Point-free affine real line?

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

Algebraic geometry defines the affine line over a field k as an affine scheme, the spectrum of k[X]. It includes a copy of k, each element a being present as the irreducible polynomial X-a, with local ring the ring of fractions got by inverting polynomials f(X) such that f(a) is non-zero.

You can carry this out for the real line R, but it is very much R as a set, and the copy of R in the underlying space of the spectrum has the discrete topology.

Does algebraic geometry provide an analogous construction that could lead to  the point-free R? Can the locally ringed space be topologized (point-free) so that the copy of R has its usual topology?

I've run into various problems.

1. It is not obvious to me that R[X] exists point-free. By that I mean that,  without presupposing a set R[X], or using non-geometric constructions, I can't see how to define a geometric theory whose models are the polynomials. The problem comes with trying to pin down the requirement that all but finitely many of the coefficients of a polynomial must be zero. You cannot continuously define the degree of a polynomial, because the function R -> N, a |-> degree(aX + 1), is not continuous.

That suggests the construction as Spec(R[X]) might have to be adjusted. Is there still some locally ringed space that does the trick?

2. The "structure sheaf" cannot be a sheaf. We hope its fibres are point-free local rings, but, whatever they are, they must be R-algebras and so cannot  have the discrete topology. The space is locally ringed by some bundle other than a sheaf (local homeomorphism).

3. The usual local rings, got as rings of fractions as described above, may be problematic point-free in the same way as R[X] is. I don't know what would  do instead. The power series ring R[[X]]? (At least as fibre over 0.) It does have the property of inverting those polynomials f for which f(0) is non-zero. And it can be defined point-free, as R^N. (However, the finitely presented approximations R[X|X^n = 0] happily exist point-free.)

Thanks for any references you can provide,

Steve.




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