Questions tagged [local-systems]

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 $.

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Local systems and Monodromy representation functor

Let sheaf $\mathcal{F}$ on $X$ be a local system, ie. for every $x$ in $X$, there is an an open set $U$ containing $x$ such that $\left.\mathcal{F}\right|_U$ is isomorphic to a constant sheaf $\...
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Terminology: Morphism inducing an isomorphism on the fundamental groups

I have a question concerning terminology: is there a name for a continuous map between topological spaces inducing an isomorphism between the fundamental groups? More generally, if $f:(X,G)\rightarrow ...
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52 views

Cohomology with local coefficients and obstruction theory

I'm reading about obstruction theory on Milnor & Stasheff and came across the following claim: If $p:E(\xi)\rightarrow B$ is a vector bundle over a CW complex $B$ and $V_k(\xi)$ is the ...
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Cohomology of weakly constructible sheaf on affine line

Let $X = \mathbb{A}^1(\mathbb{C}) \cong \mathbb{C}$ and $F$ be a sheaf on $X$. We call $F$ to be a weakly constructible sheaf (in this particular case) if there exists finite set of points say $S$, ...
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Example on Local Systems: Flat Connections

I am reading about the local system, and a local system $\mathcal{L}$ on $X$ with values in $\mathcal{C}$ is defined as a functor $$ \mathcal{L}:\Pi(X)\to \mathcal{C} $$ where $\Pi(X)$ is the ...
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Chow groups with coefficients in a local system

$\newcommand{\CH}{\mathrm{CH}} \newcommand{\F}{\mathscr{F}} $ Let $X$ be a smooth projective variety over a field $k$. Let $\F$ be a local system on $X$, i.e. a locally constant sheaf (for Zariski ...
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Local systems and connections on elliptic curves

Let $(E,O)$ be an elliptic curve over $\mathrm{Spec}(\mathbb{C})$. Then a 1-dimenstional represenation of $\pi_1(E^{\mathrm{an}}) = \mathbb{Z} \times \mathbb{Z}$ over $\mathbb{C}$ is just a pair of ...
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347 views

Differences between locally free sheaves and local systems

Is every local system with fiber a vector space a locally free sheaf? What are the main differences between these two concepts? I was playing with the sheaf of sections of $Mo \to S^1$ ($Mo=$Möbius ...
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Computing the monodromy of a local system $\mathcal{L}$

I was trying to learn a little bit about local systems and their monodromy. In the notes I'm following they define the monodromy of a local system in the following way: Let $X$ be a topological ...