Algebraic geometry over C[∞]-rings

著者

• Joyce, Dominic D.

書誌事項

Algebraic geometry over C[∞]-rings

Dominic Joyce

（Memoirs of the American Mathematical Society, no. 1256）

American Mathematical Society, c2019

タイトル別名

Algebraic geometry over C[infinity symbol]-rings

注記

[∞] is superscript

"July 2019, volume 260, number 1256(fifth of 5 numbers)"

Includes bibliographical references and index

内容説明・目次

内容説明

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''.

目次

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.

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詳細情報

• NII書誌ID(NCID): BB28927504
• ISBN:
  • 9781470436452
• 出版国コード: us
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: Providence, RI.
• ページ数/冊数: v, 139 p.
• 大きさ: 26 cm
• 親書誌ID:
  • BA00035927