References for etale cohomology and related topics (Fall 2011)

Other/better references on these topics are welcome.

Etale, flat, smooth, unramified morphisms

  • Chapter 2 of the Neron models book
  • Chapters 1 through 4 of SGA1
  • Stacks project chapter on etale morphisms
  • EGA 4, in all four parts
  • Descent theory

  • Vistoli's notes on descent theory
  • Chapter 6 of the Neron models book
  • Grothendieck's expose on flat descent
  • Stacks project chapter on descent
  • Sites and topoi

  • Artin's notes on Grothendieck topologies
  • SGA 4, tome 1; all available here
  • Stacks project chapters on hypercoverings, cohomology of sheaves, and cohomology on sites
  • Brian Conrad's notes on cohomological descent. This theory is a generalisation of Cech theory to allow fairly general morphisms, and was used originally to compute the cohomology of an arbitrary complex variety in terms of smooth projective ones by Deligne
  • Deligne's notes on cohomological descent, written by Saint-Donat in SGA 4 (tome 2, expose V, bis)
  • Artin's paper "On joins of hensel rings" is an algebra paper, but it proves (amongst other things) that Cech cohomology computes derived functor cohomology for the etale topology on reasonable schemes (needs ScienceDirect)
  • Mumford's paper describes what Grothendieck topologies are, and how one can do geometry with them, especially in the context of moduli of curves

  • Illusie's expository article on topoi
  • Jacob Lurie's "short" article on higher topoi
  • Constructible sheaves ???

    Notes, textbooks, and papers on etale cohomology proper

  • SGA 4.5; available here
  • SGA 4, tome 2 and 3; all available here
  • Milne's lecture notes
  • Milne's textbook
  • Freitag-Kiehl
  • "Introduction to etale cohomology" by Tamme

  • de Jong's lecture notes
  • Verdier's proof of the duality theorem (in "Proceedings of a conference on Local Fields" at Drierbergen)
  • Lecture notes from the Woodshole program
  • Illusie's notes on Gabber's recent finiteness theorems for etale cohomology
  • Illusie's historical notes on Grothendieck and etale cohomology
  • Weil conjectures

  • Verdier's proof of the trace formula (in "Proceedings of a conference on Local Fields" at Drierbergen)
  • Dwork's proof of the rationality of the zeta function (in "Proceedings of a conference on Local Fields" at Drierbergen)
  • Deligne's Weil 1
  • Kowalski's notes on Weil 1

  • Deligne's Weil 2
  • Katz and Messing's paper deducing purity theorems for crystalline cohomology from Deligne's theorems

  • Katz's notes on Weil 2 from the Arizona Winter School
  • Katz's notes on Weil 2 from a 1974 class at Princeton
  • "Weil conjectures, perverse sheaves and l'adic Fourier transform" by R. Kiehl and R. Wiessauer
  • Beilinson's lectures at the University of Chicago
  • Mustaţă book on zeta functions
  • Brauer groups

  • Grothendieck's exposes 1 and 2
  • Serre's exposes (especially II and II')
  • de Jong's notes provide an elegant approach to Brauer groups via the language of twisted sheaves, and also give a new proof of Gabber's result relating the Brauer group to an etale cohomology group
  • Etale fundamental groups

  • SGA1
  • Murre's notes (Tata lecture notes)
  • Grothendieck-Murre book on the tame fundamental group; available through SpringerLink as LNM 208
  • Lenstra's notes on Galois theory for schemes
  • Etale homotopy theory

  • Artin-Mazur; available through SpringerLink as LNM 100
  • Friedlander's book on etale homotopy for simplicial schemes
  • Artin-Mazur survey article (in "Proceedings of a conference on Local Fields" at Drierbergen)
  • Homological algebra

  • Gelfand-Manin
  • Some sections in the Stacks project
  • Beilinson's lectures at the University of Chicago
  • Some topological perspectives

  • Some related papers

  • Serre's paper "Analogues kahleriens de certaines conjectures de Weil." (needs Jstor)
  • Quillen's paper titled "Some remarks on etale homotopy theory and a conjecture of Adams." in Topology (needs ScienceDirect)
  • Mazur's notes on relating Tate duality for global fields to the general theory of Verdier duality in the etale topology