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.
Global Verdier duality states that for a continuous map , 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 on X and on Y we have
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
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 DX on X is defined to be
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.
It has the following properties:
- 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
Suppose X is a compact n-dimensional manifold, k is a field and kX is the constant sheaf on X with coefficients in k. Let f=p be the constant map. Global Verdier duality then states
To understand how Poincaré duality is obtained from this statement, it is perhaps easiest to understand both sides piece by piece. Let
be an injective resolution of the constant sheaf. Then by standard facts on right derived functors
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
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.
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
which is the dualizing complex for a manifold. Now we can re-express the right hand side as
We finally have obtained the statement that
By repeating this argument with the sheaf kX replaced with the same sheaf placed in degree i we get the classical Poincaré duality
- 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