In mathematics, local coefficients is an idea from algebraic topology, a kind of half-way stage between homology theory or cohomology theory with coefficients in the usual sense, in a fixed abelian group A, and general sheaf cohomology which, roughly speaking, allows coefficients to vary from point to point in a topological space X. Such a concept was introduced by Norman Steenrod in 1943.
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 is a constant sheaf.
If X is path-connected, a local system of abelian groups has the same fibre L at every point. To give such a local system is the same as to give a homomorphism
and similarly for local systems of modules,... The map giving the local system is called the monodromy representation of .
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.
Stronger definition on non-connected spaces
Another (stronger, nonequivalent) definition generalising 2, and working for non-connected X, is: a covariant functor
from the fundamental groupoid of to the category of modules over a commutative ring . Typically . What this is saying is that at every point we should assign a module with a representations of such that these representations are compatible with change of basepoint for the fundamental group.
- Constant sheaves. For instance, . This is a useful tool for computing cohomology since the sheaf cohomology
- is isomorphic to the singular cohomology of .
- . Since , there are -many linear systems on X, the one given by monodromy representation
- by sending
- Horizontal sections of vector bundles with a flat connection. If is a vector bundle with flat connection , then
- is a local system.
- 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 sections locally 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 the same as a k-linear representation of the group .
- If X is a variety, local systems are the same thing as D-modules that are in addition coherent as O-modules.
If the connection is not flat, parallel transporting a fibre around a contractible loop at x may give a nontrivial automorphism of the fibre at the base point x, so there is no chance to define a locally constant sheaf this way.
Local systems have a mild generalization to constructible sheaves. A constructible sheaf on a locally path connected topological space is a sheaf such that there exists a stratification of
where is a local system. These are typically found by taking the cohomology of the derived pushforward for some continuous map . For example, if we look at the complex points of the morphism
then the fibers over
are the smooth plane curve given by , but the fibers over are . If we take the derived pushforward then we get a constructible sheaf. Over we have the local systems
while over we have the local systems
where is the genus of the plane curve (which is ).
- "What local system really is". Stack Exchange.
- Schnell, Christian. "Computing Cohomology of Local Systems" (PDF). Discusses computing the cohomology with coefficients in a local system by using the twisted de Rham complex.
- Williamson, Geordie. "An illustrated guide to perverse sheaves" (PDF).
- Macpherson, Robert. "Intersection homology and perverse sheaves" (PDF).
- El Zein, Fouad; Snoussi, Jawad. "Local systems and constructible sheaves" (PDF).