In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact topological spaces of Alexander Grothendieck's theory ofPoincaré duality in étale cohomologyfor schemes in algebraic geometry. It is thus (together with the said étale theory and for example Grothendieck's coherent duality) one instance of Grothendieck's six operations formalism.
Verdier duality generalises the classical Poincaré duality of manifolds in two directions: it applies to continuous maps from one space to another (reducing to the classical case for the unique map from a manifold to a one-point space), and it applies to spaces that fail to be manifolds due to the presence of singularities. It is commonly encountered when studying constructible or perverse sheaves.
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\colonX\toY
Rf!
f!
lF
X
lG
Y
!l{G}). | |
RHom(Rf | |
!l{F},l{G})\congRHom(l{F},f |
Local Verdier duality states that
Rl{H}om(Rf!l{F},l{G})\congRf\astRl{H}om(l{F},f!l{G})
These results hold subject to the compactly supported direct image functor
f!
d\inN
r | |
H | |
c |
(Xy,Z)
Xy=f-1(y)
y\inY
r>d
Xy
d
d
The discussion above is about derived categories of sheaves of abelian groups. It is instead possible to consider a ring
A
A
A=Z
The dualizing complex
DX
X
\omegaX=p!(k),
where p is the map from
X
X
If
X
Db(X)
X
D\colonDb(X)\toDb(X)
defined by
D(l{F})=Rl{H}om(l{F},\omegaX).
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
kX
f=p
[Rp!kX,k]\cong
!k] | |
[k | |
X,p |
.
kX\to
\bullet | |
(I | |
X |
=
0 | |
I | |
X |
\to
1 | |
I | |
X |
\to … )
Rp!kX=p
\bullet | |
X=\Gamma |
\bullet | |
X) |
Hom\bullet
\bullet | |
(\Gamma | |
X),k)= |
… \to
\vee | |
\Gamma | |
X) |
\to
\vee | |
\Gamma | |
X) |
\to
\vee | |
\Gamma | |
X) |
\to0
[Rp!kX,k]\congH0(Hom\bullet
0 | |
(\Gamma | |
c(X;k |
\vee | |
X) |
.
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
!k=k | |
p | |
X[n], |
[kX,kX[n]]\congHn(Hom\bullet(kX,k
n(X;k | |
X). |
0 | |
H | |
c(X;k |
\vee | |
X) |
\cong
n(X;k | |
H | |
X). |
i | |
H | |
c(X;k |
\vee | |
X) |
\congHn-i(X;kX).