# Algebraic Geometry over $C^\infty$-rings

@article{joyce2009AlgebraicGO, title={Algebraic Geometry over \$C^\infty\$-rings}, author={dominic. joyce}, journal={arXiv: Algebraic Geometry}, year={2009} }

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…

## 31 Citations

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As the necessary background to construct from the aspect of Grothendieck's Algebraic Geometry dynamical fermionic D3-branes along the line of Ramond-Neveu-Schwarz superstrings in string theory, three…

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We introduce a new action $S_{standard}^{(\rho,h; \Phi,g,B,C)}$ for D-branes that is to D-branes as the Polyakov action is to fundamental strings. This `standard action' is abstractly a non-Abelian…

N ov 2 01 2 An introduction to C ∞-schemes and C ∞-algebraic geometry

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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 : R → R there is an n-fold operation Φf : C (X) → C(X) acting by Φf : c1, . .…

## References

SHOWING 1-10 OF 41 REFERENCES

Derived smooth manifolds

- Mathematics
- 2008

We define a simplicial category called the category of derived manifolds. It contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary…

Differentiable stacks and gerbes

- Mathematics, Physics
- 2006

We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study $S^1$-bundles and $S^1$-gerbes over differentiable stacks. In particular, we establish the…

Rings of smooth funcions and their localizations

- Mathematics
- 1986

Several types of rings of smooth functions, such as differentiable algebras and formal algebras, occupy a central position in singularity theory and related subjects. In this series of papers we will…

Quasi-smooth Derived Manifolds

- History
- 2007

The category Man of smooth manifolds is not closed under arbitrary fiber products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the non-transverse…

Handbook Of Categorical Algebra 1 Basic Category Theory

- Mathematics
- 2008

Category theory is the key to a clear presentation of modern abstract "Basic Category Theory for Computer Scientists" by Benjamin C. Pierce (1991). "Handbook of Categorical Algebra" by Francis…

Foundations of Topological Stacks I

- Mathematics
- 2005

This is the first in a series of papers devoted to foundations of topological stacks.
We begin developing a homotopy theory for topological stacks along the lines of classical homotopy theory of…

Lagrangian intersection floer theory : anomaly and obstruction

- Mathematics
- 2009

Part I Introduction Review: Floer cohomology The $A_\infty$ algebra associated to a Lagrangian submanifold Homotopy equivalence of $A_\infty$ algebras Homotopy equivalence of $A_\infty$ bimodules…

Algebraic geometry

- Computer ScienceGraduate texts in mathematics
- 1977

It’s better to think of Algebraic Geometry as indicating a sub-area of mathematics as a whole, rather than a very precisely defined subfield.

Handbook of Categorical Algebra

- Computer Science
- 1994

The Handbook of Categorical Algebra is intended to give, in three volumes, a rather detailed account of what, ideally, everybody working in category theory should know, whatever the specific topic of…

Synthetic Differential Geometry

- Mathematics
- 1981

Preface to the second edition (2005) Preface to the first edition (1981) Part I. The Synthetic Ttheory: 1. Basic structure on the geometric line 2. Differential calculus 3. Taylor formulae - one…