Web-based algebraic stacks project

The idea of this web-site is that there is a place for open development of mathematics next to the current scientific model. For more information on how this is supposed to work, please see the general discussion on the algebraic_geometry mailing list and its archives.

The documents on this website are published under the GFDL.

Algebraic Stacks

This is the first project started on the open source model in algebraic geometry. Its aim is mainly to explain and elucidate how to work with algebraic stacks in algebraic geometry. We will provide proofs of all results, apart from some basic facts from algebra, set theory, sheaf theory. Sometimes we will avoid stating (and proving) a result in its utmost generality to avoid the project getting too large. Sometimes as well we will use an advanced result that has a good exposition in the mathematical literature, however, in general we try to avoid this.

To contribute please start reading the documents in this project (see below). Feel free to point out typos, mistakes on the mailing list (a ``stacks'' mailing list will be instituted later). The .tex files are in the source directory. Feel free to edit and send ``patches'' and/or new files to the mailing list. The section Downloads below has a list of downloads in different formats.

Table of contents

1. Introduction to the Algebraic Stacks Documents

1.1. Overview

1. References

2. Conventions used in the Algebraic Stacks Documents

2.1. Comments

2.2. Set theory

2.3. Categories

2.4. Algebra

2. References

3. Set theory

3.1. Introduction

3.2. Everything is a set

3.2.1. The hierarchy of sets

3.2.2. Everything is is contained in some ordinal

3.3. Reflection principle

3.3.1. Statement of the theorem

3. References

4. Categories

4.1. Introduction

4.2. Categories and 2-categories

4.2.1. Categories

4.2.1.1. Additional notions

4.2.2. 2-categories

4.2.2.1. 2-fibre products

4.3. Categories fibred in groupoids

4.3.1. Definitions

4.3.2. Categories fibred in sets

4.3.3. Presheaves of groupoids

4. References

5. Sites and Sheaves

5.1. Introduction

5.2. Topologies

5.2.1. Definitions

5.2.2. More about coverings

5.3. Representable sheaves

5.4. Morphisms of sites

5.5. Topoi

5.5.1. Sites and points

5. References

6. Flat Descent for Quasi-Coherent sheaves

6.1. Introduction

6.1.1. Descent for quasi-coherent sheaves

6. References

7. The \'etale topology on schemes

7.1. Introduction

7.2. Notation and conventions

7.3. Unramified morphisms

7.3.1. Definition and sorites

7.3.2. Three other equivalent definitions

7.3.3. The functorial characterisation

7.3.4. Some topological properties

7.3.5. Examples

7.4. Flat morphisms

7.4.1. Definitions, sorites, and a theorem of Grothendieck

7.4.2. Some topological properties

7.5. \'Etale morphisms

7.5.1. Definitions and sorites

7.5.2. The structure theorem for \'etale morphisms

7.5.3. An equivalent definition

7.5.4. Some topological properties

7.5.5. The functorial characterisation

7.5.6. Permanence properties

7. References

8. Injectives

8.1. Introduction

8.2. Existence of injectives in special cases

8.2.1. Modules

8.2.1.1. Categories of modules

8.2.1.2. Projective resolutions

8.2.2. Abelian presheaves

8.2.2.1. Categories of presheaves of abelian groups

8.2.3. Abelian Sheaves

8.2.3.1. Categories of abelian sheaves

8.3. Grothendieck categories and injectives

8. References

9. Hypercoverings

9.1. Introduction

9.2. Definitions

9.2.1. Goals

9.2.2. Making simplicial objects

9.2.3. Doubly simplicial stuff

9.3. The general case

9. References

10. Stacks

10.1. Introduction

10.2. Definition

10.2.1. Explanation

10.2.2. Examples

10. References

11. Stacks and groupoids

11.1. Introduction

11. References

12. Schemes as stacks and representability

12.1. Introduction

12.2. Affine schemes, schemes, stacks representable by a scheme

12.2.1. Locally ringed spaces

12.2.2. Affine schemes

12.2.3. The category of affine schemes

12.2.3.1. Sets of affine schemes

12.2.4. Schemes

12.2.5. Stacks representable by a scheme

12.3. Morphisms representable by schemes

12.3.1. Definition

12. References

13. Algebraic stacks

13.1. Introduction

13.2. Definitions

13.2.1. Algebraic spaces

13.2.2. Morphisms representable by algebraic spaces

13.2.2.1. Algebraic stacks

13. References

14. Algebraic Stacks Desirables

14.1. Foundational and prerequisites

14.1.1. Introduction

14.1.2. Conventions

14.1.3. Set Theory

14.1.4. Categories

14.1.5. Sites and Topoi

14.1.6. Stacks

14.1.7. Algebra

14.1.8. Schemes

14.1.9. Cohomology of schemes

14.1.10. Deformation theory a la Schlessinger

14.2. Algebraic Stacks, algebraic spaces and schemes

14.2.1. Definition of schemes

14.2.1.1. Alternative

14.2.2. Definition of algebraic spaces

14.2.3. Definition of algebraic stacks

14.2.4. Examples of schemes, algebraic spaces, algebraic stacks

14.2.5. Properties of algebraic stacks

14.2.6. Lisse etale site of an algebraic stack

14.2.7. Things you always wanted to know but were afraid to ask

14.2.8. Quasi-coherent sheaves on stacks

14.3. Fundamental results

14.3.1. Flat and smooth

14.3.2. Artin's representability theorem

14.3.3. DM stacks are finitely covered by schemes

14.3.4. Martin Olson's paper on properness

14.3.5. Proper pushforward of coherent sheaves

14.3.6. Keel and Mori

14.3.7. Add more here

14. References

15. GNU Free Documentation License

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