Local system - Knowlpedia

Let $R$ be a ring. A local system of $R$ -modules on a topological space $X$ is a sheaf $\mathcal L$ of $R$ -modules that is locally isomorphic to a constant sheaf of $R$ -modules; equivalently, around each point its sections are locally constant functions with values in a fixed $R$ -module $L$ , with pointwise module operations and restriction maps given by restricting functions. More generally, a local system may take values in sets, groups, vector spaces, or another category.

Monodromy

On a path-connected, locally path-connected, and semilocally simply connected space, choosing $x\in X$ identifies local systems with fixed fiber module $L$ with representations

\pi_1(X,x)\longrightarrow \operatorname{Aut}_R(L),

up to isomorphism of the fiber module. If $L\cong R^r$ is finite free, this automorphism group is $GL_r(R)$ .

Fundamental-groupoid form

Without choosing a base point, parallel continuation gives a functor from the fundamental groupoid of $X$ . This formulation handles disconnected spaces and makes transport along paths intrinsic.

Relation to flat bundles

On a smooth manifold, a finite-rank complex local system determines a vector bundle with a flat connection. Conversely, horizontal sections of a flat bundle form a local system. This analytic correspondence should not be confused with the algebraic de Rham formulation of a $G$ -local system.

References

Alexander Grothendieck, Revêtements étales et groupe fondamental (SGA 1), Lecture Notes in Mathematics 224, Springer, 1971. DOI.