Holmstrom D10 Axiom systems for CTs in algebraic geometry

Discuss how and why CTs in AG are bigraded.

In order to give an overview of cohomology theories in algebraic geometry, we shall in later posts talk more about the following frameworks: Motivic stable homotopy theory and motives, Noncommutative motives.

In this post, I want to talk about various axiom systems that have been proposed to describe cohomology theories in algebraic geometry.

The book of Levine and More describes axioms for oriented cohomology, weak oriented cohomology, and the dual notions of oriented Borel-Moore homology and weak oriented B-M homology. It also explains universal properties of algebraic cobordism (both as a oriented cohomology and as a oriented Borel-Moore homology), as well as universal property of algebraic K-theory (or maybe $K_0[\beta, \beta^{-1}]$) and maybe also motivic cohomology.

Something by Tabuada explains how 4 versions of algebraic K-theory are universal with respect to certain axioms.

Other axiom systems

Pretheory

Mixed Weil cohomology

Twisted duality theory: See Gillet: Riemann-Roch theorems for higher algebraic K-theory (1981)

Geometric cohomology. This can be in the sense of Nekovar, with no precise axioms, or in the sense of Levine in K-th handbook.

See files in Lipman folder for the notion of bivariant cohomology theory.

Axiom systems

Gillet cohomology, Bloch-Ogus cohomology. I think some Russian referred to “Beilinson’s axioms” and one of the classical papers, maybe the Modular curves one.

nLab page on D10 Axiom systems for CTs in algebraic geometry