=Pierre | author1-link=Masaki Kashiwara | title=Sheaves on Manifolds | isbn=3540518614 | year=2002 | publisher=Springer | location=Berlin}}
• {{Citation |last1=Verdier |first1=Jean-Louis |author1-link=Jean-Louis Verdier |title=Séminaire Bourbaki |volume=9 |url=http://www.numdam.org/item?id=SB_1964-1966__9__337_0 |publisher=Société Mathématique de France |location=Paris |isbn=978-2-85629-042-2 |mr=1610971 |year=1965 |chapter=Dualité dans la cohomologie des espaces localement compacts |pages=Exp. No. 300, 337–349}}

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