Local system

In mathematics, local coefficients is an idea from algebraic topology, a kind of half-way stage between homology theory or cohomology theory with coefficients in the usual sense, in a fixed abelian group A, and general sheaf cohomology which, roughly speaking, allows coefficients to vary from point to point in a topological space X. Such a concept was introduced by Norman Steenrod in 1943.[1]

Definition

Let X be a topological space. A local system (of abelian groups/modules/...) on X is a locally constant sheaf (of abelian groups/modules...) on X. In other words, a sheaf ${\displaystyle {\mathcal {L}}}$ is a local system if every point has an open neighborhood ${\displaystyle U}$ such that ${\displaystyle {\mathcal {L}}|_{U}}$ is a constant sheaf.

Equivalent definitions

Path-connected spaces

If X is path-connected, a local system ${\displaystyle {\mathcal {L}}}$ of abelian groups has the same fibre L at every point. To give such a local system is the same as to give a homomorphism

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

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

Proof of equivalence

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

${\displaystyle {\mathcal {L}}(\rho )_{U}\ =\ \left\{{\text{sections }}s\in {\underline {L}}_{\pi ^{-1}(U)}{\text{ with }}\theta \circ s=\rho (\theta )s{\text{ for all }}\theta \in {\text{ Deck}}({\widetilde {X}}/X)=\pi _{1}(X,x)\right\}}$

where ${\displaystyle \pi$ :{\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.

Stronger definition on non-connected spaces

Another (stronger, nonequivalent) definition generalising 2, and working for non-connected X, is: a covariant functor

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

from the fundamental groupoid of ${\displaystyle X}$ to the category of modules over a commutative ring ${\displaystyle R}$. Typically ${\displaystyle R=\mathbb {Q} ,\mathbb {R} ,\mathbb {C} }$. What this is saying is that at every point ${\displaystyle x\in X}$ we should assign a module ${\displaystyle M}$ with a representations of ${\displaystyle \pi _{1}(X,x)\to {\text{Aut}}_{R}(M)}$ such that these representations are compatible with change of basepoint ${\displaystyle x\to y}$ for the fundamental group.

Examples

• Constant sheaves. For instance, ${\displaystyle {\underline {\mathbb {Q} }}_{X}}$. This is a useful tool for computing cohomology since the sheaf cohomology

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

is isomorphic to the singular cohomology of ${\displaystyle X}$.

• ${\displaystyle X=\mathbb {R} ^{2}\setminus \{(0,0)\}}$. Since ${\displaystyle \pi _{1}(\mathbb {R} ^{2}\setminus \{(0,0)\})=\mathbb {Z} }$, there are ${\displaystyle S^{1}}$-many linear systems on X, the ${\displaystyle \theta \in \mathbb {R} }$ one given by monodromy representation

${\displaystyle \pi _{x}(X)=\mathbb {Z} \to {\text{Aut}}_{\mathbb {C} }(\mathbb {C} )}$ by sending ${\displaystyle n\mapsto e^{in\theta }}$

• Horizontal sections of vector bundles with a flat connection. If ${\displaystyle E\to X}$ is a vector bundle with flat connection ${\displaystyle \nabla }$, then

${\displaystyle E_{U}^{\nabla }=\left\{{\text{sections }}s\in \Gamma (U,E){\text{ which are horizontal: }}\nabla s=0\right\}}$

is a local system.

For instance, take ${\displaystyle X=\mathbb {C} \setminus 0}$ and ${\displaystyle E=X\times \mathbb {C} ^{n}}$ the trivial bundle. Sections of E are n-tuples of functions on X, so ${\displaystyle \nabla _{0}(f_{1},\dots ,f_{n})=(df_{1},\dots ,df_{n})}$ defines a flat connection on E, as does ${\displaystyle \nabla (f_{1},\dots ,f_{n})=(df_{1},\dots ,df_{n})-\Theta (x)(f_{1},\dots ,f_{n})^{t}}$ for any matrix of one-forms ${\displaystyle \Theta }$ on X. The horizontal sections are then

${\displaystyle E_{U}^{\nabla }=\left\{(f_{1},\dots ,f_{n})\in E_{U}:(df_{1},\dots ,df_{n})=\Theta (f_{1},\dots ,f_{n})^{t}\right\}}$

i.e., the solutions to the linear differential equation ${\displaystyle df_{i}=\sum \Theta _{ij}f_{j}}$.

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

${\displaystyle \Theta ={\begin{pmatrix}0&dx/x\\dx&e^{x}dx\end{pmatrix}}}$

in which case for ${\displaystyle \nabla =d+\Theta }$,

${\displaystyle E_{U}^{\nabla }=\left\{f_{1},f_{2}:U\to \mathbb {C} \ \ {\text{ with }}f'_{1}=f_{2}/x\ \ f_{2}'=f_{1}+e^{x}f_{2}\right\}}$

• An n-sheeted covering map ${\displaystyle X\to Y}$ is a local system with sections locally the set ${\displaystyle \{1,\dots ,n\}}$. 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 the same as a k-linear representation of the group ${\displaystyle \pi _{1}(X,x)}$.
• If X is a variety, local systems are the same thing as D-modules that are in addition coherent as O-modules.

If the connection is not flat, parallel transporting a fibre around a contractible loop at x may give a nontrivial automorphism of the fibre at the base point x, so there is no chance to define a locally constant sheaf this way.

The Gauss–Manin connection is a very interesting example of a connection, whose horizontal sections occur in the study of variation of Hodge structures.

Generalization

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

${\displaystyle X=\coprod X_{\lambda }}$

where ${\displaystyle {\mathcal {L}}|_{X_{\lambda }}}$ is a local system. These are typically found by taking the cohomology of the derived pushforward for some continuous map ${\displaystyle f:X\to Y}$. For example, if we look at the complex points of the morphism

${\displaystyle f:X={\text{Proj}}\left({\frac {\mathbb {C} [s,t][x,y,z]}{(stf(x,y,z))}}\right)\to {\text{Spec}}(\mathbb {C} [s,t])}$

then the fibers over

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

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

${\displaystyle {\begin{aligned}\mathbf {R} ^{0}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {V} (st)}\\\mathbf {R} ^{2}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {V} (st)}\\\mathbf {R} ^{4}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {V} (st)}\\\mathbf {R} ^{k}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {V} (st)}&={\underline {0}}_{\mathbb {V} (st)}{\text{ otherwise}}\end{aligned}}}$

while over ${\displaystyle \mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}$ we have the local systems

${\displaystyle {\begin{aligned}\mathbf {R} ^{0}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}\\\mathbf {R} ^{1}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}^{\oplus 2g}\\\mathbf {R} ^{2}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}\\\mathbf {R} ^{k}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}&={\underline {0}}_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}{\text{ otherwise}}\end{aligned}}}$

where ${\displaystyle g}$ is the genus of the plane curve (which is ${\displaystyle g=({\text{deg}}(f)-1)({\text{deg}}(f)-2)/2}$).

Applications

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.

Topological Bivariant Theory

For maps ${\displaystyle f\colon X\to Y}$ there is a Bivariant theory similar to William Fulton's, called Topological Bivariant Theory.[2] Defining such a theory requires local systems and the six-functor formalism. Bivariant theories are characterized by the property

${\displaystyle H^{i}(X{\xrightarrow {f}}Y)={\text{Hom}}_{D^{b}(Y)}(\mathbb {R} f_{!}\mathbb {Q} _{X},\mathbb {Q} _{Y}[+i])}$

For example, this can be computed in some simple cases. If ${\displaystyle Y}$ is a point, this recovers Borel–Moore homology. If ${\displaystyle Y=X}$ and the map is the identity, then this is oridinary cohomology. Another informative class of example includes covering spaces. For example, if ${\displaystyle f\colon \mathbb {G} _{m}\to \mathbb {G} _{m}=\mathbb {C} ^{*}}$is the degree ${\displaystyle d}$ covering given by ${\displaystyle z\mapsto z^{d}}$. Then, at the stalk level the cohomology groups are of the form

${\displaystyle \mathbb {R} f_{!}(\mathbb {Q} _{\mathbb {G} _{m}})|_{x}\cong \mathbb {Q} ^{\oplus d}}$

and the monodromy for ${\displaystyle \pi _{1}(\mathbb {G} _{m})=\mathbb {Z} }$ is given by the map taking a branch to its next branch and the ${\displaystyle d}$-th branch to the first branch. That is, ${\displaystyle \phi \colon \mathbb {Z} \to {\text{Aut}}(\mathbb {Q} ^{d})}$ is generated by the matrix

${\displaystyle \phi (1)={\begin{pmatrix}0&0&0&\cdots &0&1\\1&0&0&\cdots &0&0\\0&1&0&\cdots &0&0\\\vdots &\vdots &\vdots &&\vdots &\vdots \\0&0&0&\cdots &1&0\end{pmatrix}}}$

See also

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

References

1. Steenrod, Norman E. (1943). "Homology with local coefficients". Annals of Mathematics. 44 (4): 610–627. doi:10.2307/1969099. MR 0009114.
2. de Cataldo, Mark Andrea A.; Migliorini, Luca. "The Gysin map is compatible with mixed Hodge Structures" (PDF). Archived (PDF) from the original on 16 February 2020.

External links

• "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).