Sites, topoi and étale cohomology

Nineteen lessons of a course on étale cohomology. Sites and sheaves introduces Grothendieck topologies, sheafification by the plus construction, and sheaves on the category of sets with a group action. Topoi, morphisms and points treats geometric morphisms, points as exact ways of taking germs, and what the points of a topos remember about a space or a group. Cohomology on sites proves that injective resolutions exist, compares Čech and derived-functor cohomology, identifies the first cohomology group with torsors, and develops the Leray spectral sequence and the base change map. Hypercoverings computes the cohomology of a site as a colimit over hypercoverings. Topologies on schemes compares the Zariski, étale, smooth, syntomic, fppf and fpqc topologies, shows that quasi-coherent modules are sheaves for all of them with the same cohomology as on the Zariski site, and proves Hilbert’s Theorem 90 for line bundles. The étale site and its points shows that the points of the étale topos are the geometric points, computes stalks through strict henselizations, and treats the Kummer sequence. Pushforward, pullback and finite morphisms computes higher direct images on stalks, proves that finite morphisms have no higher direct images, treats closed immersions and extension by zero, and proves the topological invariance of the étale site. Galois cohomology and the étale cohomology of a field identifies étale sheaves on a field with Galois modules, proves Hilbert 90 and the Kummer and Artin–Schreier sequences, and computes the cohomological dimension of finite fields, of the real numbers and of \(\mathbf C((t))\). Brauer groups and Tsen’s theorem identifies the Brauer group of a field with the second Galois cohomology of the multiplicative group and proves that it vanishes over finite fields and over function fields of curves over an algebraically closed field. The multiplicative group on a curve computes the cohomology of the multiplicative group and of the roots of unity on a smooth curve. Constructible sheaves and extension by zero proves the exactness and permanence properties of constructible sheaves and reduces finite monodromy to an \(\ell\)-group. Torsion sheaves on curves proves the degree bounds and the finiteness of the cohomology of torsion sheaves on curves, with Artin–Schreier theory in characteristic \(p\). The proper base change theorem proves that, with torsion coefficients, the stalks of the higher direct images of a proper morphism are the cohomology of its fibres. Cohomology with compact support constructs \(Rf_!\) through compactifications and proves that it does not depend on the compactification. Its constructibility proof gives the reduction from the affine-line provider and proves finite generation over an algebraically closed field; the relative-curve application checks the strict-local hypotheses explicitly. Smooth base change and local acyclicity proves smooth base change and deduces that the cohomology of a proper smooth family is locally constant for coefficients prime to the residue characteristics. Cohomological dimension and the Künneth formula proves the vanishing above the dimension, the stronger bound for affine schemes, and the Künneth formula. Poincaré duality for curves proves duality for constructible coefficients on a smooth curve, following M. Artin. Poincaré duality for smooth varieties constructs the trace for smooth morphisms from the curve trace, proves that the trace is a local isomorphism by induction on the relative dimension, and deduces relative and global Poincaré duality, purity for smooth pairs and Gysin maps. Comparison with singular cohomology proves Artin's comparison theorem: for complex schemes, étale cohomology with torsion coefficients and compact support agrees with the compactly supported sheaf cohomology of the complex analytic space, and ordinary étale cohomology agrees with ordinary analytic sheaf cohomology for constructible coefficients. Each lesson has complete proofs and solved exercises.

Prerequisites: categories, functors, limits and the Yoneda lemma, and homological algebra (injective resolutions and spectral sequences); from the fifth lesson on, schemes and flat, étale and smooth morphisms; from the eighth lesson on, Galois theory of fields and profinite groups; for the ninth, central simple algebras, which are treated in lessons of this collection linked from that lesson. Facts of category theory, homological algebra, commutative algebra and algebraic geometry are cited by tag from the Stacks project, through the AI-integrated edition of the Stacks project.

Reading order

The last lesson of the course, on the comparison with singular cohomology, is not yet part of this collection.

Download the lessons and their editable sources · Sources and authorship · Course record

Written by GPT-6.1 Sol (OpenAI) and self-checked by the writing AI; revised by Claude Opus 5.5 (Anthropic). Section 2 of Poincaré duality for smooth varieties was written by GPT-6 Astra (OpenAI) and checked by Claude Opus 5.5. The line below each lesson title states its checks. Public domain (CC0 1.0).