Local system

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.

Definition

A local system of modules on space X is

1. A locally constant sheaf of modules

{\mathcal {L}}

on X. That is, every point has an open neighborhood

U

such that

{\mathcal {L}}|_{U}

is a constant sheaf.

If X is path-connected, this the same as

2. A homomorphism

\rho :\pi _{1}(X,x)\to {\text{End}}_{R}(M)

(

M={\mathcal {L}}_{x}

in the above).

Another (stronger, nonequivalent) definition generalising 2, and working for nonconnected X, is

3. A functor

{\mathcal {L}}:\Pi _{1}(X)\to {\textbf {Mod}}(R)

from the fundamental groupoid of

X

to the category of modules over a commutative ring

R

. Typically

R=\mathbb {Q} ,\mathbb {R} ,\mathbb {C}

. What this is saying is that at every point

x\in X

we should assign a module

M

with a representations of

\pi _{1}(X,x)\to {\text{Aut}}_{R}(M)

such that these representations are compatible with change of basepoint

x\to y

for the fundamental group.

Here's a proof that 1 and 2 are the same if X is path-connected.

Take $\mathcal{L}$ as in 1 and a loop $\gamma$ at x. It's easy so show that any (definition 1)-local system on $[0,1]$ is constant. For instance, $\gamma ^{*}{\mathcal {L}}$ . So we get an isomorphism $(\gamma ^{*}{\mathcal {L}})_{0}\simeq \Gamma ([0,1],{\mathcal {L}})\simeq (\gamma ^{*}{\mathcal {L}})_{0}$ . But $\gamma$ gives an isomorphism between both sides and ${\mathcal {L}}_{x}$ , whence an endomorphism of ${\mathcal {L}}_{x}$ .
Take homomorphism $\rho :\pi _{1}(X,x)\to {\text{End}}_{R}(M)$ . Consider the constant sheaf ${\underline {M}}$ on the universal cover $\widetilde{X}$ of $X$ with cover $\pi :{\widetilde {X}}\to X$ , and let $\mathcal{L}$ be the deck-transform-ρ-equivariant part:

{\mathcal {L}}_{U}=\left\{{\text{sections }}s\in {\underline {M}}_{\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\}

The proof shows that (for X path-connected) another equivalent definition of a local system is

4. A sheaf

\mathcal{L}

whose pullback

\pi ^{*}{\mathcal {L}}

by the universal cover

\pi :{\widetilde {X}}\to X

is the constant sheaf.

Call the map $\pi _{1}(X,x)\to {\text{End}}_{R}(M)$ the monodromy representation of the local system.

Examples

Constant sheaves. For instance, ${\underline {\mathbb {Q} }}_{X}$ . This is a useful tool for computing cohomology since the sheaf cohomology

H^{k}(X,{\underline {\mathbb {Q} }}_{X})\cong H_{\text{sing}}^{k}(X,\mathbb {Q} )

is isomorphic to the singular cohomology of

X

.

$X=\mathbb {R} ^{2}\setminus \{0,0\}$ . Since $\pi _{1}(\mathbb {R} ^{2}\setminus \{0,0\})=\mathbb {Z}$ , there are $S^{1}$ -many linear systems on X, the $\theta \in {\mathbb {R}}$ one given by monodromy representation

\pi _{x}(X)=\mathbb {Z} \to {\text{Aut}}_{\mathbb {C} }(\mathbb {C} )

by sending

n\mapsto e^{in\theta }

Horizontal sections of vector bundles with a flat connection. If $E\to X$ is a vector bundle with flat connection $\nabla$ , then

E_{U}^{\nabla }=\left\{{\text{sections }}s\in \Gamma (U,E){\text{ which are horizontal: }}\nabla s=0\right\}

is a local system.

For instance, take

X=\mathbb {C} \setminus 0

and

E=X\times {\mathbb {C}}^{n}

the trivial bundle. Sections of E are n-tuples of functions on X, so

\nabla _{0}(f_{1},...,f_{n})=(df_{1},...,df_{n})

defines a flat connection on E, as does

\nabla (f_{1},...,f_{n})=(df_{1},...,df_{n})-\Theta (x)(f_{1},...,f_{n})^{t}

for any matrix of one-forms

\Theta

on X. The horizontal sections are then

E_{U}^{\nabla }=\left\{(f_{1},...,f_{n})\in E_{U}:(df_{1},...,df_{n})=\Theta (f_{1},...,f_{n})^{t}\right\}

i.e. the solutions to the linear differential equation

df_{i}=\sum \Theta _{ij}f_{j}

.

If

\Theta

extends to a one-form on

\mathbb{C}

the above will also define a local system on

\mathbb{C}

, so will be trivial since

\pi _{1}(\mathbb {C} )=0

. So to give an interesting example, choose one with a pole at 0:

\Theta ={\begin{pmatrix}0&dx/x\\dx&e^{x}dx\end{pmatrix}}

in which case for

\nabla =d+\Theta

,

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\}

An n-sheeted covering map $X\to Y$ is a local system with sections locally the set $\{1,...,n\}$ . 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 $\pi_1(X,x)$ .
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 $X$ is a sheaf ${\mathcal {L}}$ such that there exists a stratification of

X=\coprod X_{\lambda }

where ${\mathcal {L}}|_{X_{\lambda }}$ is a local system. These are typically found by taking the cohomology of the derived pushforward for some continuous map $f:X\to Y$ . For example, if we look at the complex points of the morphism

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])

then the fibers over

\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)

are the smooth plane curve given by $f$ , but the fibers over $\mathbb {V}$ are $\mathbb {P} ^{2}$ . If we take the derived pushforward $\mathbf {R} f_{!}({\underline {\mathbb {Q} }}_{X})$ then we get a constructible sheaf. Over $\mathbb {V}$ we have the local systems

{\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}}

while over $\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)$ we have the local systems

{\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}}

where $g$ is the genus of the plane curve (which is $g=({\text{deg}}(f)-1)({\text{deg}}(f)-2)/2$ ).

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.

External links

A discussion of the notion on Stack exchange
Computing Cohomology of Local Systems discusses computing the cohomology with coefficients in a local system by using the twisted de-Rham complex.
An Illustrated Guide to Perverse Sheaves by Geordie Williamson
Intersection Homology and Perverse Sheaves by Robert Macpherson
Local systems and Constructible Sheaves by Fouad El Zein and Jawad Snoussi.

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