The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

1.2 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 104.1.

  2. Bhargav Bhatt wrote the initial version of a chapter on étale morphisms of schemes, see Étale Morphisms, Section 40.1.

  3. Bhargav Bhatt wrote the initial version of More on Algebra, Section 15.80.

  4. Kiran Kedlaya contributed the initial writeup of Descent, Section 34.4.

  5. The initial versions of

    1. Algebra, Section 10.27,

    2. Injectives, Section 19.2, and

    3. the chapter on fields, see Fields, Section 9.1.

    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 10.94.5.

  7. Alex Perry wrote the chapter on deformation theory a la Schlessinger and Rim, see Formal Deformation Theory, Section 82.1.

  8. Thibaut Pugin, Zachary Maddock and Min Lee took notes for a course which formed the basis for a chapter on étale cohomology, see Étale Cohomology, Section 54.1.

  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 71.12 and 71.15.

  10. Burt Totaro contributed Examples, Sections 102.57, 102.58, and Properties of Stacks, Section 92.12.

  11. The chapter on pro-étale cohomology, see Pro-étale Cohomology, Section 56.1, is taken from a paper by Bhargav Bhatt and Peter Scholze.

  12. Bhargav Bhatt contributed Examples, Sections 102.64 and 102.68.

  13. Ofer Gabber found mistakes, contributed corrections and he contributed Varieties, Lemma 32.7.17, Formal Spaces, Lemma 79.9.5, the material in More on Groupoids, Section 39.15, the main result of Properties of Spaces, Section 58.17, and the proof of More on Flatness, Proposition 37.25.13.

  14. János Kollár contributed Algebra, Lemma 10.118.2 and Local Cohomology, Proposition 48.7.7.

  15. Kiran Kedlaya wrote the initial version of More on Algebra, Section 15.81.

  16. Matthew Emerton, Toby Gee, and Brandon Levin contributed some results on thickenings, in particular More on Morphisms of Stacks, Lemmas 98.3.7, 98.3.8, and 98.3.9.

  17. Lena Min Ji wrote the initial version of More on Algebra, Section 15.105.

  18. Matthew Emerton and Toby Gee wrote the initial versions of Geometry of Stacks, Sections 99.3 and 99.5.


Comments (4)

Comment #165 by Evelyn Mitchell on

The references to the sections don't resolve to a tag.

"Jarod Alper wrote \hyperref[guide-section-phantom]{Guide to Literature}", for example, should refer to http://stacks.math.columbia.edu/chapter/73 but, doesn't. Chapter 73 is all of the "Guide to Literature", but doesn't have a whole chapter tag.

How have you resolved that problem before?

Comment #167 by on

Yes, you are right. Luckily there is only one place where this precise problem happens, and it is in this section. I'm going to ask my people if we can find a better solution. We'll get back to you.

Technical discussion: Strictly speaking the chapters don't have tags themselves, but each chapter has a "phantom section" which does have a tag. I introduced these tags for exactly this reason: to be able to refer to chapters by cross-file links. Unfortunately, references to these phantom sections do not look good in the pdfs, because phantom sections do not get a number (and they shouldn't), or on the web site. See 20.1. In fact, this is currently the only one such reference. In the introduction (see above) we used the hyperref construction because it looks better in the pdf, but it doesn't work in the website, and actually I think it isn't the correct solution either.

Conclusion: What is a good way to introduce references to chapters that works both in the pdf versions (book or split out chapters) and on the web-site?

Comment #3384 by Daniel Litt on

Lines (1), (2), (7), and (11) seem not to be rendering correctly on my computer.


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