Bloch–Ogus theory

Bloch–Ogus theory
Name	Bloch–Ogus theory
Field	Algebraic geometry, Algebraic topology
Introduced	1970s
Contributors	Spencer Bloch, Arthur Ogus
Related	Étale cohomology, Motivic cohomology, Gersten conjecture

Bloch–Ogus theory describes a framework connecting sheaf cohomology, algebraic K-theory, and duality for schemes via axiomatic properties of cohomology theories, establishing comparison theorems and spectral sequences that generalize classical statements of Poincaré duality, purity, and descent. The theory links ideas from Spencer Bloch, Arthur Ogus, Alexander Grothendieck, Jean-Pierre Serre, and Alexander Beilinson and has influenced developments in Étale cohomology, Motivic cohomology, Algebraic cycles, and the proof of cases of the Gersten conjecture.

Introduction

Bloch–Ogus theory provides axioms for a Zariski or étale sheafified cohomology on schemes that yield long exact localization sequences and coniveau spectral sequences, enabling comparison between cohomology with supports and cohomology of local rings; prominent contributors include Spencer Bloch, Arthur Ogus, Alexander Grothendieck, Jean-Pierre Serre, and Pierre Deligne. The axiomatic approach formalizes compatibility conditions such as homotopy invariance, purity, and excision that appear in contexts studied by John Milnor, Benoît Mandelbrot, André Weil, Sergey Novikov, and William Fulton. The central outputs are Gersten-type resolutions and spectral sequences that bridge local-to-global principles exploited by Alexander Beilinson, Vladimir Voevodsky, and Charles Weibel.

Background and motivation

The origins trace to attempts by Spencer Bloch and Arthur Ogus to systematize cohomological properties observed in Étale cohomology, Algebraic K-theory, and de Rham cohomology for regular schemes and to give unified proofs of purity and duality results appearing in work of Alexander Grothendieck and Pierre Deligne. Tasks motivating the theory included establishing Gersten resolutions conjectured by Jean-Pierre Serre and structural features used in the study of algebraic cycles by William Fulton and Robert MacPherson. The framework was influenced by analogies with classical results from Henri Poincaré, Emmy Noether, and later categorical insights from Grothendieck, which guided applications in the work of Vladimir Voevodsky and Alexander Beilinson.

Bloch–Ogus axioms and statements

The Bloch–Ogus axioms enumerate functoriality, Mayer–Vietoris, purity, long exact localization sequences, and homotopy invariance for a cohomology theory on schemes; these axioms mirror properties exploited by Pierre Deligne, Jean-Pierre Serre, Alexander Grothendieck, and Grothendieck school-affiliated researchers. The core statements include the coniveau spectral sequence linking cohomology of a scheme to cohomology with support in its strata, and Gersten-type exact sequences for regular local rings, results utilized by Charles Weibel, Suslin, Vladimir Voevodsky, and Spencer Bloch. Key formal theorems assert that for theories satisfying the axioms one obtains purity isomorphisms and duality compatibilities reminiscent of results by Henri Cartan, Élie Cartan, and Jean Leray as adapted to the algebraic context by Grothendieck and Deligne.

Applications and consequences

Bloch–Ogus theory underpins proofs of the Gersten conjecture in many cases used by Charles Weibel and Hyman Bass, informs computations in Motivic cohomology pursued by Alexander Beilinson and Vladimir Voevodsky, and supports duality frameworks applied by Jean-Pierre Serre and Pierre Deligne. It clarifies relationships exploited in the study of algebraic cycles by Spencer Bloch and William Fulton, and plays a role in comparisons between de Rham cohomology, crystalline cohomology, and Étale cohomology in work of Pierre Berthelot, Luc Illusie, and Arthur Ogus. Consequences include explicit spectral sequences used in computations by Charles Weibel, the establishment of purity statements used by Kazuya Kato and Shuji Saito, and structural inputs to the theory of regulators and special values as pursued by Alexander Beilinson and Don Zagier.

Examples and computations

Concrete instances appear for smooth varieties over fields where Bloch–Ogus machinery yields Gersten resolutions for Algebraic K-theory shown in work of Hyman Bass and Charles Weibel, and for étale sheaf cohomology computations in treatments by Pierre Deligne and Jean-Pierre Serre. Computations of coniveau spectral sequences for projective smooth surfaces feature in analyses by Spencer Bloch and William Fulton, while comparisons with Motivic cohomology and higher Chow groups were developed by Bloch, Beilinson, and Vladimir Voevodsky. Examples in mixed characteristic exploit techniques from Pierre Berthelot, Luc Illusie, and Arthur Ogus to handle crystalline and de Rham comparisons and arithmetic applications pursued by Kazuya Kato.

Relations to other theories

Bloch–Ogus theory intersects with Étale cohomology, Algebraic K-theory, Motivic cohomology, and duality theories elaborated by Alexander Grothendieck and Pierre Deligne, and connects to conjectures of Beilinson, Bloch–Kato conjecture contexts treated by Kazuya Kato, and Vladimir Voevodsky's foundations for motivic homotopy. The axiomatic framework informs Gersten-type approaches used by Charles Weibel and Hyman Bass, and integrates with comparison theorems in crystalline cohomology developed by Pierre Berthelot and Luc Illusie. Interactions with regulator maps and special value conjectures appear in the work of Alexander Beilinson and Don Zagier.

Historical development and contributors

The theory originated in joint work by Spencer Bloch and Arthur Ogus in the 1970s, building on foundations laid by Alexander Grothendieck, Jean-Pierre Serre, and Pierre Deligne. Subsequent contributors who advanced applications and generalizations include Alexander Beilinson, Vladimir Voevodsky, Charles Weibel, Hyman Bass, Kazuya Kato, Pierre Berthelot, Luc Illusie, and William Fulton. Later developments connecting Bloch–Ogus ideas to Motivic cohomology and modern homotopical techniques were advanced by Vladimir Voevodsky, Alexander Beilinson, and researchers in the Grothendieck-style school, influencing contemporary work in algebraic cycles, arithmetic geometry, and cohomological methods.

Category:Algebraic geometry