• Corpus ID: 115159482

Algebraic Geometry over $C^\infty$-rings

  title={Algebraic Geometry over \$C^\infty\$-rings},
  author={dominic. joyce},
  journal={arXiv: Algebraic Geometry},
  • D. joyce
  • Published 31 December 2009
  • Mathematics
  • arXiv: Algebraic Geometry
If $X$ is a smooth manifold then the $\mathbb R$-algebra $C^\infty(X)$ of smooth functions $c:X\to\mathbb R$ is a $C^\infty$-$ring$. That is, for each smooth function $f:{\mathbb R}^n\to\mathbb R$ there is an $n$-fold operation $\Phi_f:C^\infty(X)^n\to C^\infty(X)$ acting by $\Phi_f:(c_1,\ldots,c_n)\mapsto f(c_1,...,c_n)$, 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. We… 
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