A local system on a topological space is a locally constant sheaf. A finite-rank local system of vector spaces over a fixed field on a connected, locally path-connected, locally simply connected topological space is equivalently a group representation of its fundamental group over that field, after choosing a basepoint. Transport around loops gives the group representation, called monodromy. On a simply connected topological space satisfying these local conditions, such a local system is constant.

In the étale setting, a lisse sheaf is the analogous -adic coefficient object. On a connected normal variety over a field of characteristic different from , a lisse sheaf with coefficient field a finite extension of corresponds to a continuous group representation of the étale fundamental group on a finite-dimensional -vector space (Milne 2013, pp. 124-126). The geometric Langlands correspondence uses local systems for the Langlands dual group as spectral parameters.

See also

Etale Fundamental Group, Fundamental Group, Geometric Langlands Correspondence, Lisse Sheaf, Local Coefficient System, Sheaf

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References

Milne, J. S. "Lectures on Étale Cohomology." Version 2.21, Mar. 22, 2013. https://www.jmilne.org/math/CourseNotes/LEC.pdf.Simpson, C. T. "Moduli of Representations of the Fundamental Group of a Smooth Projective Variety I." Publ. Math. I.H.E.S. 79, 47-129, 1994. https://doi.org/10.1007/BF02698887.

Cite this as:

Weisstein, Eric W. "Local System." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LocalSystem.html

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