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On analytic models of synthetic differential geometry. (English) Zbl 0790.32009
This article is a sequel of previous works of the first author and G. Taubin [Cah. Topologie Geom. Differ. 24, 225-265 (1983; Zbl 0575.32004)] and the second author [ibid. 31, No. 1, 21-52 (1990; Zbl 0705.32002)], where we introduced the notions of analytic and (local) analytic rings, and developed their spectral theory. These works culminate here, where the construction of a model of Synthetic Differential Geometry (S.D.G.) well adapted to the study of complex analytic varieties and spaces is achieved.
Let \({\mathcal M}\) be a category of algebraic varieties, or (complex) analytic or (real) differentiable manifolds. Objects in \({\mathcal M}\) are built up pasting together basic objects. This implies that a well adapted model of DSG, \(i:{\mathcal M}\to{\mathcal T}\), as a functor into a topos \({\mathcal T}\) should preserve all open covers.
Besides the preservation of open covers and the original axiom of Jet Representability [the first author, ‘Integración de campos vectoriales y geometría diferencial sintética’, Proc. VII Sem. Nac. Mat., Univ. Nac. Cordoba (1984) and M. Bunge and the first author (loc. cit.)], the stronger axiom of Germ Representability [A. Kock, J. Pure Appl. Algebra 20, 55-70 (1981; Zbl 0487.18006) and M. Bunge and the first author, Lect. Notes Pure Appl. Math. 106, 93-159 (1987; Zbl 0658.18004)] has to hold for the applicability of SDG to results of local (and not only infinitesimal) character. The validity of the axiom of germ representability is reflected in the model by the fact that the object of infinitesimals \(\Delta=[[x\in\text{Line}^ n|\neg\neg(x=0)]]\) should be a representable sheaf. This fact is achieved in the differentiable case by means of the notion of germ determined ideal (introduced by the first author in Am. J. Math. 103, 683-690 (1981; Zbl 0483.58003), the axiom of germ representability proved later by him in J. Pure Appl. Algebra 64, No. 2, 131-144 (1990; Zbl 0702.18008)). This solution does not apply to the analytic case.
We introduce here the notion of Analytic Scheme (definition 1.3), key to the solution of the problem. Analytic Schemes are (strictly) more general than usual Analytic Spaces, and as such they have no counterpart in the differentiable case (they are determined by an open set in \(\mathbb{C}^ n\) and two coherent sheaves of ideals).

MSC:
32B05 Analytic algebras and generalizations, preparation theorems
51K10 Synthetic differential geometry
18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
18B25 Topoi
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References:
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