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**The structure of smooth mappings over Weil algebras and the category of manifolds over algebras.**
*(English)*
Zbl 0985.58001

The general theory of manifolds over algebras was studied by A. P. Shirokov, J. Vanžura, V. V. Vishnevskiĭ, G. I. Kruchkovich, and others. A. P. Shirokov [Tr. Geom. Semin. 1, 425-456 (1966; Zbl 0198.54601)] proved that structures of finite-dimensional manifolds over local algebras arise on tangent bundles and bundles of infinitely near points in the sense of A. Weil. Various types of manifolds over algebras were defined in the case of infinite-dimensional manifolds and algebras. Consider a finite-dimensional, local, associative, commutative and unital algebra A and finite-dimensional vector spaces, which are also A-modules. A map between such modules is called A-smooth if it is smooth and its differential is A-linear at every point of its domain. A smooth manifold modeled on an A-module is called an A-manifold if it has an atlas with A-smooth coordinate transformations.

In this paper, the author studies the local expression of an A-smooth mapping from an open subset of the module \({\mathbf A}^n\) to the module \({\mathbf A}^k\). Next, in the case when a local algebra A is the semidirect sum \(\overline{\mathbf A} \oplus {\mathbf I}\) of a subalgebra \(\overline{\mathbf A}\) and an ideal I, the expression of an \(\overline{\mathbf A}\)-smooth mapping from an open subset \(\overline{\mathbf U}\) of \(\overline{\mathbf A}^n\) to \({\mathbf A}^k\) is established and it is proven that such a mapping can be prolonged to an A-smooth mapping between domains of \({\mathbf A}^n\) and \({\mathbf A}^k\). This result allows to construct a functor from the category of \(\overline{\mathbf A}\)-manifolds to the category of A-manifolds that is a generalization of the Weil functor of A-prolongation. Also, the local expression of an A-smooth mapping between A-modules of the form \({\mathbf A}^n \oplus \mathbb{R}^m\) is given.

The author presents a number of examples of A-manifolds and functors defined on the category of A-manifolds. Furthermore, the author defines the canonical osculating fiber bundle \({\mathbf O}^V_{\mathbf I}(M^{\mathbf A})\) and the homotopy groupoid \(\Pi_{\mathbf I}M^{\mathbf A}\) associated with the canonical \({\mathbf I}^n\)-foliation \({\mathcal F}^{\mathbf I}\) induced by an ideal I of a local algebra A on an \(n\)-dimensional A-manifold \(M^{\mathbf A}\) modeled on \({\mathbf A}^n\). It is shown that \({\mathbf O}^V_{\mathbf I}(M^{\mathbf A})\) and \(\Pi_{\mathbf I}M^{\mathbf A}\) carry natural structures of smooth manifolds over the algebra \(\widehat{{\mathbf A}_{\mathbf I}}=(p\oplus p)^{-1}(\Delta(\overline{\mathbf A}))\), where \(\Delta(\overline{\mathbf A})\) is the diagonal of \(\overline {\mathbf A}={\mathbf A}/{\mathbf I}\) and \(p: {\mathbf A} \to \overline{\mathbf A}\) is the canonical projection. Also, if \(M^{\mathbf A}\) is complete, then \({\mathcal F}^{\mathbf I}\) has no vanishing cycles. For each leaf of the canonical \({\mathbf I}^n\)-foliation \({\mathcal F}^{\mathbf I}\) on \(M^{\mathbf A}\) the author associates the I-holonomy representation and defines the I-holonomy groupoid \(\Gamma_{\mathbf I}M^{\mathbf A}\) which also carries a natural structure of a smooth manifold over the algebra \(\widehat{{\mathbf A}_{\mathbf I}}\). Finally, the author proves that \(\Gamma_{\mathbf I}M^{\mathbf A}\) is a Hausdorff topological space if and only if \(M^{\mathbf A}\) has no I-holonomy vanishing cycles.

In this paper, the author studies the local expression of an A-smooth mapping from an open subset of the module \({\mathbf A}^n\) to the module \({\mathbf A}^k\). Next, in the case when a local algebra A is the semidirect sum \(\overline{\mathbf A} \oplus {\mathbf I}\) of a subalgebra \(\overline{\mathbf A}\) and an ideal I, the expression of an \(\overline{\mathbf A}\)-smooth mapping from an open subset \(\overline{\mathbf U}\) of \(\overline{\mathbf A}^n\) to \({\mathbf A}^k\) is established and it is proven that such a mapping can be prolonged to an A-smooth mapping between domains of \({\mathbf A}^n\) and \({\mathbf A}^k\). This result allows to construct a functor from the category of \(\overline{\mathbf A}\)-manifolds to the category of A-manifolds that is a generalization of the Weil functor of A-prolongation. Also, the local expression of an A-smooth mapping between A-modules of the form \({\mathbf A}^n \oplus \mathbb{R}^m\) is given.

The author presents a number of examples of A-manifolds and functors defined on the category of A-manifolds. Furthermore, the author defines the canonical osculating fiber bundle \({\mathbf O}^V_{\mathbf I}(M^{\mathbf A})\) and the homotopy groupoid \(\Pi_{\mathbf I}M^{\mathbf A}\) associated with the canonical \({\mathbf I}^n\)-foliation \({\mathcal F}^{\mathbf I}\) induced by an ideal I of a local algebra A on an \(n\)-dimensional A-manifold \(M^{\mathbf A}\) modeled on \({\mathbf A}^n\). It is shown that \({\mathbf O}^V_{\mathbf I}(M^{\mathbf A})\) and \(\Pi_{\mathbf I}M^{\mathbf A}\) carry natural structures of smooth manifolds over the algebra \(\widehat{{\mathbf A}_{\mathbf I}}=(p\oplus p)^{-1}(\Delta(\overline{\mathbf A}))\), where \(\Delta(\overline{\mathbf A})\) is the diagonal of \(\overline {\mathbf A}={\mathbf A}/{\mathbf I}\) and \(p: {\mathbf A} \to \overline{\mathbf A}\) is the canonical projection. Also, if \(M^{\mathbf A}\) is complete, then \({\mathcal F}^{\mathbf I}\) has no vanishing cycles. For each leaf of the canonical \({\mathbf I}^n\)-foliation \({\mathcal F}^{\mathbf I}\) on \(M^{\mathbf A}\) the author associates the I-holonomy representation and defines the I-holonomy groupoid \(\Gamma_{\mathbf I}M^{\mathbf A}\) which also carries a natural structure of a smooth manifold over the algebra \(\widehat{{\mathbf A}_{\mathbf I}}\). Finally, the author proves that \(\Gamma_{\mathbf I}M^{\mathbf A}\) is a Hausdorff topological space if and only if \(M^{\mathbf A}\) has no I-holonomy vanishing cycles.

Reviewer: Andrew Bucki (Oklahoma City)