The Stacks project

1.1 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 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.

Comments (8)

Comment #1 by Pieter Belmans on

Testing the comments system on 1.1

Comment #2786 by Stanisław Szawiel on

This project is a brilliant, invaluable resource, but why is everything a lemma?

Not really everything, but almost 94% -- I count about 39000 instances of "Lemma" in the text. Comparatively "Theorem" is on the verge of extinction with a paltry 1072 instances, with "Proposition" barely doing any better at 1500 instances. I counted references as well, but still... O_O

Don't worry about it, it's only slightly jarring.

Comment #2894 by on

Because I think there is no difference between lemmas, propositions, theorems, corollaries. Each of those is a mathematical result we can use in further mathematical results.

Comment #4861 by Matthieu Romagny on

For some reason, in the book.pdf version, the numbering of chapters does not increment when passing from last chapter of "Part 1: Preliminaries" to first chapter of "Part 2: Schemes". Check page 6 of book.pdf : (25) Hypercoverings (25) Schemes

Comment #7501 by Hao Peng on

I am sorry to comment here but I want to know how to cite tags or comment tags that is clickable in the comments?

Comment #7513 by David Holmes on

You can use \ref; for example removing the spaces from \ref { 0001 } yields 1.1. For more, see the 'Markdown' link in the text above the comments box.

Comment #7868 by Anonymous on

Sorry that I have to give a comment here since there is no other place. When I download the book, bold font is not shown in the pdf file (for example: Lemma, section, tag,...). Can you fix it ? Thanks.

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