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The Bloch-Ogus-Gabber theorem. (English) Zbl 0911.14004
Snaith, Victor P. (ed.), Algebraic K-theory. Papers from the 2nd Great Lakes conference, Canada, March 1996, in memory of Robert Wayne Thomason. Providence, RI: American Mathematical Society. Fields Inst. Commun. 16, 31-94 (1997).
Given a smooth variety, $$X$$, and a suitable cohomology theory, $$h^*$$, filtration by codimension of support yields the coniveau spectral sequence convergent to $$h^*(X)$$, whose $$E_1$$-term is the Cousin complex. The restriction of the Cousin complexes to open subsets of $$X$$ defines complexes of flasque Zariski sheaves. The Bloch-Ogus theorem says that these complexes of sheaves are acyclic and thereby identifies the $$E_2$$-term as $$H^*_{\text{Zar}} (X; \underline h^*)$$.
This is a central theorem in modern algebraic geometry. The original proof appeared in a paper by S. Bloch and A. Ogus [Ann. Sci. Éc. Norm. Supér., IV. Sér. 7 (1974), 181-201 (1975; Zbl 0307.14008)]. Subsequently O. Gabber gave a proof [Manuscr. Math. 85, No. 3-4, 323-343 (1994; Zbl 0827.19002)] which makes it clear that the result holds very generally. For this reason the authors have given an axiomatic treatment of the proof, which yields a number of extensions en route.
For the entire collection see [Zbl 0871.00037].

##### MSC:
 14F20 Étale and other Grothendieck topologies and (co)homologies 19E08 $$K$$-theory of schemes 19E20 Relations of $$K$$-theory with cohomology theories
Cousin complex