1. A locally constant sheaf of modules on X. That is, every point has an open neighborhood such that is a constant sheaf.
If X is path-connected, this the same as
2. A homomorphism ( in the above).
Another (stronger, nonequivalent) definition generalising 2, and working for nonconnected X, is
3. A 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.
Here's a proof that 1 and 2 are the same if X is path-connected.
Take as in 1 and a loop at x. It's easy so show that any (definition 1)-local system on is constant. For instance, . So we get an isomorphism . But gives an isomorphism between both sides and , whence an endomorphism of .
Take homomorphism . Consider the constant sheaf on the universal cover of with cover , and let be the deck-transform-ρ-equivariant part:
The proof shows that (for X path-connected) another equivalent definition of a local system is
4. A sheaf whose pullback by the universal cover is the constant sheaf.
Call the map the monodromy representation of the local system.
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
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.