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Algebraic Geometry over $C^/infty $-Rings (Paperback)
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Algebraic Geometry over $C^/infty $-Rings (Paperback)
Series: Memoirs of the American Mathematical Society
Expected to ship within 12 - 19 working days
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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''.
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