Let $X$ be a topological space, and let $\mathcal{F}$ be a (pre-)sheaf of sets (groups, rings, etc.) on $X$. It seems to me that several authors in algebraic geometry prefer to use the global section functor $\Gamma$ to express the set (resp. group, ring, etc.) of sections $\mathcal{F}(U)$ as $\Gamma(U,\mathcal{F})$, for an open subset $U$ in $X$. For instance, see the fifth line in this Tag on the Stacks Project.

Is there a good reason to prefer the expression $\Gamma(U,\mathcal{F})$ to $\mathcal{F}(U)$, apart from emphasising being contravariant in $U$ and covariant in $\mathcal{F}$?


One reason may be that the notation $\mathcal F(U)$ isn't always standard to mean sections of $\mathcal F$ over $U$. For instance, Serre's FAC is an example of an important paper where $\mathcal F(U)$ means something totally different; it actually refers to the restriction of the sheaf $\mathcal F$ to $U$, which a lot of people now would write $\mathcal F|_U$.

As for using the notation $\Gamma$ in general: You may or may not be aware that sheaves $\mathcal F$ on $X$ are in bijection with surjective maps $\pi:\mathscr F\to X$, where $\mathscr F$ is a topological space, such that for each $f\in\mathscr F$ there is a neighborhood $U$ of $f$ and $V$ of $\pi(f)$ such that $\pi|_U:U\to V$ is a homeomorphism. In this sense, "sections of a sheaf $\mathcal F$" are literally sections of the map $\pi:\mathscr F\to X$, i.e. functions $s:X\to\mathscr F$ such that $\pi\circ s=\text{id}$. Furthermore, this is analogous to sections of a vector bundle $\pi:E\to X$, and when talking about vector bundles we always denote global sections by $\Gamma(E)$. So I guess this notation just carried over to sheaves.

  • $\begingroup$ Would you know the answer to this question maybe? $\endgroup$ – Watson Jul 22 '17 at 19:17
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    $\begingroup$ @Watson I've always assumed the $\Gamma$ stood for "global" in "global sections" like you say in your comment, and have never really questioned it. Could be wrong though.. $\endgroup$ – Alex Mathers Jul 24 '17 at 2:34

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