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]
The category of perverse sheaves on a manifold is equivalent to the category of local systems on the manifold.[2]
Definition
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
Path-connected spaces
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
![{\displaystyle \rho :\pi _{1}(X,x)\to {\text{Aut}}(L)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25222d1daf255ee447c0ccd571c3351d74c86257)
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
![{\displaystyle {\mathcal {L}}(\rho )_{U}\ =\ \left\{{\text{sections }}s\in {\underline {L}}_{\pi ^{-1}(U)}{\text{ with }}\theta \circ s=\rho (\theta )s{\text{ for all }}\theta \in {\text{ Deck}}({\widetilde {X}}/X)=\pi _{1}(X,x)\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/deeac1a1971af023e4a5962a37892a1f3f7a8bef)
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
A stronger nonequivalent definition that works for non-connected X is: the following: a local system is a covariant functor
![{\displaystyle {\mathcal {L}}\colon \Pi _{1}(X)\to {\textbf {Mod}}(R)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2023b166e30dd1373d2f006468550361ce151c81)
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
- 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:
![{\displaystyle H^{k}(X,{\underline {\mathbb {Q} }}_{X})\cong H_{\text{sing}}^{k}(X,\mathbb {Q} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f87c5c2fd3e95a59fbfff625cc7a7b76b444d8a)
- Let
. Since
, there is an
family of local systems on X corresponding to the maps
:
![{\displaystyle \rho _{\theta }:\pi _{1}(X;x_{0})\cong \mathbb {Z} \to {\text{Aut}}_{\mathbb {C} }(\mathbb {C} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aff6755df6b9929697ef587e10c7cb2f78b25fd)
- 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 ![{\displaystyle E_{U}^{\nabla }=\left\{{\text{sections }}s\in \Gamma (U,E){\text{ which are horizontal: }}\nabla s=0\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa565f8380d0628ee02c2b8811fd9d5928448b93)
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 ![{\displaystyle E_{U}^{\nabla }=\left\{(f_{1},\dots ,f_{n})\in E_{U}:(df_{1},\dots ,df_{n})=\Theta (f_{1},\dots ,f_{n})^{t}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/464a86f0573f730f115c34e9cd758cfda9c3b558)
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:
![{\displaystyle \Theta ={\begin{pmatrix}0&dx/x\\dx&e^{x}dx\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81fbc1ee7e2e68321c1f7a87788cf9e54c0ba9be)
in which case for
, ![{\displaystyle E_{U}^{\nabla }=\left\{f_{1},f_{2}:U\to \mathbb {C} \ \ {\text{ with }}f'_{1}=f_{2}/x\ \ f_{2}'=f_{1}+e^{x}f_{2}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34685b9162894c8118ec97a6773a5e6dc334ac37)
- 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.
Generalization
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
![{\displaystyle X=\coprod X_{\lambda }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea8d8b46a646435a299bd9d8cbdd24ab7bd2c210)
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
![{\displaystyle \mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d9f087dd643e7301bd56332ec12e54bfe3ee41a)
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
![{\displaystyle {\begin{aligned}\mathbf {R} ^{0}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {V} (st)}\\\mathbf {R} ^{2}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {V} (st)}\\\mathbf {R} ^{4}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {V} (st)}\\\mathbf {R} ^{k}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {V} (st)}&={\underline {0}}_{\mathbb {V} (st)}{\text{ otherwise}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aab7637a381b79944e364e8d4e12d0834d7bef5a)
while over
we have the local systems
![{\displaystyle {\begin{aligned}\mathbf {R} ^{0}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}\\\mathbf {R} ^{1}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}^{\oplus 2g}\\\mathbf {R} ^{2}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}\\\mathbf {R} ^{k}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}&={\underline {0}}_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}{\text{ otherwise}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/407ef0affdceaeef1e95b5141bf7a440e821caeb)
where
is the genus of the plane curve (which is
).
Applications
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
References
External links