Holmstrom Etale cohomology

Etale cohomology

For Thomason’s descent spectral sequence of Atiyah-Hirzebruch type, see Mitchell.

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Etale cohomology

See Gereon Quick.

Isaksen: Etale realization on the A^1-homotopy theory of schemes

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Etale cohomology

Jardine: Homotopy and homotopical algebra: The etale cohomology rings of the classifying simplicial schemes $BG$ fo various group schemes $G$ have been calculated. (p. 660)

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Etale cohomology

In Kahn, there some construction needed for “anti-Chern classes” from etale cohomology to algebraic K-theory (see “On the Lichtenbaum-Quillen conjecture”, Algebraic K-theory and algebraic topology (P.G Goerss, J.F. Jardine, eds), NATO ASI Series, Ser. C 407 (1993), 147-166).

Soule’s K-theory Chern class map:

Here I think $A$ is a ring containing a primitive $\ell$-th root of unity.

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Etale cohomology

MathSciNet

Google Scholar

Google

arXiv: Experimental full text search

arXiv: Abstract search

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Etale cohomology

AG (Algebraic geometry)

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Etale cohomology

Pure, Mixed (?)

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Etale cohomology

See also Sheaf cohomology, Weil cohomology, l-adic cohomology, Non-standard étale cohomology

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Etale cohomology

Notes from Katz in Motives vol

Cohomology groups $H^i(X, F)$ defined for any scheme $X$, and any abelian étale sheaf on the category of schemes étale over $X$. This construction is functorial in the pair $(X, F)$.

In order to be able to say anything, let $k$ be a field and let $X$ be separated and of finite type over $k$.

We could consider $H^i(X, \mathbb{Z} / N \mathbb{Z})$, but instead we look at $H^i(X \otimes_k \bar{k}, \mathbb{Z} / N \mathbb{Z})$, which are $N$-torsion groups, with a Galois action.

Want: groups with compact support. Nagata shows that there is an open $k$-immersion of $X$ into some $X'$ which is proper over $k$. For any étale sheaf on $X$, on can define the extension by zero of $F$ on $X'$, and pull this back to $X' \otimes \bar{k}$. This defines $H^i_c(X \otimes \bar{k}, F)$. These are abelian groups with a Galois action.

If $U$ is open in $X$ with closed complement $Z$, then there is an excision l.e.s.

Theorem: For $X$ separated and of finite type over a field $k$, the cohomology groups with compact support and coeffs $\mathbb{Z} / N \mathbb{Z}$ are all finite and invariant under extension of $\bar{k}$ to an algebraically closed overfield. Moreover, they vanish for $i \geq 2 \ dim(X)$.

Finiteness and invariance, but not the vanishing, fails in general for the groups $H^i(X \otimes_k \bar{k}, \mathbb{Z} / N \mathbb{Z})$, for example if $X$ is the affine line and $N$ is the characteristic of $k$. However, there is the following theorem.

Thm: For $X$ separated and of finite type over a field $k$, and $N$ an integer invertible in $k$, the ordinary cohomology groups $H^i(X \otimes_k \bar{k}, \mathbb{Z} / N \mathbb{Z})$ are all finite, invariant as above, and vanish as above. If $X$ is affine, they vanish for $i \geq dim(X)$.

Intuition: When $N$ is invertible, the above groups should behave just like ordinary topological cohomology, and cohomology with compact support. For example, comparison thm for separated $\mathbb{C}$-schemes of finite type, with topological cohomology.

Def of Tate twist $\mathbb{Z} / N \mathbb{Z}(i)$.

Thm (PD): Let $X$ and $N$ be as above. (1) If $X$ is geometrically irreducible $k$-scheme of dim $d$, then $H^i_c(X \otimes_k \bar{k}, \mathbb{Z} / N \mathbb{Z})(d) \cong \mathbb{Z} / N \mathbb{Z}$. (2) If in addition $X/k$ is smooth, then for any integers $a,b$ with $a+b=d$, and any $i$, the cup-product pairing $H^i_c(a) \times H^{2d-i}(b) \to H^{2d}_c(d)$ is a Galois equivariant pairing which identifies each of the two groups with the $\mathbb{Z} / N \mathbb{Z}$-dual of the other. (Here the argument of the cohomology groups is the same in all three places, and omitted).

Application of PD: Can define the cohomology class of a subvariety. Detials omitted here.

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Notation

Etale cohomology of the scheme $X$ with coefficients in the sheaf $\mathcal{F}$:

$H_{et}^i (X, \mathcal{F})$

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Etale cohomology

Jardine: Cup products in sheaf cohomology

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Etale cohomology

Tamme

SGA4 1/2, and SGA4

Milne’s book

Motives volumes article

EMS Vol 35 (Alg geom II), ed Shafarevich. Covers cohomology of coherent sheaves and etale cohomology. In Cohomology folder under AG.

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Etale cohomology

Regarding what happens when p is nnot invertible, see maybe this MO question. http://mathoverflow.net/questions/49887/does-one-need-l-to-be-invertible-in-s-in-order-to-consider-the-l-adic-cohomology

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Etale cohomology

http://mathoverflow.net/questions/6070/etale-cohomology-why-study-it

http://mathoverflow.net/questions/16195/applications-of-etale-cohomology

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Etale cohomology

arXiv:1207.3648 Travaux de Gabber sur l’uniformisation locale et la cohomologie etale des schemas quasi-excellents. Seminaire a l’Ecole polytechnique 2006–2008 fra arXiv Front: math.AG av Luc Illusie, Yves Laszlo, Fabrice Orgogozo This book contains notes of a seminar on Ofer Gabber’s work on the etale cohomology and uniformization of quasi-excellent schemes. His main results include (cf. introduction) constructibility theorems (for abelian or non-abelian coefficients), vanishing theorems (e.g. affine Lefschetz), uniformization for the “prime-to-l alteration topology”, rigidity for non-abelian coefficients, a new proof of the absolute purity conjecture, duality, etc.

For the Bloch-Ogus-Gabber theorem, see http://www.math.uiuc.edu/K-theory/0169

Kahn: We prove some finiteness theorems for the 'étale cohomology, Borel-Moore homology and cohomology with proper supports of schemes of finite type over a finite or p-adic field. This yields vanishing results for their l-adic cohomology, proving part of a conjecture of Jannsen.

K-theory of semi-local rings with finite coefficients and étale cohomology, by Bruno Kahn

Variations on the Bloch-Ogus Theorem, by Ivan Panin and Kirill Zainoulline: http://www.math.uiuc.edu/K-theory/0556

Hoobler on the étale and Galois cohomology of semi-local rings

Probably not so interesting, some thing about mod 2 étale cohomology of real varieties: http://www.springerlink.com/content/x66u20605w22/?p=3c80b59be1374aea82a8a70beebece54&pi=324

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Etale cohomology

My own essay (put this online)

Milne’s online notes

A great list of links by Bhatt I think.

http://mathoverflow.net/questions/80633/textbook-for-etale-cohomology

nLab page on Etale cohomology