Holmstrom Bloch-Ogus cohomology

Bloch-Ogus cohomology

The Bloch-Ogus–Gabber Theorem, by Jean-Louis Colliot-Thélène, Raymond T. Hoobler, and Bruno Kahn, http://www.math.uiuc.edu/K-theory/0169:

In this paper, we give an exposition of Gabber’s proof of the Bloch-Ogus theorem for étale cohomology with locally constructible torsion coefficients. We then abstract the ingredients of the proof to give an axiomatic treatment of it. The axioms involved are much less demanding than those of Bloch and Ogus and therefore apply to a vaster array of cohomology theories. We also give a detailed treatment of universal exactness à la Grayson, as well as several applications.

This paper has appeared in Fields Institute for Research in Mathematical Sciences Communications Series 16, A.M.S., 1997, 31-94, so the dvi files have been removed. See http://www.fields.utoronto.ca/pubs.html.

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Bloch-Ogus cohomology

This is a set of axioms, for details see Levine, page 24.

The axioms describe a functor which to each object of (the big Zariski site) $Sm_k$ associates a bigraded object $\oplus_{p,q} H^p_{\Gamma}(X, q)$. The axioms require among other things:

• Some kind of “relative” version for closed subsets of $X$
• A product
• Homotopy invariance
• Purity
• Cycle classes $[W] \in H^{2q}_{\Gamma, W}(X,q)$ where $q = codim (W)$

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Bloch-Ogus cohomology

Can define Chern classes $c^{q,p}_{\Gamma}: K_{2q-p}(X) \to H^p_{\Gamma}(X, q)$

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Bloch-Ogus cohomology

Memo notes from Bloch and Ogus: Gersten’s conjecture and the homology of schemes

The whole article is online here.

0. Intro

Let $H^i(X)$ be some cohomology group of a smooth algebraic variety. “The deepest conjectures in algebraic geometry (Weil, Hodge, Tate) are attempts to calculate the ”arithmetic filtration“ of such a group”. This is also called the coniveau filtration, and is defined by

The above conjectures assert that this mysterious filtration is contained in or equal to some other filtration which can actually be computed.

The coniveau filtration is the filtration of a natural spectral sequence, whose $E_1$ term is (Grothendieck)

the sum taken over the points of codimension $p$.

Our main result is an expression for the $E_2$-term. Regard $H^{q-p}(k(x))$ as a constant sheaf on $\bar{ \{ x \} }$ and extend it by zero to $X$. The differentials of the spectral sequence give a complex of sheaves on $X$:

where $\mathcal{H}^q$ is the sheaf associated to $U \mapsto H^q(U)$. Our theorem says that this sequence is exact.

Some consequences of this:

• $E^{pq}_2 = H^p(X, \mathcal{H}^q)$.
• $H^p(X, \mathcal{H}^q) = 0$ if $p > q$.
• In the case of de Rham cohomology over a field of characteristic zero, the spectral sequence coincides from $E_2$ on with “the second spectral sequence of hypercohomology”. In particular, the two filtrations are the same.
• We have $H^p(X, \mathcal{H}^p) = A^p(X) \otimes H^0(pt)$ in the etale and de Rham theories, where $A^p(X)$ is the group of cycles mod algebraic equivalence.
• $H^0(X, \mathcal{H}^q)$ can be identified with the space of cohomology classes of the second kind, in the sense of Lefschetz.

Organization of the paper.

Our axioms will hold for “etale cohomology, de Rham cohomology, and simgular cohomology of associated analytic space”.

Intellectual debt to Gersten and Quillen.

1. Poincare duality with supports

Let $V$ be a category of schemes of finite type over a fixed ground field $k$, containing all quasi-projective $k$-schemes. We assume that locally closed subschemes of objects in $V$ are in $V$.

We shall describe the axioms a cohomology functor on $V$ must satisfy in order for a “theory of coniveau” to exist. Ref to Grothendieck: Le groupe de Brauer I, II, III, in Dix exposes; for a “theory of coniveau”. These are consequences of a satisfactory theory of $f_{!}$ and $f^{!}$.

For the details, see the original article. Here only very brief notes:

Def $V^*$ as the category of closed immersions and cartesian squares.

Def: Twisted cohomology theory with support, twisted homology theory. PD theory with supports.

Some basic properties of a PD theory.

2. Examples

Etale cohomology. De Rham cohomology. Betti cohomology (with coeffs in any ring).

3.Filtration by niveau and coniveau

Defs of filtrations. Homology spectral sequence (“analogous to that of a simplicial complex by skeletons”). Cohomological spectral sequence.

4. The arithmetic resolution

Including the main result.

5. Proof of “4.5”

6. Applications: the filtration by coniveau

7. Algebraic cycles

8. Differentials of the second kind

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Bloch-Ogus cohomology

Bloch and Ogus

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Bloch-Ogus cohomology

It might be the case that Gersten’s conjecture holds for all Bloch-Ogus theories. See section 2.5.7 in Gillet: K-theory and Intersection theory

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Bloch-Ogus cohomology

MathSciNet

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Bloch-Ogus cohomology

AG (Algebraic geometry)

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Bloch-Ogus cohomology

Mixed

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Bloch-Ogus cohomology

An axiomatic framework for cohomology theories in algebraic geometry. Examples constructed using Sheaf cohomology

Examples

I wrote these examples a long time ago: l-adic cohomology, Singular cohomology, de Rham cohomology, Deligne cohomology, Motivic cohomology, but should the first three really be there?

The universal example is supposed to be Motivic cohomology. See also Motivic homology, Motivic cohomology with compact supports

Morphic cohomology / Topological cycle cohomology, Lawson homology

Deligne cohomology, Deligne homology. See also Absolute Hodge cohomology, Deligne-Beilinson cohomology

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Bloch-Ogus cohomology

There are several articles of Barbieri-Viale on H-cohomology and other things, in which he uses the formalism of Bloch-Ogus cohomology. See MathSciNet reviews - most of these articles are not online in Feb 2009.

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Bloch-Ogus cohomology

Kato defined Bloch-Ogus like complexes for any excellent scheme, see http://www.ams.org/mathscinet-getitem?mr=0833016

nLab page on Bloch-Ogus cohomology