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. (December 2009)
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 .
Proof of equivalence
Take local system and a loop at x. It's easy so 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.
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.
The Gauss–Manin connection is a very interesting example of a connection, whose horizontal sections occur in the study of variation of Hodge structures.
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 ).
The cohomology with local coefficients in the module corresponding to the orientation covering can be used to formulate Poincaré duality for non-orientable manifolds: see Twisted Poincaré duality.
Topological Bivariant Theory
For maps there is a Bivariant theory similar to William Fulton's, called Topological Bivariant Theory. Defining such a theory requires local systems and the six-functor formalism. Bivariant theories are characterized by the property
For example, this can be computed in some simple cases. If is a point, this recovers Borel–Moore homology. If and the map is the identity, then this is oridinary cohomology. Another informative class of example includes covering spaces. For example, if is the degree covering given by . Then, at the stalk level the cohomology groups are of the form
and the monodromy for is given by the map taking a branch to its next branch and the -th branch to the first branch. That is, is generated by the matrix