# Local system

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{{Short description|Locally constant sheaf of abelian groups on topological space}}
{{More citations needed|date=January 2021}}
In [mathematics](/source/mathematics), a '''local system''' (or a system of '''local coefficients''') on a [topological space](/source/topological_space) ''X'' is a tool from [algebraic topology](/source/algebraic_topology) which interpolates between [cohomology](/source/homology_theory) with coefficients in a fixed [abelian group](/source/abelian_group) ''A'', and general [sheaf cohomology](/source/sheaf_cohomology) in which coefficients vary from point to point.  Local coefficient systems were introduced by [Norman Steenrod](/source/Norman_Steenrod) in 1943.<ref>{{cite journal | last=Steenrod | first=Norman E. | author-link=Norman Steenrod| title=Homology with local coefficients | journal=[Annals of Mathematics](/source/Annals_of_Mathematics) | volume=44 | issue=4 | year=1943 | doi=10.2307/1969099 | pages=610–627 | jstor=1969099 | mr=9114}}</ref>

Local systems are the building blocks of more general tools, such as [constructible](/source/constructible_sheaf) and [perverse](/source/perverse_sheaf) sheaves.

==Definition==
Let ''X'' be a [topological space](/source/topological_space). A '''local system''' (of [abelian group](/source/abelian_group)s/[modules](/source/Module_over_a_ring)...) on ''X'' is a [locally constant sheaf](/source/locally_constant_sheaf) (of abelian groups/[of modules](/source/sheaf_of_modules)...) on ''X''. In other words, a sheaf <math>\mathcal{L}</math> is a local system if every point has an open neighborhood <math>U</math> such that the restricted sheaf <math>\mathcal{L}|_U</math> is isomorphic to the sheafification of some constant presheaf. {{clarify| there is some ambiguity of constant sheaf, some uses the definition that restriction is identity; others uses the convention that isomorphic to such a sheaf suffices.|date=September 2022}}

===Equivalent definitions===
====Path-connected spaces====
If ''X'' is [path-connected](/source/Connected_space),{{clarify|Don't you also need "[locally simply connected](/source/locally_simply_connected)?"|date=August 2022}} a local system <math>\mathcal{L}</math> of abelian groups has the same stalk <math>L</math> at every point. There is a bijective correspondence between local systems on ''X'' and group homomorphisms
: <math> \rho: \pi_1(X,x) \to \text{Aut}(L)</math>
and similarly for local systems of modules. The map <math> \pi_1(X,x) \to \text{Aut}(L) </math> giving the local system <math>\mathcal{L}</math> is called the '''[monodromy](/source/monodromy) representation''' of <math>\mathcal{L}</math>.

{{math proof|title=''Proof of equivalence''|proof=Take local system <math> \mathcal{L} </math> and a loop <math>\gamma</math> at ''x''.  It's easy to show that any local system on <math> [0,1] </math> is constant. For instance, <math> \gamma^* \mathcal{L} </math> is constant. This gives an isomorphism <math> (\gamma^*\mathcal{L})_0\simeq \Gamma([0,1], \mathcal{L}) \simeq (\gamma^*\mathcal{L})_1 </math>, i.e. between <math>L</math> and itself. 
Conversely, given a homomorphism <math> \rho: \pi_1(X,x)\to \text{Aut}(L)</math>, consider the ''constant'' sheaf <math>\underline{L} </math> on the universal cover <math>\widetilde{X}</math> of ''X''. The deck-transform-invariant sections of <math>\underline{L} </math> 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
: <math> \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\} </math>
where <math>\pi:\widetilde{X}\to X</math> is the universal covering.
}}

This shows that (for ''X'' path-connected) a local system is precisely a sheaf whose pullback to the [universal cover](/source/universal_cover) of ''X'' is a constant sheaf.

This correspondence can be upgraded to an [equivalence of categories](/source/equivalence_of_categories) between the category of local systems of abelian groups on ''X'' and the [category of abelian groups](/source/category_of_abelian_groups) endowed with an action of <math>\pi_1(X,x)</math> (equivalently, <math>\mathbb{Z}[\pi_1(X,x)]</math>-modules).<ref> [Milne, James S.](/source/James_Milne_(mathematician)) (2017). ''[https://www.jmilne.org/math/xnotes/svi.pdf Introduction to Shimura Varieties]''. Proposition 14.7.</ref>

====Stronger definition on non-connected spaces====
A stronger nonequivalent definition that works for non-connected ''X'' is the following: a local system is a [covariant functor](/source/covariant_functor)
:<math> \mathcal{L}\colon \Pi_1(X) \to \textbf{Mod}(R)</math>
from the fundamental groupoid of <math>X</math> to the category of modules over a commutative ring <math>R</math>, where typically <math>R = \Q,\R,\Complex</math>. This is equivalently the data of an assignment to every point <math>x\in X</math> a module <math>M</math> along with a group representation <math>\rho_x: \pi_1(X,x) \to \text{Aut}_R(M)</math> such that the various <math>\rho_x</math> are compatible with change of basepoint <math>x \to y</math> and the induced map <math>\pi_1(X, x) \to \pi_1(X, y)</math> on [fundamental groups](/source/fundamental_groups).

==Examples==

* [Constant sheaves](/source/Constant_sheaf) such as <math>\underline{\Q}_X</math>. This is a useful tool for computing cohomology since in good situations, there is an isomorphism between sheaf cohomology and singular cohomology: 

<math display="block">H^k(X,\underline{\Q}_X) \cong H^k_\text{sing}(X,\Q)</math>

* Let <math>X=\R^2 \setminus \{(0,0)\}</math>. Since <math> \pi_1(\R^2 \setminus \{(0,0)\})=\mathbb{Z}</math>, there is an <math> S^1</math> family of local systems on ''X'' corresponding to the maps <math>n \mapsto e^{in\theta}</math>:

<math display="block">\rho_\theta: \pi_1(X; x_0) \cong \Z \to \text{Aut}_\Complex(\Complex)</math>

* Horizontal sections of vector bundles with a flat connection. If <math> E\to X </math> is a vector bundle with flat connection <math> \nabla</math>, then there is a local system given by <math display="block"> E^\nabla_U=\left\{\text{sections }s\in \Gamma(U,E) \text{ which are horizontal: }\nabla s=0\right\}</math>  For instance, take <math>X=\Complex \setminus 0</math> and <math>E = X \times \Complex^n</math>, the trivial bundle. Sections of ''E'' are ''n''-tuples of functions on ''X'', so <math> \nabla_0(f_1,\dots,f_n)= (df_1,\dots,df_n)</math> defines a flat connection on ''E'', as does <math> \nabla(f_1,\dots,f_n)= (df_1,\dots,df_n)-\Theta(x)(f_1,\dots,f_n)^t</math> for any matrix of one-forms <math> \Theta </math> on ''X''. The horizontal sections are then</p> <math display="block"> E^\nabla_U= \left\{(f_1,\dots,f_n)\in E_U: (df_1,\dots,df_n)=\Theta (f_1,\dots,f_n)^t\right\}</math> i.e., the solutions to the linear differential equation <math> df_i = \sum \Theta_{ij} f_j</math>.<p>If <math> \Theta </math> extends to a one-form on <math> \Complex</math> the above will also define a local system on <math> \Complex</math>, so will be trivial since <math>\pi_1(\Complex) = 0</math>. So to give an interesting example, choose one with a pole at ''0'':</p> <math display="block"> \Theta= \begin{pmatrix} 0 & dx/x  \\ dx & e^x dx \end{pmatrix}</math> in which case for <math>  \nabla= d+ \Theta </math>, <math display="block"> E^\nabla_U =\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\} </math>

* An ''n''-sheeted covering map <math> X\to Y</math> is a local system with fibers given by the set <math> \{1,\dots,n\} </math>. 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 equivalent to a ''k''-linear [representation](/source/Group_representation) of <math>\pi_1(X,x)</math>.

* If ''X'' is a variety, local systems are the same thing as [D-modules](/source/D-modules) which are additionally coherent ''O_X''-modules (see [O modules](/source/Sheaf_of_modules)).

* If the connection is not flat (i.e. its [curvature](/source/curvature) is nonzero), then [parallel transport](/source/parallel_transport) of a fibre ''F_x'' over ''x'' around a contractible loop based at ''x''_0 may give a nontrivial automorphism of ''F_x'', so locally constant sheaves can not necessarily be defined for non-flat connections.

* The [Gauss–Manin connection](/source/Gauss%E2%80%93Manin_connection) is a prominent example of a connection whose horizontal sections are studied in relation to [variation of Hodge structures](/source/Hodge_structure).

==Cohomology==

There are several ways to define the cohomology of a local system, called '''cohomology with local coefficients''', which become equivalent under mild assumptions on ''X''.

* Given a locally constant sheaf <math>\mathcal{L}</math> of abelian groups on ''X'', we have the [sheaf cohomology](/source/sheaf_cohomology) groups <math>H^j(X,\mathcal{L})</math> with coefficients in <math>\mathcal{L}</math>.

* Given a locally constant sheaf <math>\mathcal{L}</math> of abelian groups on ''X'', let <math>C^n(X;\mathcal{L})</math> be the group of all functions ''f'' which map each [singular ''n''-simplex](/source/Singular_homology) <math>\sigma\colon\Delta^n\to X</math> to a global section <math>f(\sigma)</math> of the [inverse-image sheaf](/source/Inverse_image_functor) <math>\sigma^{-1}\mathcal{L}</math>. These groups can be made into a cochain complex with differentials constructed as in usual singular cohomology. Define <math>H^j_\mathrm{sing}(X;\mathcal{L})</math> to be the cohomology of this complex.

* The group <math>C_n(\widetilde{X})</math> of singular ''n''-chains on the universal cover of ''X'' has an action of <math>\pi_1(X,x)</math> by [deck transformations](/source/Covering_space). Explicitly, a deck transformation <math>\gamma\colon\widetilde{X}\to\widetilde{X}</math> takes a singular ''n''-simplex <math>\sigma\colon\Delta^n\to\widetilde{X}</math> to <math>\gamma\circ\sigma</math>. Then, given an abelian group ''L'' equipped with an action of <math>\pi_1(X,x)</math>, one can form a cochain complex from the groups <math>\operatorname{Hom}_{\pi_1(X,x)}(C_n(\widetilde{X}),L)</math> of <math>\pi_1(X,x)</math>-equivariant homomorphisms as above. Define <math>H^j_\mathrm{sing}(X;L)</math> to be the cohomology of this complex.

If ''X'' is [paracompact](/source/Paracompact_space) and [locally contractible](/source/Contractible_space), then <math>H^j(X,\mathcal{L})\cong H^j_\mathrm{sing}(X;\mathcal{L})</math>.<ref> [Bredon, Glen E.](/source/Glen_Bredon) (1997). ''Sheaf Theory'', Second Edition, Graduate Texts in Mathematics, vol. 25, [Springer-Verlag](/source/Springer_Science%2BBusiness_Media). Chapter III, Theorem 1.1.</ref> If <math>\mathcal{L}</math> is the local system corresponding to ''L'', then there is an identification <math>C^n(X;\mathcal{L})\cong\operatorname{Hom}_{\pi_1(X,x)}(C_n(\widetilde{X}),L)</math> compatible with the differentials,<ref> [Hatcher, Allen](/source/Allen_Hatcher) (2001). ''Algebraic Topology'', [Cambridge University Press](/source/Cambridge_University_Press). Section 3.H.</ref> so <math>H^j_\mathrm{sing}(X;\mathcal{L})\cong H^j_\mathrm{sing}(X;L)</math>.

==Generalization==
Local systems have a mild generalization to [constructible sheaves](/source/constructible_sheaf) -- a constructible sheaf on a locally path connected topological space <math>X</math> is a sheaf <math>\mathcal{L}</math> such that there exists a stratification of
:<math>X = \coprod X_\lambda</math>
where <math>\mathcal{L}|_{X_\lambda}</math> is a local system. These are typically found by taking the cohomology of the derived pushforward for some continuous map <math>f:X \to Y</math>. For example, if we look at the complex points of the morphism
:<math>f:X = \text{Proj}\left(\frac{\Complex[s,t][x,y,z]}{(st\cdot h(x,y,z))}\right) \to \text{Spec}(\Complex[s,t])</math>
then the fibers over
:<math>\mathbb{A}^2_{s,t} - \mathbb{V}(st)</math>
are the plane curve given by <math>h</math>, but the fibers over <math>\mathbb{V}= \mathbb{V}(st)</math> are <math>\mathbb{P}^2</math>. If we take the derived pushforward <math>\mathbf{R}f_!(\underline{\Q}_X)</math> then we get a constructible sheaf. Over <math>\mathbb{V}</math> we have the local systems
:<math>
\begin{align}
\mathbf{R}^0f_!(\underline{\mathbb{Q}}_X)|_{\mathbb{V}(st)} &= \underline{\Q}_{\mathbb{V}(st)} \\
\mathbf{R}^2f_!(\underline{\mathbb{Q}}_X)|_{\mathbb{V}(st)} &= \underline{\Q}_{\mathbb{V}(st)} \\
\mathbf{R}^4f_!(\underline{\mathbb{Q}}_X)|_{\mathbb{V}(st)} &= \underline{\Q}_{\mathbb{V}(st)} \\
\mathbf{R}^kf_!(\underline{\mathbb{Q}}_X)|_{\mathbb{V}(st)} &= \underline{0}_{\mathbb{V}(st)} \text{ otherwise}
\end{align}
</math>
while over <math>\mathbb{A}^2_{s,t} - \mathbb{V}(st)</math> we have the local systems
:<math>\begin{align}
\mathbf{R}^0f_!(\underline{\Q}_X)|_{\mathbb{A}^2_{s,t} - \mathbb{V}(st)} &= \underline{\Q}_{\mathbb{A}^2_{s,t} - \mathbb{V}(st)} \\
\mathbf{R}^1f_!(\underline{\Q}_X)|_{\mathbb{A}^2_{s,t} - \mathbb{V}(st)} &= \underline{\Q}_{\mathbb{A}^2_{s,t} - \mathbb{V}(st)}^{\oplus 2g} \\
\mathbf{R}^2f_!(\underline{\Q}_X)|_{\mathbb{A}^2_{s,t} - \mathbb{V}(st)} &= \underline{\Q}_{\mathbb{A}^2_{s,t} - \mathbb{V}(st)} \\
\mathbf{R}^kf_!(\underline{\Q}_X)|_{\mathbb{A}^2_{s,t} - \mathbb{V}(st)} &= \underline{0}_{\mathbb{A}^2_{s,t} - \mathbb{V}(st)} \text{ otherwise}
\end{align}
</math>
where <math>g</math> is the genus of the plane curve (which is <math>g = (\deg(f) - 1)(\deg(f) - 2)/2</math>).

==Applications==

The cohomology with local coefficients in the module corresponding to the [orientation covering](/source/orientation_covering) can be used to formulate [Poincaré duality](/source/Poincar%C3%A9_duality) for non-orientable manifolds: see [Twisted Poincaré duality](/source/Twisted_Poincar%C3%A9_duality).

==See also==
* [Leray spectral sequence](/source/Leray_spectral_sequence)
* [Gauss–Manin connection](/source/Gauss%E2%80%93Manin_connection)
* [D-module](/source/D-module)
* [Intersection homology](/source/Intersection_homology)
* [Perverse sheaf](/source/Perverse_sheaf)

==References==
{{reflist}}

==External links==
* {{cite web|url=https://math.stackexchange.com/q/13332|title=What local system really is|publisher= [Stack Exchange](/source/Stack_Exchange)}}
* {{cite web|url=https://www.math.stonybrook.edu/~cschnell/pdf/notes/locsys.pdf|title= Computing Cohomology of Local Systems|first=Christian|last=Schnell}} Discusses computing the cohomology with coefficients in a local system by using the twisted de Rham complex.
* {{cite web|url=http://people.mpim-bonn.mpg.de/geordie/perverse_course/lectures.pdf|title= An illustrated guide to perverse sheaves|first= Geordie|last= Williamson|author-link= Geordie Williamson}}
* {{cite web|url=http://faculty.tcu.edu/gfriedman/notes/ih.pdf |title=Intersection homology and perverse sheaves| first=Robert | last=MacPherson | author-link=Robert MacPherson (mathematician)|date=December 15, 1990}}
* {{cite web|url=https://webusers.imj-prg.fr/~fouad.elzein/elzein-snoussif.pdf|title= Local systems and constructible sheaves|first1= Fouad|last1= El Zein|first2= Jawad |last2=Snoussi}}

{{DEFAULTSORT:Local System}}
Category:Sheaf theory
Category:Algebraic topology

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