Local system

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In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman Steenrod in 1943.^[1]

Local systems are the building blocks of more general tools, such as constructible and perverse sheaves.

Let X be a topological space. A local system (of abelian groups/modules...) on X is a locally constant sheaf (of abelian groups/of modules...) on X. In other words, a sheaf L {\displaystyle {\mathcal {L}}} is a local system if every point has an open neighborhood U {\displaystyle U} such that the restricted sheaf L | U {\displaystyle {\mathcal {L}}|_{U}} is isomorphic to the sheafification of some constant presheaf. ^{[clarification needed]}

If X is path-connected,^{[clarification needed]} a local system L {\displaystyle {\mathcal {L}}} of abelian groups has the same stalk L {\displaystyle L} at every point. There is a bijective correspondence between local systems on X and group homomorphisms

ρ : π 1 ( X , x ) → Aut ( L ) {\displaystyle \rho :\pi _{1}(X,x)\to {\text{Aut}}(L)}

and similarly for local systems of modules. The map π 1 ( X , x ) → Aut ( L ) {\displaystyle \pi _{1}(X,x)\to {\text{Aut}}(L)} giving the local system L {\displaystyle {\mathcal {L}}} is called the monodromy representation of L {\displaystyle {\mathcal {L}}} .

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

This correspondence can be upgraded to an equivalence of categories between the category of local systems of abelian groups on X and the category of abelian groups endowed with an action of π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} (equivalently, Z [ π 1 ( X , x ) ] {\displaystyle \mathbb {Z} [\pi _{1}(X,x)]} -modules).^[2]

A stronger nonequivalent definition that works for non-connected X is the following: a local system is a covariant functor

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

from the fundamental groupoid of X {\displaystyle X} to the category of modules over a commutative ring R {\displaystyle R} , where typically R = Q , R , C {\displaystyle R=\mathbb {Q} ,\mathbb {R} ,\mathbb {C} } . This is equivalently the data of an assignment to every point x ∈ X {\displaystyle x\in X} a module M {\displaystyle M} along with a group representation ρ x : π 1 ( X , x ) → Aut R ( M ) {\displaystyle \rho _{x}:\pi _{1}(X,x)\to {\text{Aut}}_{R}(M)} such that the various ρ x {\displaystyle \rho _{x}} are compatible with change of basepoint x → y {\displaystyle x\to y} and the induced map π 1 ( X , x ) → π 1 ( X , y ) {\displaystyle \pi _{1}(X,x)\to \pi _{1}(X,y)} on fundamental groups.

• Constant sheaves such as Q _ X {\displaystyle {\underline {\mathbb {Q} }}_{X}} . This is a useful tool for computing cohomology since in good situations, there is an isomorphism between sheaf cohomology and singular cohomology:

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

• Let X = R 2 ∖ { ( 0 , 0 ) } {\displaystyle X=\mathbb {R} ^{2}\setminus \{(0,0)\}} . Since π 1 ( R 2 ∖ { ( 0 , 0 ) } ) = Z {\displaystyle \pi _{1}(\mathbb {R} ^{2}\setminus \{(0,0)\})=\mathbb {Z} } , there is an S 1 {\displaystyle S^{1}} family of local systems on X corresponding to the maps n ↦ e i n θ {\displaystyle n\mapsto e^{in\theta }} :

ρ θ : π 1 ( X ; x 0 ) ≅ Z → Aut C ( C ) {\displaystyle \rho _{\theta }:\pi _{1}(X;x_{0})\cong \mathbb {Z} \to {\text{Aut}}_{\mathbb {C} }(\mathbb {C} )}

• Horizontal sections of vector bundles with a flat connection. If E → X {\displaystyle E\to X} is a vector bundle with flat connection ∇ {\displaystyle \nabla } , then there is a local system given by E U ∇ = { sections  s ∈ Γ ( U , E )  which are horizontal:  ∇ s = 0 } {\displaystyle E_{U}^{\nabla }=\left\{{\text{sections }}s\in \Gamma (U,E){\text{ which are horizontal: }}\nabla s=0\right\}} For instance, take X = C ∖ 0 {\displaystyle X=\mathbb {C} \setminus 0} and E = X × C n {\displaystyle E=X\times \mathbb {C} ^{n}} , the trivial bundle. Sections of E are n-tuples of functions on X, so ∇ 0 ( f 1 , … , f n ) = ( d f 1 , … , d f n ) {\displaystyle \nabla _{0}(f_{1},\dots ,f_{n})=(df_{1},\dots ,df_{n})} defines a flat connection on E, as does ∇ ( f 1 , … , f n ) = ( d f 1 , … , d f n ) − Θ ( x ) ( f 1 , … , f n ) t {\displaystyle \nabla (f_{1},\dots ,f_{n})=(df_{1},\dots ,df_{n})-\Theta (x)(f_{1},\dots ,f_{n})^{t}} for any matrix of one-forms Θ {\displaystyle \Theta } on X. The horizontal sections are then
  E U ∇ = { ( f 1 , … , f n ) ∈ E U : ( d f 1 , … , d f n ) = Θ ( f 1 , … , f n ) t } {\displaystyle E_{U}^{\nabla }=\left\{(f_{1},\dots ,f_{n})\in E_{U}:(df_{1},\dots ,df_{n})=\Theta (f_{1},\dots ,f_{n})^{t}\right\}} i.e., the solutions to the linear differential equation d f i = ∑ Θ i j f j {\displaystyle df_{i}=\sum \Theta _{ij}f_{j}} .

  If Θ {\displaystyle \Theta } extends to a one-form on C {\displaystyle \mathbb {C} } the above will also define a local system on C {\displaystyle \mathbb {C} } , so will be trivial since π 1 ( C ) = 0 {\displaystyle \pi _{1}(\mathbb {C} )=0} . So to give an interesting example, choose one with a pole at 0:

  Θ = ( 0 d x / x d x e x d x ) {\displaystyle \Theta ={\begin{pmatrix}0&dx/x\\dx&e^{x}dx\end{pmatrix}}} in which case for ∇ = d + Θ {\displaystyle \nabla =d+\Theta } , E U ∇ = { f 1 , f 2 : U → C      with  f 1 ′ = f 2 / x     f 2 ′ = f 1 + e x f 2 } {\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 X → Y {\displaystyle X\to Y} is a local system with fibers given by the set { 1 , … , n } {\displaystyle \{1,\dots ,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 equivalent to a k-linear representation of π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} .

• If X is a variety, local systems are the same thing as D-modules which are additionally coherent O_X-modules (see O modules).

• If the connection is not flat (i.e. its curvature is nonzero), then 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 is a prominent example of a connection whose horizontal sections are studied in relation to variation of Hodge structures.

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 L {\displaystyle {\mathcal {L}}} of abelian groups on X, we have the sheaf cohomology groups H j ( X , L ) {\displaystyle H^{j}(X,{\mathcal {L}})} with coefficients in L {\displaystyle {\mathcal {L}}} .

• Given a locally constant sheaf L {\displaystyle {\mathcal {L}}} of abelian groups on X, let C n ( X ; L ) {\displaystyle C^{n}(X;{\mathcal {L}})} be the group of all functions f which map each singular n-simplex σ : Δ n → X {\displaystyle \sigma \colon \Delta ^{n}\to X} to a global section f ( σ ) {\displaystyle f(\sigma )} of the inverse-image sheaf σ − 1 L {\displaystyle \sigma ^{-1}{\mathcal {L}}} . These groups can be made into a cochain complex with differentials constructed as in usual singular cohomology. Define H s i n g j ( X ; L ) {\displaystyle H_{\mathrm {sing} }^{j}(X;{\mathcal {L}})} to be the cohomology of this complex.

• The group C n ( X ~ ) {\displaystyle C_{n}({\widetilde {X}})} of singular n-chains on the universal cover of X has an action of π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} by deck transformations. Explicitly, a deck transformation γ : X ~ → X ~ {\displaystyle \gamma \colon {\widetilde {X}}\to {\widetilde {X}}} takes a singular n-simplex σ : Δ n → X ~ {\displaystyle \sigma \colon \Delta ^{n}\to {\widetilde {X}}} to γ ∘ σ {\displaystyle \gamma \circ \sigma } . Then, given an abelian group L equipped with an action of π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} , one can form a cochain complex from the groups Hom π 1 ( X , x ) ⁡ ( C n ( X ~ ) , L ) {\displaystyle \operatorname {Hom} _{\pi _{1}(X,x)}(C_{n}({\widetilde {X}}),L)} of π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} -equivariant homomorphisms as above. Define H s i n g j ( X ; L ) {\displaystyle H_{\mathrm {sing} }^{j}(X;L)} to be the cohomology of this complex.

If X is paracompact and locally contractible, then H j ( X , L ) ≅ H s i n g j ( X ; L ) {\displaystyle H^{j}(X,{\mathcal {L}})\cong H_{\mathrm {sing} }^{j}(X;{\mathcal {L}})} .^[3] If L {\displaystyle {\mathcal {L}}} is the local system corresponding to L, then there is an identification C n ( X ; L ) ≅ Hom π 1 ( X , x ) ⁡ ( C n ( X ~ ) , L ) {\displaystyle C^{n}(X;{\mathcal {L}})\cong \operatorname {Hom} _{\pi _{1}(X,x)}(C_{n}({\widetilde {X}}),L)} compatible with the differentials,^[4] so H s i n g j ( X ; L ) ≅ H s i n g j ( X ; L ) {\displaystyle H_{\mathrm {sing} }^{j}(X;{\mathcal {L}})\cong H_{\mathrm {sing} }^{j}(X;L)} .

Local systems have a mild generalization to constructible sheaves -- a constructible sheaf on a locally path connected topological space X {\displaystyle X} is a sheaf L {\displaystyle {\mathcal {L}}} such that there exists a stratification of

X = ∐ X λ {\displaystyle X=\coprod X_{\lambda }}

where L | X λ {\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 f : X → Y {\displaystyle f:X\to Y} . For example, if we look at the complex points of the morphism

f : X = Proj ( C [ s , t ] [ x , y , z ] ( s t ⋅ h ( x , y , z ) ) ) → Spec ( C [ s , t ] ) {\displaystyle f:X={\text{Proj}}\left({\frac {\mathbb {C} [s,t][x,y,z]}{(st\cdot h(x,y,z))}}\right)\to {\text{Spec}}(\mathbb {C} [s,t])}

then the fibers over

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

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

R 0 f ! ( Q _ X ) | V ( s t ) = Q _ V ( s t ) R 2 f ! ( Q _ X ) | V ( s t ) = Q _ V ( s t ) R 4 f ! ( Q _ X ) | V ( s t ) = Q _ V ( s t ) R k f ! ( Q _ X ) | V ( s t ) = 0 _ V ( s t )  otherwise {\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 A s , t 2 − V ( s t ) {\displaystyle \mathbb {A} _{s,t}^{2}-\mathbb {V} (st)} we have the local systems

R 0 f ! ( Q _ X ) | A s , t 2 − V ( s t ) = Q _ A s , t 2 − V ( s t ) R 1 f ! ( Q _ X ) | A s , t 2 − V ( s t ) = Q _ A s , t 2 − V ( s t ) ⊕ 2 g R 2 f ! ( Q _ X ) | A s , t 2 − V ( s t ) = Q _ A s , t 2 − V ( s t ) R k f ! ( Q _ X ) | A s , t 2 − V ( s t ) = 0 _ A s , t 2 − V ( s t )  otherwise {\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 g {\displaystyle g} is the genus of the plane curve (which is g = ( deg ⁡ ( f ) − 1 ) ( deg ⁡ ( f ) − 2 ) / 2 {\displaystyle g=(\deg(f)-1)(\deg(f)-2)/2} ).

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.

• Leray spectral sequence
• Gauss–Manin connection
• D-module
• Intersection homology
• Perverse sheaf

• "What local system really is". Stack Exchange.
• Schnell, Christian. "Computing Cohomology of Local Systems" (PDF). Discusses computing the cohomology with coefficients in a local system by using the twisted de Rham complex.
• Williamson, Geordie. "An illustrated guide to perverse sheaves" (PDF).
• MacPherson, Robert (December 15, 1990). "Intersection homology and perverse sheaves" (PDF).
• El Zein, Fouad; Snoussi, Jawad. "Local systems and constructible sheaves" (PDF).