16

I've been looking at some old notes (1970s) on Riemann surfaces, trying to match up terminology with modern definitions (at least going by Wikipedia). The notes use the same terms as Gunning's Lectures on Riemann Surfaces (1966). I'm wondering if (a) I have the translations correct, and (b) Is modern terminology completely standardized?

The old notes define a presheaf $S$ to be (essentially) a contravariant functor from the poset of open sets on a topological space to the category Set, so that if $U\subset V$ then we have a morphism from $S(V)$ to $S(U)$ ("restriction"). A canonical (or complete) presheaf satisfies two additional requirements, the same as locality and gluing in Wikipedia.

A sheaf is defined as $\pi:E\rightarrow B$ with $\pi$ a local homeomorphism; also, $\pi^{-1}(x)$, the stalk over $x$, is required to have the appropriate algebraic structures (e.g., sheaf of groups, sheaf of rings) and the operations are required to be continuous.

Given a sheaf, we can define the presheaf of sections, and given a presheaf, we can define the sheaf of germs with a direct limit construction. The completion of a presheaf is obtained by going presheaf $\Rightarrow$ sheaf $\Rightarrow$ presheaf; the completion is a complete presheaf, and is isomorphic to the original presheaf iff it was complete.

The modern terminology appears to follow this dictionary: $$\begin{array}{|l|l|}\hline{\bf old} & {\bf new} \\ \hline \text{presheaf} & \text{presheaf} \\ \text{canonical/complete presheaf} & \text{sheaf} \\ \text{sheaf} & \text{etale space} \\ \text{completion} & \text{sheafification} \\ \hline\end{array}$$OK, my questions:

  1. Is this correct?
  2. Is the modern terminology completely standarized?
  3. "Stalk" appears to be synonym for "fiber"; true? Why have two terms, if so?
  4. (Less important) What's the history of the shift in terminology?
Michael Weiss
  • 8,813
  • 1
    Fibers are defined for arbitrary maps $\pi : E \to B$, not necessarily etale spaces. But stalk seems more appropriate if you aren't thinking of a sheaf in terms of its etale space. – Qiaochu Yuan May 28 '14 at 16:43

1 Answers1

9
  1. First let me congratulate you on your dictionary: it is absolutely correct and will be useful to other users as well.
  2. Yes, the terminology is essentially standardized and it is nice to keep the concepts of étalé space and sheaf distinct.
  3. Stalk and fiber are different concepts for locally ringed spaces, the arena for most of sheaf theory.
    The most important examples of such locally ringed spaces $(X,\mathcal O_X)$ are differential manifolds, schemes and analytic spaces.
    Given a sheaf $\mathcal F$ of $\mathcal O_X$-modules on $X$, we have at each $x\in X$ its stalk $\mathcal F_{x} $, an $\mathcal O_{X,x} $- module described by the inductive limit process you alluded to.
    But we also have its fiber $\mathcal F(x)=\mathcal F_{x}/\mathfrak m_x\mathcal F_{x}$, a $\kappa (x)=\mathcal O_{X,x}/\mathfrak m_x$-vector space.
    For example if $p:E\to X$ is a vector bundle on the real differential manifold $(X,\mathcal C^\infty _X)$, it has an associated sheaf of sections $\mathcal E$.
    The stalk $\mathcal E_x$ is an infinite dimensional real vector space but the fiber $\mathcal E(x)$ is the finite-dimensional real vector space $\mathcal E(x)=p^{-1}(x)\subset E$ .
  4. Sheaf theory has an exciting, sometimes poignant, history starting with its invention in captivity by Jean Leray, a French officer made prisoner by the German military in WWII.
    The changes in terminology result from the development of sheaf theory in various branches of mathematics in the hands of luminaries like Henri Cartan, Koszul, Serre, Godement, Grothendieck in the fifteen years following the end of the war.
    Here and here are somewhat related posts on the history of the subject.
Community
  • 1
Georges Elencwajg
  • 150,790
  • Thanks! I don't completely follow (3): how is $\mathcal{E}_x$ infinite dimensional? Would it be correct to say that the stalks of the sheaf of sections $\mathcal{E}$ are fibers of the étalé map $Et(\mathcal{E})\rightarrow X$? – Michael Weiss May 28 '14 at 19:58
  • Dear Michael, take for $E$ the trivial rank-one vector bundle $E=X\times \mathbb R$. Then $\mathcal E=\mathcal C^\infty$ and $\mathcal E_x=\mathcal C^\infty_x$. Do you see that $\mathcal C^\infty_x$ is an infinite-dimensional real vector space ? Just think of the simplest case, the manifold $X=\mathbb R$ ! As to your second question, yes what you write is quite correct. – Georges Elencwajg May 28 '14 at 20:14
  • 1
    I see, we need the word "stalk" so we can talk about the fibers of the associated etale space without explicitly mentioning it. thanks! – Michael Weiss May 29 '14 at 12:03