Stacks / ∞
English · AI-integrated / 001

Introduction

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Unofficial AI-integrated English snapshot, not the official Stacks Project and not human peer review

Unofficial AI-integrated English snapshot, not the official Stacks Project and not human peer review. It includes corrections and additions absent from the translation snapshots. Language switching preserves locations, not mathematical-version identity.

In this chapterOverview
Attribution
Copyright (C) 2005 -- 2025 Johan de Jong

Permission is granted to copy, distribute and/or modify this

document under the terms of the GNU Free Documentation License,

Version 1.2 or any later version published by the Free Software

Foundation; with no Invariant Sections, no Front-Cover Texts,

and no Back-Cover Texts. A copy of the license is included in

the section entitled "GNU Free Documentation License".

Overview

Besides the book by Laumon and Moret-Bailly, see [LM-B], and the work (in progress) by Fulton et al, we think there is a place for an open source textbook on algebraic stacks and the algebraic geometry that is needed to define them. The Stacks Project attempts to do this by building the foundations starting with commutative algebra and proceeding via the theory of schemes and algebraic spaces to a comprehensive foundation for the theory of algebraic stacks.

We expect this material to be read online as a key feature are the hyperlinks giving quick access to internal references spread over many different pages. If you use an embedded pdf or dvi viewer in your browser, the cross file links should work.

This project is a collaborative effort and we encourage you to help out. Please email any typos or errors you find while reading or any suggestions, additional material, or examples you have to stacks.project@gmail.com. You can download a tarball containing all source files, extract, run make, and use a dvi or pdf viewer locally. Please feel free to edit the LaTeX files and email your improvements.

Attribution

The scope of this work is such that it is a daunting task to attribute correctly and succinctly all of those mathematicians whose work has led to the development of the theory we try to explain here. We hope eventually to generate enough community interest to find contributors willing to write sections with historical remarks for each and every chapter.

Those who contributed to this work are listed on the title page of the book version of this work and online. Here we would like to name a selection of major contributions:

  1. Jarod Alper contributed a chapter discussing the literature on algebraic stacks, see Guide to Literature, Section 03B1.

  2. Bhargav Bhatt wrote the initial version of a chapter on étale morphisms of schemes, see Étale Morphisms, Section 024K.

  3. Bhargav Bhatt wrote the initial version of More on Algebra, Section 05E5.

  4. Kiran Kedlaya contributed the initial writeup of Descent, Section 08WE.

  5. The initial versions of

    1. Algebra, Section 05K7,

    2. Injectives, Section 05NM, and

    3. the chapter on fields, see Fields, Section 09FB.

    are from The CRing Project, courtesy of Akhil Mathew et al.

  6. Alex Perry wrote the material on projective modules, Mittag-Leffler modules, including the proof of Algebra, Theorem 05A9.

  7. Alex Perry wrote the chapter on deformation theory a la Schlessinger and Rim, see Formal Deformation Theory, Section 06G8.

  8. Thibaut Pugin, Zachary Maddock and Min Lee took notes for a course which formed the basis for a chapter on étale cohomology and a chapter on the trace formula. See Étale Cohomology, Section 03N2 and The Trace Formula, Section 0F5Q.

  9. David Rydh has contributed many helpful comments, pointed out several mistakes, helped out in an essential way with the material on residual gerbes, and was the originator for the material in More on Groupoids in Spaces, Sections 04PB and 04RJ.

  10. Burt Totaro contributed Examples, Sections 08J0, 08KE, and Properties of Stacks, Section 0AFL.

  11. The chapter on pro-étale cohomology, see Pro-étale Cohomology, Section 0966, is taken from a paper by Bhargav Bhatt and Peter Scholze.

  12. Bhargav Bhatt contributed Examples, Sections 0AF8 and 0ATE.

  13. Ofer Gabber found mistakes, contributed corrections and he contributed Varieties, Lemma 04PZ, Formal Spaces, Lemma 0APW, the material in More on Groupoids, Section 0APG, the main result of Properties of Spaces, Section 03W8, and the proof of More on Flatness, Proposition 05U9.

  14. János Kollár contributed Algebra, Lemma 0BHZ and Local Cohomology, Proposition 0BK3.

  15. Kiran Kedlaya wrote the initial version of More on Algebra, Section 0BNI.

  16. Matthew Emerton, Toby Gee, and Brandon Levin contributed some results on thickenings, in particular More on Morphisms of Stacks, Lemmas 0BPU, 0BPV, and 0BPW.

  17. Lena Min Ji wrote the initial version of More on Algebra, Section 0BWR.

  18. Matthew Emerton and Toby Gee wrote the initial versions of Geometry of Stacks, Sections 0DR4 and 0DRE.