Algebraic Geometry over C∞-Rings

Released on by Dominic Joyce
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  • Algebraic Geometry over C∞-Rings Book Detail

  • Author : Dominic Joyce
  • Release Date : 2019-09-05
  • Publisher : American Mathematical Soc.
  • Genre : Mathematics
  • Pages : 152
  • ISBN 13 : 1470436450
  • File Size : 27,27 MB
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Algebraic Geometry over C∞-Rings by Dominic Joyce PDF Summary

Book Description: If X is a manifold then the R-algebra C∞(X) of smooth functions c:X→R is a C∞-ring. That is, for each smooth function f:Rn→R there is an n-fold operation Φf:C∞(X)n→C∞(X) acting by Φf:(c1,…,cn)↦f(c1,…,cn), and these operations Φf satisfy many natural identities. Thus, C∞(X) actually has a far richer structure than the obvious R-algebra structure. The author explains the foundations of a version of algebraic geometry in which rings or algebras are replaced by C∞-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are C∞-schemes, a category of geometric objects which generalize manifolds and whose morphisms generalize smooth maps. The author also studies quasicoherent sheaves on C∞-schemes, and C∞-stacks, in particular Deligne-Mumford C∞-stacks, a 2-category of geometric objects generalizing orbifolds. Many of these ideas are not new: C∞-rings and C∞ -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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Algebraic Geometry over C∞-Rings

Algebraic Geometry over C∞-Rings

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If X is a manifold then the R-algebra C∞(X) of smooth functions c:X→R is a C∞-ring. That is, for each smooth function f:Rn→R there is an n-fold operatio

Algebraic Geometry

Algebraic Geometry

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An introduction to abstract algebraic geometry, with the only prerequisites being results from commutative algebra, which are stated as needed, and some element