# 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 ${\displaystyle {\mathcal {L}}}$ on X. That is, every point has an open neighborhood ${\displaystyle U}$ such that ${\displaystyle {\mathcal {L}}|_{U}}$ is a constant sheaf.

If X is path-connected, this the same as

2. A homomorphism ${\displaystyle \rho :\pi _{1}(X,x)\to {\text{End}}_{R}(M)}$ (${\displaystyle M={\mathcal {L}}_{x}}$ in the above).

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

3. A functor
${\displaystyle {\mathcal {L}}:\Pi _{1}(X)\to {\textbf {Mod}}(R)}$
from the fundamental groupoid of ${\displaystyle X}$ to the category of modules over a commutative ring ${\displaystyle R}$. Typically ${\displaystyle R=\mathbb {Q} ,\mathbb {R} ,\mathbb {C} }$. What this is saying is that at every point ${\displaystyle x\in X}$ we should assign a module :${\displaystyle M}$ with a representations of ${\displaystyle \pi _{1}(X,x)\to {\text{Aut}}_{R}(M)}$ such that these representations are compatible with change of basepoint ${\displaystyle x\to y}$ for the fundamental group.

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

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

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

4. A sheaf ${\displaystyle {\mathcal {L}}}$ whose pullback ${\displaystyle \pi ^{*}{\mathcal {L}}}$ by the universal cover ${\displaystyle \pi :{\widetilde {X}}\to X}$ is the constant sheaf.

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

## Examples

• Constant sheaves. For instance, ${\displaystyle {\underline {\mathbb {Q} }}_{X}}$. This is a useful tool for computing cohomology since the sheaf cohomology
${\displaystyle H^{k}(X,{\underline {\mathbb {Q} }}_{X})\cong H_{\text{sing}}^{k}(X,\mathbb {Q} )}$
is isomorphic to the singular cohomology of ${\displaystyle X}$.
• ${\displaystyle X=\mathbb {R} ^{2}\setminus \{0,0\}}$. Since ${\displaystyle \pi _{1}(\mathbb {R} ^{2}\setminus \{0,0\})=\mathbb {Z} }$, there are ${\displaystyle S^{1}}$-many linear systems on X, the ${\displaystyle \theta \in \mathbb {R} }$ one given by monodromy representation
${\displaystyle \pi _{x}(X)=\mathbb {Z} \to {\text{Aut}}_{\mathbb {C} }(\mathbb {C} )}$ by sending ${\displaystyle n\mapsto e^{in\theta }}$
• Horizontal sections of vector bundles with a flat connection. If ${\displaystyle E\to X}$ is a vector bundle with flat connection ${\displaystyle \nabla }$, then
${\displaystyle E_{U}^{\nabla }=\left\{{\text{sections }}s\in \Gamma (U,E){\text{ which are horizontal: }}\nabla s=0\right\}}$
is a local system.
For instance, take ${\displaystyle X=\mathbb {C} \setminus 0}$ and ${\displaystyle E=X\times \mathbb {C} ^{n}}$ the trivial bundle. Sections of E are n-tuples of functions on X, so ${\displaystyle \nabla _{0}(f_{1},...,f_{n})=(df_{1},...,df_{n})}$ defines a flat connection on E, as does ${\displaystyle \nabla (f_{1},...,f_{n})=(df_{1},...,df_{n})-\Theta (f_{1},...,f_{n})}$ for any matrix of one-forms ${\displaystyle \Theta }$ on X. The horizontal sections are then
${\displaystyle E_{U}^{\nabla }=\left\{(f_{1},...,f_{n})\in E_{U}:(df_{1},...,df_{n})=\Theta (f_{1},...,f_{n})\right\}}$
i.e. the solutions to the linear differential equation ${\displaystyle df_{i}=\sum \Theta _{ij}f_{j}}$.
If ${\displaystyle \Theta }$ extends to a one-form on ${\displaystyle \mathbb {C} }$ the above will also define a local system on ${\displaystyle \mathbb {C} }$, so will be trivial since ${\displaystyle \pi _{1}(\mathbb {C} )=0}$. So to give an interesting example, choose one with a pole at 0:
${\displaystyle \Theta ={\begin{pmatrix}0&dx/x\\dx&e^{x}dx\end{pmatrix}}}$
in which case for ${\displaystyle \nabla =d+\Theta }$,
${\displaystyle E_{U}^{\nabla }=\left\{f_{1},f_{2}:U\to \mathbb {C} \ \ {\text{ with }}f'_{1}=f_{2}/x\ \ f_{2}'=f_{1}+e^{x}f_{2}\right\}}$
• An n-sheeted covering map ${\displaystyle X\to Y}$ is a local system with sections locally the set ${\displaystyle \{1,...,n\}}$. 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 ${\displaystyle \pi _{1}(X,x)}$.
• 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 ${\displaystyle X}$ is a sheaf ${\displaystyle {\mathcal {L}}}$ such that there exists a stratification of

${\displaystyle X=\coprod X_{\lambda }}$

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

${\displaystyle f:X={\text{Proj}}\left({\frac {\mathbb {C} [s,t][x,y,z]}{(stf(x,y,z))}}\right)\to {\text{Spec}}(\mathbb {C} [s,t])}$

then the fibers over

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

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

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

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

## 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.