Verdier duality

In mathematics, Verdier duality is a duality in sheaf theory that generalizes Poincaré duality for manifolds. Verdier duality was introduced by Verdier (1967, 1995) as an analog for locally compact spaces of the coherent duality for schemes due to Grothendieck. It is commonly encountered when studying constructible or perverse sheaves.

Verdier duality

Verdier duality states that certain image functors for sheaves are actually adjoint functors. There are two versions.

Global Verdier duality states that for a continuous map $f: X \to Y$ , the derived functor of the direct image with proper supports Rf_! has a right adjoint f^! in the derived category of sheaves, in other words, for a sheaf ${\mathcal {F}}$ on X and ${\mathcal {G}}$ on Y we have

[Rf_{!}{\mathcal {F}},{\mathcal {G}}]\cong [{\mathcal {F}},f^{!}{\mathcal {G}}].

The exclamation mark is often pronounced "shriek" (slang for exclamation mark), and the maps called "f shriek" or "f lower shriek" and "f upper shriek" – see also shriek map.

Local Verdier duality states that

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 of k modules over X. It is important to note that the distinction between the global and local versions is that the former relates maps between sheaves, whereas the latter relates (complexes of) sheaves directly and so can be evaluated locally. Taking global sections of both sides in the local statement gives global Verdier duality.

The dualizing complex D_X on X is defined to be

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

defined by

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

It has the following properties:

$D^2(\mathcal{F})\cong \mathcal{F}$ for sheaves with constructible cohomology.
(Intertwining of functors f_* and f_!) If f is a continuous map from X to Y then there is an isomorphism
$D(Rf_{\ast}(\mathcal{F})) \cong Rf_!D(\mathcal{F})$ .

Poincaré duality

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 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. Global Verdier duality then states

[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

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

Rp_!k_X=p_!I^{\bullet}_X=\Gamma_c(X;I^{\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

\mathrm{Hom}^{\bullet}(\Gamma_c(X;I^{\bullet}_X),k)= \cdots \to \Gamma_c(X;I^2_X)^{\vee}\to \Gamma_c(X;I^1_X)^{\vee}\to \Gamma_c(X;I^0_X)^{\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.

[Rp_!k_X,k]\cong H^0(\mathrm{Hom}^{\bullet}(\Gamma_c(X;I^{\bullet}_X),k))=H^0_c(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 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]]\cong H^n(\mathrm{Hom}^{\bullet}(k_X,k_X))=H^n(X;k_X).

We finally have obtained the statement that

H^0_c(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

H^i_c(X;k_X)^{\vee}\cong H^{n-i}(X;k_X).

References

Borel, Armand (1984), Intersection cohomology, Progress in Mathematics, Basel, Boston, Berlin: Birkhäuser, ISBN 978-0-8176-3274-8
Gelfand, Sergei I.; Manin, Yuri Ivanovich (1999), Homological algebra, Berlin: Springer, ISBN 978-3-540-65378-3
Grothendieck, Alexandre (1977), Séminaire de Géométrie Algébrique du Bois Marie - 1965-66 - Cohomologie l-adique et Fonctions L - (SGA 5), Lecture notes in mathematics, 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
Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3, MR 842190
Kashiwara, Masaki; Schapira, Pierre (2002), Sheaves on Manifolds, Berlin: Springer, ISBN 3540518614
Verdier, Jean-Louis (1967), "A duality theorem in the etale cohomology of schemes", in Springer, Tonny Albert, Proceedings of a Conference on Local Fields: NUFFIC Summer School held at Driebergen (The Netherlands) in 1966, Berlin, New York: Springer-Verlag, pp. 184–198, ISBN 978-3-540-03953-2, MR 0230732
Verdier, Jean-Louis (1995), "Dualité dans la cohomologie des espaces localement compacts", Séminaire Bourbaki, Vol. 9, Paris: Société Mathématique de France, pp. Exp. No. 300, 337–349, ISBN 978-2-85629-042-2, MR 1610971

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Verdier duality

Verdier duality

Poincaré duality

See also

References