Curves are algebraic $K(\pi,1)$: theoretical and practical aspects

Author:

Christophe Levrat

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Number Theory (math.NT)

journal:

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date:

2023-06-04 16:00:00

Abstract

We prove that any connected curve with a rational point $x$ over a field $k$ is an algebraic $K(\pi,1)$, as soon as all of its irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible \'etale sheaf of $\mathbb{Z}/n\mathbb{Z}$-modules, with $n$ invertible in $k$, is canonically isomorphic to the cohomology of its corresponding $\pi_1(X,\bar x)$-module, where $\bar x$ is a geometric point of $X$ above $x$. When $k$ is algebraically closed, we explicitly describe finite quotients of $\pi_1(X,\bar x)$ that allow to compute the cohomology groups of the sheaf, as well as the cup products $H^1\times H^1\to H^2$.