The category of perverse sheaves on a manifold is equivalent to the category of local systems on the manifold.
Let X be a topological space. A local system (of abelian groups/modules/...) on X is a locally constant sheaf (of abelian groups/modules...) on X. In other words, a sheaf is a local system if every point has an open neighborhood such that the restricted sheaf is isomorphic to the sheafification of some constant presheaf.[clarification needed]
If X is path-connected,[clarification needed] a local system of abelian groups has the same stalk L at every point. There is a bijective correspondence between local systems on X and group homomorphisms
and similarly for local systems of modules. The map giving the local system is called the monodromy representation of .
Proof of equivalence
Take local system and a loop at x. It's easy to show that any local system on is constant. For instance, is constant. This gives an isomorphism , i.e. between L and itself.
Conversely, given a homomorphism , consider the constant sheaf on the universal cover of X. The deck-transform-invariant sections of gives a local system on X. Similarly, the deck-transform-ρ-equivariant sections give another local system on X: for a small enough open set U, it is defined as
where is the universal covering.
This shows that (for X path-connected) a local system is precisely a sheaf whose pullback to the universal cover of X is a constant sheaf.
This correspondence can be upgraded to an equivalence of categories between the category of local systems of abelian groups on X and the category of abelian groups endowed with an action of (equivalently, -modules).
Stronger definition on non-connected spacesEdit
A stronger nonequivalent definition that works for non-connected X is: the following: a local system is a covariant functor
from the fundamental groupoid of to the category of modules over a commutative ring , where typically . This is equivalently the data of an assignment to every point a module along with a group representation such that the various are compatible with change of basepoint and the induced map on fundamental groups.
Constant sheaves such as . This is a useful tool for computing cohomology since in good situations, there is an isomorphism between sheaf cohomology and singular cohomology:
Let . Since , there is an family of local systems on X corresponding to the maps :
Horizontal sections of vector bundles with a flat connection. If is a vector bundle with flat connection , then there is a local system given by
For instance, take and the trivial bundle. Sections of E are n-tuples of functions on X, so defines a flat connection on E, as does for any matrix of one-forms on X. The horizontal sections are then
i.e., the solutions to the linear differential equation .
If extends to a one-form on the above will also define a local system on , so will be trivial since . So to give an interesting example, choose one with a pole at 0:
in which case for ,
An n-sheeted covering map is a local system with fibers given by the set . Similarly, a fibre bundle with discrete fibre is a local system, because each path lifts uniquely to a given lift of its basepoint. (The definition adjusts to include set-valued local systems in the obvious way).
A local system of k-vector spaces on X is equivalent to a k-linear representation of .
If X is a variety, local systems are the same thing as D-modules which are additionally coherent O_X-modules (see O modules).
If the connection is not flat (i.e. its curvature is nonzero), then parallel transport of a fibre F_x over x around a contractible loop based at x_0 may give a nontrivial automorphism of F_x, so locally constant sheaves can not necessarily be defined for non-flat connections.
There are several ways to define the cohomology of a local system, called cohomology with local coefficients, which become equivalent under mild assumptions on X.
Given a locally constant sheaf of abelian groups on X, we have the sheaf cohomology groups with coefficients in .
Given a locally constant sheaf of abelian groups on X, let be the group of all functions f which map each singular n-simplex to a global section of the inverse-image sheaf. These groups can be made into a cochain complex with differentials constructed as in usual singular cohomology. Define to be the cohomology of this complex.
The group of singular n-chains on the universal cover of X has an action of by deck transformations. Explicitly, a deck transformation takes a singular n-simplex to . Then, given an abelian group L equipped with an action of , one can form a cochain complex from the groups of -equivariant homomorphisms as above. Define to be the cohomology of this complex.