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Functions on the zeroes of \(dx\). (English) Zbl 1090.58001

R. Bryant and P. Griffiths [J. Am. Math. Soc. 8, No. 3, 507–596 (1995; Zbl 0845.58004)] introduced the notion of the characteristic cohomology of an exterior differential system and showed that in the local involutive case the characteristic cohomology \(\overline{H}^q\) vanishes if \(0 < q < n - \ell\), where \(\ell\) is some geometric invariant of the system. The question whether there is an analog for characteristic cohomology to the De Rham theorem for the usual De Rham cohomology was not answered. The problem is difficult in part because characteristic cohomology is defined as an inverse limit and it is not formed directly from looking at differential forms on spaces. In this paper, the author investigates this question using synthetic differential geometry.
The objects in a synthetic differential geometry are the smooth spaces including some more general spaces than the \(C^\infty\) manifolds. This extension allows to form function spaces, quotient spaces, the spaces of zeroes of arbitrary smooth functions, etc. A differential \(1\)-form on a smooth space \(M\) may be viewed as a map from \(M\) into a suitable universal space (a generalized space). The forms under consideration in characteristic cohomology are one-forms and their derivatives. The ideals are generated (as closed differential ideals) by one-forms. The zeros of the maps from \(M\) into a generalized space form a generalized subspace of the original manifold. The final goal is to show that De Rham’s theorem holds on this generalized space.
In this paper, the author considers the simplest case where the De Rham cohomology of the generalized subspace is the characteristic cohomology of the differential ideal. In this case, the ideal consists of forms on the real line generated by the standard one form \(dx\). The cohomology group consists of smooth functions on the real line whose exterior derivatives are multiples of \(dx\). Within synthetic differential geometry, the author forms the subobject \(S \subset R\) of the zeros of the “amazing right adjoint” of the differential form \(dx\) and proves that in the model \(\mathcal F\) of synthetic differential geometry consisting of sheaves (with respect to open covers) over the opposite category \(\mathbb F\) of the category of closed finitely generated \(C^\infty\)-rings, any morphism from \(S\) to the real line \(\mathbb{R}\) extends to a morphism from \(\mathbb{R}\) to \(\mathbb{R}\). This shows that the De Rham cohomology of the space \(S\) is the same as the characteristic cohomology of the ideal generated by \(dx\).

MSC:

58A03 Topos-theoretic approach to differentiable manifolds
18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
58A10 Differential forms in global analysis
58A15 Exterior differential systems (Cartan theory)
51K10 Synthetic differential geometry

Citations:

Zbl 0845.58004
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References:

[1] Briançon, J.; Skoda, H., Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de \(C^n\), C. R. Acad. Sci. Paris Ser. A, 278, 949-951 (1974) · Zbl 0307.32007
[2] Bryant, R. L.; Griffiths, P. A., Characteristic cohomology of differential systems. I. General theory, J. Amer. Math. Soc., 8, 3, 507-596 (1995) · Zbl 0845.58004
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