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I have read 4 chapters of Hartshorne's Algebraic Geometry, when I go back to the beginning of scheme and the definition of a section, I am kind of confused why we call such a element "section".

Let me quote the definition: If $\mathcal{F}$ is a presheaf, we say $\mathcal{F}(U)$ the sections of the presheaf $\mathcal{F}$ over the open set $U$.

In the case of structure sheaf(on a variety or affine scheme), we say $\mathcal{O}(U)$ the set of (regular) functions. But how about other sheaves? What should I view these "sections" as?(functions? rational functions? maps?) When those sections can become rational functions?

(In basic algebra, for a short exact sequence $0 \to A \to B \to C \to 0$, a "section" is a map $C \to B$ such that the composition $C \to B \to C$ is identity. In algebraic geometry, are these "sections" have similar property?)

Any point of view is welcome! Thanks a lot!

finiteness
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  • You have read first 4 chapters of Hartshorne? Do you want to join in reading group on Hartshorne.. –  Apr 06 '18 at 13:50

1 Answers1

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The origin of the terminology, I think, comes from bundles in topology. Given a base space $B$ and a bundle over $B$ — i.e. a continuous map $E \xrightarrow{\pi} B$ — a section of this bundle is a continuous map $s : B \to E$ with $\pi \circ s = 1_B$.

This definition has the obvious generalization to very arbitrary situations; e.g. the definition you cite from basic algebra.

More generally we can consider local sections; given an open subset $U \subseteq B$, a section of the bundle $E \xrightarrow{\pi} B$ over $U$ is a continuous map $s : U \to E$ such that $\pi \circ s$ is the inclusion map $U \to B$.

Every sheaf of sets $F$ on a topological space $B$ has an associated étale space: a bundle $E_F \xrightarrow{\pi} B$ with the property that $F(U)$ is naturally isomorphic to the set of sections over $U$ of this bundle.

In fact, the category of sheaves of sets on $B$ is equivalent to the category of bundles on $B$ for which $\pi$ is a local homeomorphism.

Alternatively, by the Yoneda embedding, any open set $U$ determines a sheaf $yU$ defined by

$$ yU(V) = \begin{cases} * & V \subseteq U \\ \varnothing & V \not\subseteq U \end{cases} $$

where $*$ is an arbitrarily chosen one-element set. Any sheaf $F$ determines a unique natural transformation $\pi : F \to yB$; the sections (i.e. natural transformations $s : yB \to F$ with $\pi \circ s = 1_{yB}$) of this map are bijective with $F(B)$.

In fact, $F(U)$ is naturally isomorphic to the set of natural transformations $yU \to F$. (the composite $yU \to F \to yB$ is automatically equal to the inclusion $yU \to yB$)

  • This I have realised when reading these notes Introduction to language of gerbes and stacks https://arxiv.org/pdf/math/0212266.pdf .. This is discussed in pages 2,3,4 –  Apr 06 '18 at 13:45