Dear Steve, I have been working on very similar ideas for my PhD thesis. If R is the real line, then R[X] has at least 3 reasonable topologies: the compact-open topology, the Whitney topology and the coarsest topology making the coefficient maps continuous. None of these gives a localic ring since they are not complete. Their completions are, of course, the ring of continuous functions on R, the ring of smooth functions on R and the ring of formal power series R[[X]], respectively. Perhaps this indicates that these are more appropriate than the polynomial ring when R has a non-discrete topology. It seems to me that the classical Zariski spectrum is only correct for discrete rings. The most obvious example where it gives pathological results is for the ring of continuous functions on a compact Hausdorff space. For Gelfand duality we must consider not all the ideals, but only the closed ideals. The Zariski spectrum is given by taking the localic reflection of the quantale of ideals of a discrete ring. For a localic ring, I believe we should instead take the localic reflection of the quantale of overt ideals. This should give the expected results for both discrete rings and for rings of continuous functions. For the localic rings above, I expect R to appear inside the spectrum with its usual topology, but I've only checked this in the first two cases. I don't know if the construction of the quantale of overt ideals from a localic ring is geometric. I think it might not be. There is a geometric theory that gives the elements of this quantale as its points. This tentatively suggests to me that the resulting spectrum will not be a locale, but a colocale, but you know much more about these than I do. I have thought less about the corresponding 'sheaf'. In particular, I do not know what the correct notion of localic local ring is. I found some work on localisation of topological rings (see here <https://www.sciencedirect.com/science/article/pii/0166864194900353>), but I haven't thought how this would work in the pointfree setting. Best regards, Graham On Fri, 1 Jun 2018 at 01:02 Steve Vickers <s.j.vickers@cs.bham.ac.uk> wrote:
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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