Algebraic Geometry Over $C^/infty $-Rings
Seiten
2019
American Mathematical Society (Verlag)
978-1-4704-3645-2 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-3645-2 (ISBN)
Explains the foundations of a version of algebraic geometry in which rings or algebras are replaced by $C^/infty $-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are $C^/infty $-schemes, a category of geometric objects which generalize manifolds and whose morphisms generalize smooth maps.
If $X$ is a manifold then the $/mathbb R$-algebra $C^/infty (X)$ of smooth functions $c:X/rightarrow /mathbb R$ is a $C^/infty $-ring. That is, for each smooth function $f:/mathbb R^n/rightarrow /mathbb R$ there is an $n$-fold operation $/Phi _f:C^/infty (X)^n/rightarrow C^/infty (X)$ acting by $/Phi _f:(c_1,/ldots ,c_n)/mapsto f(c_1,/ldots ,c_n)$, and these operations $/Phi _f$ satisfy many natural identities. Thus, $C^/infty (X)$ actually has a far richer structure than the obvious $/mathbb R$-algebra structure.
The author explains the foundations of a version of algebraic geometry in which rings or algebras are replaced by $C^/infty $-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are $C^/infty $-schemes, a category of geometric objects which generalize manifolds and whose morphisms generalize smooth maps. The author also studies quasicoherent sheaves on $C^/infty $-schemes, and $C^/infty $-stacks, in particular Deligne-Mumford $C^/infty$-stacks, a 2-category of geometric objects generalizing orbifolds.
Many of these ideas are not new: $C^/infty$-rings and $C^/infty $-schemes have long been part of synthetic differential geometry. But the author develops them in new directions. In earlier publications, the author used these tools to define d-manifolds and d-orbifolds, ``derived'' versions of manifolds and orbifolds related to Spivak's ``derived manifolds''.
If $X$ is a manifold then the $/mathbb R$-algebra $C^/infty (X)$ of smooth functions $c:X/rightarrow /mathbb R$ is a $C^/infty $-ring. That is, for each smooth function $f:/mathbb R^n/rightarrow /mathbb R$ there is an $n$-fold operation $/Phi _f:C^/infty (X)^n/rightarrow C^/infty (X)$ acting by $/Phi _f:(c_1,/ldots ,c_n)/mapsto f(c_1,/ldots ,c_n)$, and these operations $/Phi _f$ satisfy many natural identities. Thus, $C^/infty (X)$ actually has a far richer structure than the obvious $/mathbb R$-algebra structure.
The author explains the foundations of a version of algebraic geometry in which rings or algebras are replaced by $C^/infty $-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are $C^/infty $-schemes, a category of geometric objects which generalize manifolds and whose morphisms generalize smooth maps. The author also studies quasicoherent sheaves on $C^/infty $-schemes, and $C^/infty $-stacks, in particular Deligne-Mumford $C^/infty$-stacks, a 2-category of geometric objects generalizing orbifolds.
Many of these ideas are not new: $C^/infty$-rings and $C^/infty $-schemes have long been part of synthetic differential geometry. But the author develops them in new directions. In earlier publications, the author used these tools to define d-manifolds and d-orbifolds, ``derived'' versions of manifolds and orbifolds related to Spivak's ``derived manifolds''.
Dominic Joyce, University of Oxford, United Kingdom.
Introduction
$C^/infty$-rings
The $C^/infty$-ring $C^/infty (X)$ of a manifold $X$
$C^/infty $-ringed spaces and $C^/infty $-schemes
Modules over $C^/infty$-rings and $C^/infty $-schemes
$C^/infty $-stacks
Deligne-Mumford $C^/infty $-stacks
Sheaves on Deligne-Mumford $C^/infty $-stacks
Orbifold strata of $C^/infty $-stacks
Appendix A. Background material on stacks
Bibliography
Glossary of Notation
Index.
| Erscheinungsdatum | 02.09.2019 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 228 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| ISBN-10 | 1-4704-3645-0 / 1470436450 |
| ISBN-13 | 978-1-4704-3645-2 / 9781470436452 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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