Algebraic Geometry Over

Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. ^{\iy}-rings -- 2.1. Two definitions of ^{\iy}-ring -- 2.2. ^{\iy}-rings as commutative \R-algebras, and ideals -- 2.3. Local ^{\iy}-rings, and localization -- 2.4. Fair ^{\iy}-rings -- 2.5. Pushouts of ^{\iy}-rings -- 2.6. Flat ideals -- Chapter 3. The ^{\iy}-ring ^{\iy}( ) of a manifold -- Chapter 4. ^{\iy}-ringed spaces and ^{\iy}-schemes -- 4.1. Some basic topology -- 4.2. Sheaves on topological spaces -- 4.3. ^{\iy}-ringed spaces and local ^{\iy}-ringed spaces -- 4.4. The spectrum functor -- 4.5. Affine ^{\iy}-schemes and ^{\iy}-schemes -- 4.6. Complete ^{\iy}-rings -- 4.7. Partitions of unity -- 4.8. A criterion for affine ^{\iy}-schemes -- 4.9. Quotients of ^{\iy}-schemes by finite groups -- Chapter 5. Modules over ^{\iy}-rings and ^{\iy}-schemes -- 5.1. Modules over ^{\iy}-rings -- 5.2. Cotangent modules of ^{\iy}-rings -- 5.3. Sheaves of Ø_{ }-modules on a ^{\iy}-ringed space ( ,Ø_{ }) -- 5.4. Sheaves on affine ^{\iy}-schemes, \MSpec and \Ga -- 5.5. Complete modules over ^{\iy}-rings -- 5.6. Cotangent sheaves of ^{\iy}-schemes -- Chapter 6. ^{\iy}-stacks -- 6.1. ^{\iy}-stacks -- 6.2. Properties of 1-morphisms of ^{\iy}-stacks -- 6.3. Open ^{\iy}-substacks and open covers -- 6.4. The underlying topological space of a ^{\iy}-stack -- 6.5. Gluing ^{\iy}-stacks by equivalences -- Chapter 7. Deligne-Mumford ^{\iy}-stacks -- 7.1. Quotient ^{\iy}-stacks, 1-morphisms, and 2-morphisms -- 7.2. Deligne-Mumford ^{\iy}-stacks -- 7.3. Characterizing Deligne-Mumford ^{\iy}-stacks -- 7.4. Quotient ^{\iy}-stacks, 1- and 2-morphisms as local models for objects, 1- and 2-morphisms in \DMCSta -- 7.5. Effective Deligne-Mumford ^{\iy}-stacks -- 7.6. Orbifolds as Deligne-Mumford ^{\iy}-stacks -- Chapter 8. Sheaves on Deligne-Mumford ^{\iy}-stacks.

8.1. Quasicoherent sheaves -- 8.2. Writing sheaves in terms of a groupoid presentation -- 8.3. Pullback of sheaves as a weak 2-functor -- 8.4. Cotangent sheaves of Deligne-Mumford ^{\iy}-stacks -- Chapter 9. Orbifold strata of ^{\iy}-stacks -- 9.1. The definition of orbifold strata \cX^{\Ga},…,\hcX^{\Ga}_{\ci} -- 9.2. Lifting 1- and 2-morphisms to orbifold strata -- 9.3. Orbifold strata of quotient ^{\iy}-stacks [\uX/ ] -- 9.4. Sheaves on orbifold strata -- 9.5. Sheaves on orbifold strata of quotients [\uX/ ] -- 9.6. Cotangent sheaves of orbifold strata -- Appendix A. Background material on stacks -- A.1. Introduction to 2-categories -- A.2. Grothendieck topologies, sites, prestacks, and stacks -- A.3. Descent theory on a site -- A.4. Properties of 1-morphisms -- A.5. Geometric stacks, and stacks associated to groupoids -- Bibliography -- Glossary of Notation -- Index -- Back Cover.

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 Deligne-Mumford C^\infty-stacks, a 2-category 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 d-manifolds and d-orbifolds, "derived" versions of manifolds and orbifolds related to Spivak's "derived manifolds".

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