| This article needs additional citations for verification. Please help by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: – · · · scholar · JSTOR (January 2021) (Learn how and when to remove this template message) |
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.[1]
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
is a constant sheaf.
Equivalent definitions[]
Path-connected spaces[]
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
![{\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[]
Another (stronger, nonequivalent) definition generalising 2, and working for non-connected X, 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
. Typically
. What this is saying is that at every point
we should assign a module
with a representation
such that these representations are compatible with change of basepoint
for the fundamental group.
Examples[]
- Constant sheaves. For instance,
. This is a useful tool for computing cohomology since the sheaf 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)
is isomorphic to the singular cohomology of
.
. Since
, there are
-many linear systems on X, the
one given by monodromy representation ![{\displaystyle \pi _{x}(X)=\mathbb {Z} \to {\text{Aut}}_{\mathbb {C} }(\mathbb {C} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23d477a378c793c61fee1d7bd11a7c060c56ad72)
by sending ![{\displaystyle n\mapsto e^{in\theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd11c6ea07d9930ab0f09b0e5c83791ac40e9e30)
Horizontal sections of vector bundles with a flat connection. If
is a vector bundle with flat connection
, then
![{\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)
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
![{\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 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.
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[]