Sanity check about Wikipedia definition of differentiable manifold as a locally ringed space

Question

Most textbooks introduce differentiable manifolds via atlases and charts. This has the advantage of being concrete, but the disadvantage that the local coordinates are usually completely irrelevant- the choice of atlas and chart is arbitrary, and rarely if ever seems to play any role in differential geometry/topology.

There is a much better definition of differentiable manifolds, which I don't know a good textbook reference for, via sheaves of local rings. This definition does not involve any strange arbitrary choices, and is coordinate free. Paragraph 3 in Wikipedia (which is the actual definition) states:

A differentiable manifold (of class $C_k$) consists of a pair $(M, \mathcal{O}_M)$ where $M$ is a topological space, and $\mathcal{O}_M$ is a sheaf of local $R$-algebras defined on $M$, such that the locally ringed space $(M,\mathcal{O}_M)$ is locally isomorphic to $(\mathbb{R}^n, \mathcal{O})$.

This confuses me, because I don't see why such a sheaf should be acyclic, or where conditions like "paracompact" or "complete metric space" or "second countable Hausdorff" are implicit. So either:

1. The wikipedia entry has a mistake (I would want to sanity-check this before editing the entry, because this is such a fundamental definition which thousands must have read).
2. Somewhere in that definition, the condition that $M$ be paracompact is implicit.

Question: Should the definition above indeed require that $M$ be second-countable Hausdorff or paracompact or whatever? Or is it somehow implicit somewhere, and if so, where?

Also, is this definition given carefully in any textbook?

Update: I have editted the Wikipedia article to require that $M$ be second-countable Hausdorff. But I'm still wondering if there is a textbook covering this stuff, and whether requiring the sheaf to be acyclic might have worked instead as an alternative.

Answer 1

To expand a bit on my comment above:

Being isomorphic as a locally ringed space to $(\mathbb{R}^n,\mathcal{O})$ doesn't impose additional conditions on the underlying topological space of a locally ringed space beyond requiring it to be locally homeomorphic to $\mathbb{R}^n$. (Well, that's a lie: a differentiable structure does of course place limitations on the topology of a manifold, but only very subtle ones: it doesn't impose either of the limitations you are asking about. See below!)

Thus, if you want your definition of a manifold to include Hausdorff and second countable and/or paracompact, you had better put that in explicitly. (And, although it's a matter of taste and terminology, in my opinion you do want this.)

I think you will find these lecture notes enlightening on these points. In particular, on page 4 I give an example (taken from Thurston's book on 3-manifolds!) of a Galois covering map where the total space is a manifold but the quotient space is not Hausdorff. (When I gave this example I mentioned that I wish someone had told me that covering maps could destroy the Hausdorff property! And indeed the audience looked suitably shaken.)

With regard to your other question ("Also, is this definition given carefully in any textbook?")...I completely sympathize. When I was giving these lectures I found that I really wanted to speak in terms of locally ringed spaces! See in particular Theorem 9 in my notes, which contains the unpleasantly anemic statement: "If $X$ has extra local structure, then $\Gamma \backslash X$ canonically inherits this structure." What I really wanted to say is that if $\pi: X \rightarrow \Gamma \backslash X$, then $\mathcal{O}_{\Gamma \backslash X} = \pi_* \mathcal{O}_X$! (I am actually not the kind of arithmetic geometer who has to express everything in sheaf-theoretic language, but come on -- this is clearly the way to go in this instance: that one little equation is worth a thousand words and a lot of hand waving about "local structure".)

What is even more ironic is that my course is being taken by students almost all of whom have taken a full course on sheaves in the context of algebraic geometry. But whatever differential / complex geometry / topology they know, they know in the classical language of coordinate charts and matrices of partial derivatives. It's really kind of a strange situation.

I fantasize about teaching a year long graduate course called "modern geometry" where we start off with locally ringed spaces and use them in the topological / smooth / complex analytic / Riemannian categories as well as just for technical, foundational things in a third course in algebraic geometry. (As for most graduate courses I want to teach, improving my own understanding is a not-so-secret ulterior motive.) In recent years many similar fantasies have come true, but this time there are two additional hurdles: (i) this course cuts transversally across several disciplines so implicitly "competes" with other graduate courses we offer and (ii) this should be a course for early career students, and at a less than completely fancy place like UGA such a highbrow approach would, um, raise many eyebrows.

Answer 2

I'm afraid I don't know the answer to your main question, but I would like to mention a textbook that approaches manifolds from the sheaf-theoretic perspective: Ramanan's Global Calculus.

He explicitly includes the Hausdorff + second-countable conditions, defining a manifold as follows:

Definition. A differential manifold $M$ (of dimension $n$) consists of

a) a topological space which is Hausdorff and admits a countable base for open sets, and

b) a sheaf $\mathcal{A}^M=\mathcal{A}$ of subalgebras of the sheaf of continuous functions on $M$.

These are required to satisfy the following local condition. For any $x\in M$, there is an open neighborhood $U$ of $x$ and a homeomorphism of $U$ with an open set $V$ in $\mathbb{R}^n$ such that the restriction of $\mathcal{A}$ to $U$ is the inverse image of the sheaf of differentiable functions on $V$.

Answer 3

1) Godement has written a book on Lie groups where he defines and uses manifolds through sheaves.
This is not surprising since he wrote a treatise on sheaf theory nearly sixty years ago, which surprisingly is still the standard reference on the subject.
The sheaf theory is very easy, since the structural sheaf is a ring of functions and thus automatically separated (= satisfies first axiom for a presheaf to be a sheaf).

The book has no English translation (to my knowledge), but if you can overcome that hurdle you will be able to savour Godelment's inimitably idiosyncratic style, as well as the expertise of this great mathematician .

2) Since you mention acyclicity, let me remark that it does not follow from either definition but is a theorem.
It is a consequence of the existence of partitions of unity, which implies that the structural sheaf $\mathcal C^k_M$ is fine, hence acyclic.
However partitions of unity require $M$ to be paracompact, which might be an argument for including paracompactness (or equivalent conditions) in the definition.