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nLab > Latest Changes: embedding of smooth manifolds into formal duals of R-algebras

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2015

    I finally gave this statement its own entry, in order to be able to conveniently point to it:

    embedding of smooth manifolds into formal duals of R-algebras

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 10th 2015

    Such a statement is true for manifolds with finitely many or countably many connected components.

    As Theo Johnson-Freyd once pointed out to me, and later expanded in this answer on MO: http://mathoverflow.net/a/91445 it is false for arbitrary paracompact Hausdorff manifolds, in particular, given two uncountable (discrete) sets S and T one can find a morphism of real algebras C^∞(T)→C^∞(S) that is not induced by a function S→T.

    However, the construction is a very subtle set-theoretical argument that uses measurable cardinals.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2015

    Thanks. I have made the standard regularity assumptions explicit in the entry now and added pointer to this MO discussioon.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 11th 2015
    • (edited Mar 11th 2015)

    Ah, I was just wondering what sort of things break for uncountable disjoint unions of second countable manifolds.

    EDIT: I was thinking continuum-many summands, which is still better behaved than for general uncountable coproducts.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 18th 2020
    • (edited Jul 18th 2020)

    added doi to

    and

    diff, v11, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 18th 2020

    added pointer to Milnor’s original statement

    and to these proofs for the case of isomorphisms:

    diff, v12, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 18th 2021

    Added redirect: Milnor duality. To satisfy a link at duality between geometry and algebra.

    diff, v14, current

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 7th 2022

    Added:

    The case of the category of smooth manifolds and diffeomorphisms is proved in

    • Lyle Eugene Pursell, Algebraic structures associated with smooth manifolds, PhD dissertation, Purdue University, 1952. 93 pp. ISBN: 978-1392-88143-9. PDF.

    diff, v16, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 8th 2022
    • (edited Dec 8th 2022)

    Interesting that such an early reference exists.

    In trying to check it out on my phone, I only get to see the first 19 pages. Do you mean to say Pursell’s proof covers only diffeos/ring-isos, but not non-invertible maps?

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 8th 2022

    Yes. There is also this announcement by Pursell’s PhD advisor M. E. Shanks: https://www.ams.org/journals/bull/1951-57-04/S0002-9904-1951-09521-X/S0002-9904-1951-09521-X.pdf, see page 295.

    They never published any of this, except that they have a similar paper about Lie algebras: https://www.ams.org/journals/bull/1951-57-04/S0002-9904-1951-09521-X/S0002-9904-1951-09521-X.pdf.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeDec 8th 2022

    Thanks. I have added pointer also to Shanks’s announcement.

    Also, I adjusted the wording of the attribution paragraph (here).

    diff, v18, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeOct 21st 2023
    • (edited Oct 21st 2023)

    added pointer to:

    diff, v22, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2024
    • (edited Sep 12th 2024)

    Made the statement of Milnor’s exercise an actual numbered Lemma (now here)

    and then spelled out (here) the proof, from this lemma, of the full embedding statement (following Cor. 35.10 in Kolář, Michor & Slovák 1993)

    diff, v24, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2024

    Seeing that the entry still didn’t have any Idea-section, I added some paragraphs (here).

    diff, v24, current

    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 16th 2024
    • CommentRowNumber16.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 16th 2024
    • CommentRowNumber17.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 21st 2024

    Added a short simple proof of “Milnor’s exercise” (known before Milnor, it seems), since the proof in Michor et al. is too complicated:

    \begin{proof} We provide a simplified proof using Hadamard’s lemma. Suppose MM is a smooth manifold and ϕ:C ∞(M)→R\phi\colon C^\infty(M)\to\mathbf{R} is a homomorphism of real algebras.

    If M=R nM=\mathbf{R}^n for some n≥0n\ge0, then set y i=ϕ(x i)y_i=\phi(x_i), where x i:R n→Rx_i\colon\mathbf{R}^n\to\mathbf{R} is the iith coordinate function. We have

    ϕ(f)=ϕ(f(y)+∑ i(x i−y i)⋅g i)=f(y)+∑ i(ϕ(x i)−y i)⋅ϕ(g i)=f(y),\phi(f)=\phi(f(y)+\sum_i (x_i-y_i)\cdot g_i)=f(y)+\sum_i (\phi(x_i)-y_i)\cdot \phi(g_i)=f(y),

    where the functions g ig_i are provided by Hadamard’s lemma.

    For a general MM, use Whitney’s theorem to embed MM into some R n\mathbf{R}^n and consider the composition

    C ∞(R n)→C ∞(M)→R,C^\infty(\mathbf{R}^n)\to C^\infty(M)\to \mathbf{R},

    where the first homomorphism ρ\rho is given by restricting along the embedding. The composition is given by evaluating at some point p∈R np\in\mathbf{R}^n. We have p∈Mp\in M because every composition ϕρ\phi\rho must vanish on the kernel of ρ\rho, but the evaluation homomorphism at p∉Mp\notin M does not. \end{proof}

    diff, v30, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeSep 21st 2024

    That’s a good proof!

    Interesting about this and similar proofs is that it manages to use the coordinate functions x ix^i as if they were generators of C ∞(ℝ n)C^\infty(\mathbb{R}^n), even though they are not (not in the sense of ℝ\mathbb{R}-algebras). While it is the “C ∞C^\infty-ring“-structure on C ∞(ℝ n)C^\infty(\mathbb{R}^n) that does make the coordinate functions be actual generators, these kinds of proofs show that/why for many purposes this exctra structure is not actually necessary.

    • CommentRowNumber19.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 21st 2024

    Mention that the image of the embedding must be closed. Clarified the last paragraph of the proof.

    diff, v31, current

    • CommentRowNumber20.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 21st 2024

    Re #18: Yes, and in fact, Hadamard’s lemma is key for the initial development of differential geometry.

    It’s worth pointing out that analogous proofs work in the algebraic and complex holomorphic cases, where they show, in exactly the same manner, that the C-spectrum of the algebra of holomorphic functions on a Stein manifold MM is isomorphic to MM. (Recall that Stein manifolds can be defined as complex manifolds admitting a proper holomorphic embedding into C n\mathbf{C}^n, which is precisely what we need for the proof.) Likewise, in the algebraic case we recover Nullstellensatz.

    The statement could probably be generalized to algebras over Fermat theories.

    • CommentRowNumber21.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 24th 2024

    Uploading Pursell’s dissertation…

    diff, v32, current

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