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
is a local system if every point has an open neighborhood
such that
is a constant sheaf.
Equivalent definitions
Path-connected spaces
If X is path-connected, a local system
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25222d1daf255ee447c0ccd571c3351d74c86257)
and similarly for local systems of modules,... The map
giving the local system
is called the monodromy representation of
.
Proof of equivalence
Take local system
and a loop
at x. It's easy to show that any local system on
is constant. For instance,
is constant. This gives an isomorphism
, i.e. between L and itself.
Conversely, given a homomorphism
, consider the constant sheaf
on the universal cover
of X. The deck-transform-invariant sections of
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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/deeac1a1971af023e4a5962a37892a1f3f7a8bef)
where
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2023b166e30dd1373d2f006468550361ce151c81)
from the fundamental groupoid of
to the category of modules over a commutative ring
. Typically
. What this is saying is that at every point
we should assign a module
with a representations of
such that these representations are compatible with change of basepoint
for the fundamental group.
Examples
- Constant sheaves. For instance,
. 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} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f87c5c2fd3e95a59fbfff625cc7a7b76b444d8a)
- is isomorphic to the singular cohomology of
.
. Since
, there are
-many linear systems on X, the
one given by monodromy representation
by sending ![{\displaystyle n\mapsto e^{in\theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd11c6ea07d9930ab0f09b0e5c83791ac40e9e30)
- Horizontal sections of vector bundles with a flat connection. If
is a vector bundle with flat connection
, then
![{\displaystyle E_{U}^{\nabla }=\left\{{\text{sections }}s\in \Gamma (U,E){\text{ which are horizontal: }}\nabla s=0\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa565f8380d0628ee02c2b8811fd9d5928448b93)
- is a local system.
- For instance, take
and
the trivial bundle. Sections of E are n-tuples of functions on X, so
defines a flat connection on E, as does
for any matrix of one-forms
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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/464a86f0573f730f115c34e9cd758cfda9c3b558)
- i.e., the solutions to the linear differential equation
.
- If
extends to a one-form on
the above will also define a local system on
, so will be trivial since
. 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81fbc1ee7e2e68321c1f7a87788cf9e54c0ba9be)
- in which case for
,
![{\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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34685b9162894c8118ec97a6773a5e6dc334ac37)
- An n-sheeted covering map
is a local system with sections locally the set
. 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
.
- 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
is a sheaf
such that there exists a stratification of
![{\displaystyle X=\coprod X_{\lambda }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea8d8b46a646435a299bd9d8cbdd24ab7bd2c210)
where
is a local system. These are typically found by taking the cohomology of the derived pushforward for some continuous map
. 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])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24576c5532e31ccb650d2f589fddfbc1e543948e)
then the fibers over
![{\displaystyle \mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d9f087dd643e7301bd56332ec12e54bfe3ee41a)
are the smooth plane curve given by
, but the fibers over
are
. If we take the derived pushforward
then we get a constructible sheaf. Over
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aab7637a381b79944e364e8d4e12d0834d7bef5a)
while over
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/407ef0affdceaeef1e95b5141bf7a440e821caeb)
where
is the genus of the plane curve (which is
).
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
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])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2d9ab786eb46083984428a0f3bc736ec6933c9b)
For example, this can be computed in some simple cases. If
is a point, this recovers Borel–Moore homology. If
and the map is the identity, then this is oridinary cohomology. Another informative class of example includes covering spaces. For example, if
is the degree
covering given by
. 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39decba583dcb77b94c01132b5160dc6c4df2f37)
and the monodromy for
is given by the map taking a branch to its next branch and the
-th branch to the first branch. That is,
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f8dfc8e8d01f8faf59b9f9d9a5b0c7d2767307c)
See also
References
External links