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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
(FMCS and BLAST attendees will have seen this before.) Apropos of affine spaces over a field (the affine real line being a one-dimensional example), here are some axioms for a language with one binary operation ab, with (ab)c abbreviated to abc (left-associative convention as for combinatory algebra etc.), satisfying A1. aa = a A2. abb = a A3. abc = ac(bc) A4. abcd = adcb (Side remark: Axiom Ai requires i variables.) The intended model of ab is 2b - a over an arbitrary abelian group, e.g. the integers. (A4 forces abelian. Obviously the free algebra on n generators for n < 2 has n elements. Identify the free algebra F2 on {a,b} with the integers ..., -3, -2, -1, 0, 1, 2, 3, 4, ... as follows. ... baba, aba, ba, a, b, ab, bab, abab, ... (Exercise: This are the only elements of F2.) As usual the elements of F2 can be understood as binary operations on F2. Abbreviate the binary operation identified with n as a[n]b; thus a[0]b = a, a[1]b = b, a[2]b = ab, a[-1]b = ba, etc. Interpreting ab as 2b - a over the integers, a[n]b = a + n(b - a), and 0[n]1 = n, making F2 yet another binary notation for the integers, albeit exponentially longer. Now expand this one-operation language with a family c1, c2, c3, c4, ... of operations of respective finite arities 1, 2, 3, 4, ... satisfying c1(a) = a cm(a1, ..., am)[n]cn(a1, ..., an) = an for all n > 1 where m = n-1 cn(a1, ..., am, cm(a1, ..., am))[n]b = b ditto The intended model of cn(a1, ..., an) is the centroid or mean of a1, ..., an over an arbitrary field, i.e. (a1 + ... + an)/n. Claim. The variety defined by this equational theory, including its homomorphisms as standardly defined, is equivalent to the category of affine spaces over the rationals. (Affine transformations over the rationals are linear combinations whose coefficients sum to 1, e.g. centroid. The idea is that other models such as the real line can be pulled apart by homomorphisms into uncountably many copies of the rational line because the operations of the theory can only make rational connections between points of the line.) I mention this in case it has any bearing on Steve's question about a point-free version of the real affine line. Not that I see one myself. Vaughan Pratt On Thu, May 31, 2018 at 2:40 AM, 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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Steve, On Thu, May 31, 2018 at 10:40:41AM +0100, Steve Vickers wrote:
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?
very interesting question. I feel that the following remarks have some relevance. * Algebraic geometers do consider an analogue of the usual spectrum construction for topological rings: the "formal spectrum". Unfortunately, the theory is only developed for actual topological rings, not point-free versions of them. * One can define a point-free notion of a spectrum for apartness rings (ring objects in the category of sets-equipped-with-apartness-relation). This secretly comes up in algebraic geometry: Let X be a scheme. Inside the topos Sh(X), there is the ring O_X to which we can apply the usual point-free spectrum construction. We might hope that this just yields the one-point space; but unless X was zero-dimensional to begin with, this hope is false. In fact, the externalization of this construction is a locally ringed locale which comes equpped with a morphism of ringed locales to (X,O_X); but this morphism is not a morphism of *locally* ringed locales. The solution to this problem is to observe that O_X, being a local ring, has canonically the structure of an apartness ring. The variant of the spectrum construction for apartness rings applied to it yields the one-point space (i.e. (X,O_X) itself from the external point of view), as one would expect. * Both the problem and the solution can be generalized a bit. Let X be a scheme. Let A be an O_X-algebra. Then algebraic geometers consider the "relative spectrum of A", which is always a locally ringed locale over X and will be a scheme if A is quasicoherent. As before, it's not true that one can obtain the relative spectrum simply by carrying out the usual point-free spectrum construction internally in Sh(X). A variant is needed. A description of these variants of the usual construction can be found in Section 12 of https://rawgit.com/iblech/internal-methods/master/notes.pdf. However, I don't know yet a natural generalization of these constructions which could be directly helpful to your project -- they are tailored to apartness rings and to algebras over local rings. While it's certainly better to view the reals as an apartness ring instead of an ordinary ring, information has still been lost from the point-free version of the reals. Cheers, Ingo [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
-
Graham Manuell -
Ingo Blechschmidt -
Steve Vickers -
Vaughan Pratt