24.1 KB · ⇊ 15 parts · ⤓ Micron 24.1 KB · ⤓ HTML 171.0 KB

top

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.cite-ref-1[1]

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

• Definition

• Equivalent definitions

• Examples

• Cohomology

• Generalization

• Applications

• See also

• References

• External links

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.

If X is path-connected, 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 ρ : π 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 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}}} .

Proof of equivalence

Take local system L {\displaystyle {\mathcal {L}}} and a loop γ γ {\displaystyle \gamma } at x. It's easy to show that any local system on [ 0 , 1 ] {\displaystyle [0,1]} is constant. For instance, γ γ ∗ ∗ L {\displaystyle \gamma ^{*}{\mathcal {L}}} is constant. This gives an isomorphism ( γ γ ∗ ∗ L ) 0 ≃ ≃ Γ Γ ( [ 0 , 1 ] , L ) ≃ ≃ ( γ γ ∗ ∗ L ) 1 {\displaystyle (\gamma ^{*}{\mathcal {L}})_{0}\simeq \Gamma ([0,1],{\mathcal {L}})\simeq (\gamma ^{*}{\mathcal {L}})_{1}} , i.e. between L {\displaystyle L} and itself. Conversely, given a homomorphism ρ ρ : π π 1 ( X , x ) → → Aut ( L ) {\displaystyle \rho :\pi _{1}(X,x)\to {\text{Aut}}(L)} , consider the constant sheaf L _ _ {\displaystyle {\underline {L}}} on the universal cover X ~ ~ {\displaystyle {\widetilde {X}}} of X. The deck-transform-invariant sections of L _ _ {\displaystyle {\underline {L}}} 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

L ( ρ ρ ) U = { sections s ∈ ∈ L _ _ π π − − 1 ( U ) with θ θ ∘ ∘ s = ρ ρ ( θ θ ) s for all θ θ ∈ ∈ Deck ( X ~ ~ / X ) = π π 1 ( X , x ) } {\displaystyle {\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\}}

where π π : X ~ ~ → → X {\displaystyle \pi :{\widetilde {X}}\to X} is the universal covering.

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 π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} (equivalently, Z [ π π 1 ( X , x ) ] {\displaystyle \mathbb {Z} [\pi _{1}(X,x)]} -modules).cite-ref-2[2]

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

L : : Π 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 ρ x : π π 1 ( X , x ) → → Aut R ( M ) {\displaystyle \rho _{x}:\pi _{1}(X,x)\to {\text{Aut}}_{R}(M)} such that the various ρ ρ x {\displaystyle \rho _{x}} are compatible with change of basepoint x → → y {\displaystyle x\to y} and the induced map π π 1 ( X , x ) → → π 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 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 θ {\displaystyle n\mapsto e^{in\theta }} :

ρ ρ θ θ : π 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 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 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 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 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 Θ i j f j {\displaystyle df_{i}=\sum \Theta _{ij}f_{j}} .If Θ 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 π 1 ( C ) = 0 {\displaystyle \pi _{1}(\mathbb {C} )=0} . So to give an interesting example, choose one with a pole at 0: Θ 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 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 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 σ : : Δ Δ n → → X {\displaystyle \sigma \colon \Delta ^{n}\to X} to a global section f ( σ σ ) {\displaystyle f(\sigma )} of the inverse-image sheaf σ 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 π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} by deck transformations. Explicitly, a deck transformation γ γ : : X ~ ~ → → X ~ ~ {\displaystyle \gamma \colon {\widetilde {X}}\to {\widetilde {X}}} takes a singular n-simplex σ σ : : Δ Δ n → → X ~ ~ {\displaystyle \sigma \colon \Delta ^{n}\to {\widetilde {X}}} to γ γ ∘ ∘ σ σ {\displaystyle \gamma \circ \sigma } . Then, given an abelian group L equipped with an action of π π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} , one can form a cochain complex from the groups Hom π π 1 ( X , x ) ⁡ ⁡ ( C n ( X ~ ~ ) , L ) {\displaystyle \operatorname {Hom} _{\pi _{1}(X,x)}(C_{n}({\widetilde {X}}),L)} of π 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}})} .cite-ref-3[3] If L {\displaystyle {\mathcal {L}}} is the local system corresponding to L, then there is an identification C n ( X ; L ) ≅ ≅ Hom π 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,cite-ref-4[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 λ {\displaystyle X=\coprod X_{\lambda }}

where L | X λ 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

cite-note-11. ↑ citerefsteenrod1943Steenrod, Norman E. (1943). "Homology with local coefficients". Annals of Mathematics. 44 (4): 610–627. doi:10.2307/1969099. JSTOR 1969099. MR 0009114.

cite-note-22. ↑ Milne, James S. (2017). Introduction to Shimura Varieties. Proposition 14.7.

cite-note-33. ↑ Bredon, Glen E. (1997). Sheaf Theory, Second Edition, Graduate Texts in Mathematics, vol. 25, Springer-Verlag. Chapter III, Theorem 1.1.

cite-note-44. ↑ Hatcher, Allen (2001). Algebraic Topology, Cambridge University Press. Section 3.H.

• "What local system really is". Stack Exchange.

• citerefschnellSchnell, Christian. "Computing Cohomology of Local Systems" (PDF). Discusses computing the cohomology with coefficients in a local system by using the twisted de Rham complex.

• citerefwilliamsonWilliamson, Geordie. "An illustrated guide to perverse sheaves" (PDF).

• citerefmacpherson1990MacPherson, Robert (December 15, 1990). "Intersection homology and perverse sheaves" (PDF).

• citerefel-zeinsnoussiEl Zein, Fouad; Snoussi, Jawad. "Local systems and constructible sheaves" (PDF).