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Softcover ISBN:  9781470436452 
Product Code:  MEMO/260/1256 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 
eBook ISBN:  9781470453367 
Product Code:  MEMO/260/1256.E 
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MAA Member Price:  $72.90 
AMS Member Price:  $48.60 
Softcover ISBN:  9781470436452 
eBook ISBN:  9781470453367 
Product Code:  MEMO/260/1256.B 
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MAA Member Price:  $145.80 $109.35 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 260; 2019; 139 ppMSC: Primary 58; Secondary 14; 46; 51
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 DeligneMumford \(C^\infty\)stacks, a 2category 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 dmanifolds and dorbifolds, “derived” versions of manifolds and orbifolds related to Spivak's “derived manifolds”.

Table of Contents

Chapters

1. Introduction

2. $C^\infty $rings

3. The $C^\infty $ring $C^\infty (X)$ of a manifold $X$

4. $C^\infty $ringed spaces and $C^\infty $schemes

5. Modules over $C^\infty $rings and $C^\infty $schemes

6. $C^\infty $stacks

7. Deligne–Mumford $C^\infty $stacks

8. Sheaves on Deligne–Mumford $C^\infty $stacks

9. Orbifold strata of $C^\infty $stacks

A. Background material on stacks


Additional Material

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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 DeligneMumford \(C^\infty\)stacks, a 2category 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 dmanifolds and dorbifolds, “derived” versions of manifolds and orbifolds related to Spivak's “derived manifolds”.

Chapters

1. Introduction

2. $C^\infty $rings

3. The $C^\infty $ring $C^\infty (X)$ of a manifold $X$

4. $C^\infty $ringed spaces and $C^\infty $schemes

5. Modules over $C^\infty $rings and $C^\infty $schemes

6. $C^\infty $stacks

7. Deligne–Mumford $C^\infty $stacks

8. Sheaves on Deligne–Mumford $C^\infty $stacks

9. Orbifold strata of $C^\infty $stacks

A. Background material on stacks