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I have recently been interested in studying an extension of 'usual' algebraic geometry to take into account the topology of $R$ in the definition of the affine scheme $\mathrm{Spec}\, (R)$ when the ring comes equipped with a given topology. For example, I am curious to investigate schemes over $C ^\infty (\mathbb{R}^d)$, where it is obvious from the beginning that it would be 'wrong' to ignore the canonical Fréchet space topology (for example, the 'correct' definition of $\mathrm{Spec}\, (R)$ in this case should probably be the collection of all closed prime ideals). (Part of the motivation for this is, in the spirit of Grothendieck's famous quote, to replace a bad category of only good objects (the smooth category) with a good category containing some bad objects.)

I can't imagine that people haven't investigated things like this before. On the other hand, I don't know what this subject would be called, and so I don't know where to begin looking to read up on the subject. Could someone help point me in the right direction?

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    $\begingroup$ There is a notion of smooth algebra (ncatlab.org/nlab/show/smooth+algebra) which lets one avoid making use of the Frechet topology; instead one thinks of all smooth functions as determining an algebraic theory providing operations, instead of just addition, multiplication, etc. $\endgroup$
    – Qiaochu Yuan
    Commented Feb 5, 2015 at 1:42
  • $\begingroup$ Maybe this article: arxiv.org/pdf/1502.01401v1.pdf is relevant for you? $\endgroup$
    – Lennart Meier
    Commented Feb 6, 2015 at 13:32

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Although I agree with your statement that it would be wrong to forget the topology of $R$, I would say that there is little evidence to support the idea that the set of closed prime ideals is the `right' definition for Spec(R). This is certainly not the case for analytic geometry over a non-Archimedean field, which has many interesting points not related to prime ideals at all.

In general what you often do is try to define a topos and then identify which objects in the topos are sufficiently `geometric' to deserve your attention. If you really must, you can also try to identify a good family of points. Some nice examples are Spivak's derived manifolds and $C^\infty$-schemes (which are dual to the smooth algebras that Qiaochu mentioned). Both of these are essentially some purely formal extension of the category of manifolds, and do not really involve any understanding of the topological algebra of $C^\infty$ functions.

In a different direction, Alain Connes's flavour of non-commutative geometry really depends on functional analysis of Dirac operators on nuclear Fréchet spaces - but as far as I know, there is no underlying topological space (or topos) in this case. Perhaps the the theory of schemes you are looking for will eventually involve a combination of these ideas.

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  • $\begingroup$ I think the reason I first thought to only consider closed prime ideals is because this is what one tends to do when taking quotients in functional analysis. I hadn't thought about why that was the case in awhile, but I suspect that this is so because of the result: If $G$ is a $T_1$ topological group and $H$ is a sub-group, then $G/H$ is $T_1$ iff $H$ is closed. (In addition, a topological group being $T_1$ is equivalent to it being $T_3$, and in particular, Hausdorff.) Though I guess if we don't care whether or not $R/P$ is Hausdorff (should we?), then we need not require $P$ to be closed. $\endgroup$
    – Jonathan Gleason
    Commented Feb 6, 2015 at 2:27
  • $\begingroup$ As a matter of fact, I think we certainly want to consider non-Hausdorff spaces as it seems that the canonical topology on the space of germs $C_0^\infty (\mathbb{R}^d)$ at the origin in $\mathbb{R}^d$ is non-Hausdorff. $\endgroup$
    – Jonathan Gleason
    Commented Feb 6, 2015 at 3:58
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    $\begingroup$ Maybe I was unclear: my objection was not to the restriction to closed prime ideals among prime ideals, but rather to the idea that points should be determined by any kind of prime ideals. My motivation for this is non-Archimedean analytic geometry, in which points are given by valuations. In some cases, (zero-dimensional points), the valuation is determined by its kernel, a (closed) prime ideal, but in `most' cases not. $\endgroup$
    – Andrew Macpherson
    Commented Feb 6, 2015 at 9:12

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