Locally constant sheaf of abelian groups on topological space
In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman Steenrod in 1943.[1]
Local systems are the building blocks of more general tools, such as constructible and perverse sheaves.
Definition[edit]
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]
Equivalent definitions[edit]
Path-connected spaces[edit]
If X is path-connected,[clarification needed] a local system
of abelian groups has the same stalk
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
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).[2]
Stronger definition on non-connected spaces[edit]
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.
Examples[edit]
- 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.
Cohomology[edit]
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.
If X is paracompact and locally contractible, then
.[3] If
is the local system corresponding to L, then there is an identification
compatible with the differentials,[4] so
.
Generalization[edit]
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
![{\displaystyle f:X={\text{Proj}}\left({\frac {\mathbb {C} [s,t][x,y,z]}{(stf(x,y,z))}}\right)\to {\text{Spec}}(\mathbb {C} [s,t])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24576c5532e31ccb650d2f589fddfbc1e543948e)
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
).
Applications[edit]
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.
See also[edit]
References[edit]
- ^ Steenrod, Norman E. (1943). "Homology with local coefficients". Annals of Mathematics. 44 (4): 610–627. doi:10.2307/1969099. MR 0009114.
- ^ Milne, James S. (2017). Introduction to Shimura Varieties. Proposition 14.7.
- ^ Bredon, Glen E. (1997). Sheaf Theory, Second Edition, Graduate Texts in Mathematics, vol. 25, Springer-Verlag. Chapter III, Theorem 1.1.
- ^ Hatcher, Allen (2001). Algebraic Topology, Cambridge University Press. Section 3.H.
External links[edit]