Holmstrom G20 Cohomology of algebraic objects

Some key sources:

• Weibel’s historical survey
• Tabuada’s noncommutative motives
• Maybe operadic approaches
• Barr: Algebraic cohomology - the early days

Amitsur cohomology, Andre-Quillen cohomology, Andre-Quillen homology, CCG-homology (for crossed modules)

Eilenberg-MacLane cohomology

MacLane homology and MacLane cohomology

Cohomology of (algebras over) operads Cohomology of algebraic theories Cohomology of associative algebras

Cohomology of commutative rings

Cohomology of homotopy Lie algebras Cohomology of monoids Cohomology of posets Cohomology of rings Comonadic cohomology

Chevalley-Eilenberg cohomology Chevalley-Hochschild cohomology

Chamber homology

Dialgebra cohomology

Dihedral cohomology

Lie algebra cohomology, Lie-Rinehart cohomology

Cohomology of groups, topological groups and algebraic groups

Group cohomology

Generic cohomology, Rational cohomology (see Inventiones paper linked to from Generic cohomology page)

Galois cohomology

Gamma homology

Harrison cohomology

Hochschild cohomology, Hochschild homology

Hochschild-Witt homology (noncommutative analogue of the de Rham-Witt complex?)

Homology of linear groups

Left invariant cohomology

Continuous group cohomology, Continuous bounded cohomology and Continuous cohomology (I think these are for topological groups, or maybe algebraic groups/group schemes. Create separate section for these, or join in with group cohomology?)

Local Hochschild homology

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arXiv:1206.0178 Algebraic Kasparov K-theory. II from arXiv Front: math.AT by Grigory Garkusha A kind of motivic stable homotopy theory of algebras is developed. Explicit fibrant replacements for the S^1-spectrum and (S^1,\G)-bispectrum of an algebra are constructed. As an application, unstable, Morita stable and stable universal bivariant theories are recovered. These are shown to be embedded by means of contravariant equivalences as full triangulated subcategories of compact generators of some compactly generated triangulated categories. Another application is to introduce and study the symmetric monoidal compactly generated triangulated category of K-motives. It is established that the triangulated category kk of Cortinas-Thom can be identified with K-motives of algebras. It is proved that the triangulated category of K-motives is a localization of the triangulated category of (S^1,\G)-bispectra. Also, explicit fibrant (S^1,\G)-bispectra representing stable algebraic Kasparov K-theory and algebraic homotopy K-theory are constructed.

nLab page on G20 Cohomology of algebraic objects