• Corpus ID: 115159482

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

@article{joyce2009AlgebraicGO,
title={Algebraic Geometry over \$C^\infty\$-rings},
author={dominic. joyce},
journal={arXiv: Algebraic Geometry},
year={2009}
}
• 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…
31 Citations
Manifolds with analytic corners
Manifolds with boundary and with corners form categories ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}$. A manifold with corners $X$ has two notions of tangent bundle: the tangent bundle $TX$, and
Vector fields on differentiable schemes and derivations on differentiable rings
• 2017
This is a survey paper which was presented by the author in the RIMS at Kyoto University. This paper contains several recent results which obtained by the author. Let us mention on the motivations of
More on the admissible condition on differentiable maps $\varphi: (X^{\!A\!z},E;\nabla)\rightarrow Y$ in the construction of the non-Abelian Dirac-Born-Infeld action $S_{DBI}(\varphi,\nabla)$
• Physics, Mathematics
• 2016
In D(13.1) (arXiv:1606.08529 [hep-th]), we introduced an admissible condition on differentiable maps $\varphi: (X^{\!A\!z}, E;\nabla)\rightarrow Y$ from an Azumaya/matrix manifold $X^{\!A\!z}$ (with
Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds
• Mathematics
• 2015
Let $({\bf X},\omega_{\bf X}^*)$ be a separated, $-2$-shifted symplectic derived $\mathbb C$-scheme, in the sense of Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209, of complex virtual dimension
On the Universal Property of Derived Manifolds
• Mathematics
• 2019
It is well known that any model for derived manifolds must form a higher category. In this paper, we propose a universal property for this higher category, classifying it up to equivalence. Namely,
Physicists' $d=3+1$, $N=1$ superspace-time and supersymmetric QFTs from a tower construction in complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry and a purge-evaluation/index-contracting map
• Physics, Mathematics
• 2019
The complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry aspect of a superspace(-time) $\widehat{X}$ in Sec.\,1 of D(14.1) (arXiv:1808.05011 [math.DG]) together with the Spin-Statistics
Poisson structure on manifolds with corners
Since a Poisson Structure is a smooth bivector field, we use a ring-valued sheaf $\OO_{X}$ on a manifold with corners $X$, we can interpret $\OO_{X}(U)$ as the ring of admissible smooth functions
$N=1$ fermionic D3-branes in RNS formulation I. $C^\infty$-Algebrogeometric foundations of $d=4$, $N=1$ supersymmetry, SUSY-rep compatible hybrid connections, and $\widehat{D}$-chiral maps from a $d=4$ $N=1$ Azumaya/matrix superspace
• Mathematics, Physics
• 2018
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
Dynamics of D-branes II. The standard action --- an analogue of the Polyakov action for (fundamental, stacked) D-branes
• Physics, Mathematics
• 2017
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
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
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
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
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
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
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
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
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
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