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Rings of smooth functions and their localizations. I. (English) Zbl 0592.18005
The algebraic study of rings of smooth functions as a way of extending the methods of modern algebraic geometry to differential geometry was given a new direction by Lawvere in 1967 (unpublished), who proposed that one should study such rings from the viewpoint that any smooth function \({\mathbb{R}}^ n\to {\mathbb{R}}\) is to be counted as an n-ary operation \((''C^{\infty}\)-rings”).
A serious start on the project was made by E. J. Dubuc in 1981 [Am. J. Math. 103, 683-690 (1981; Zbl 0483.58003)]. The present paper provides some further results in this direction, notably concerning formation of fraction rings in the category of \(C^{\infty}\)-rings, and concerning Henselian-ness of local \(C^{\infty}\)-rings. Thus the authors expand on a previous result of A. Joyal and the second author (”Separably real closed rings”, Sydney category seminar reports 1982, in Proceedings of the 5th Symposium of Logic in Latin America, Bogota 1981, to appear) to the effect that local \(C^{\infty}\)-rings (or ring-objects) are separably real closed [in the sense of the reviewer, Var. Publ. Ser., Aarhus Univ. 30, 123-136 (1979; Zbl 0428.03056)]; this is a result which holds in any topos.
A ”Part II” of the paper is promised, in which the authors will ”define and study the spectrum of a \(C^{\infty}\)-ring” (in a way which is different from the spectrum of Dubuc, loc. cit.).
Reviewer: A.Kock

18F99 Categories in geometry and topology
13J99 Topological rings and modules
18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
46E25 Rings and algebras of continuous, differentiable or analytic functions
54C40 Algebraic properties of function spaces in general topology
Full Text: DOI
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