# HOMOLOGY THEORIES FOR MULTIPLICATIVE SYSTEMS

@article{Eilenberg1951HOMOLOGYTF, title={HOMOLOGY THEORIES FOR MULTIPLICATIVE SYSTEMS}, author={Samuel Eilenberg and Saunders Maclane}, journal={Transactions of the American Mathematical Society}, year={1951}, volume={71}, pages={294-330} }

with d[x]=0. It is convenient to augment ^4°(II) by regarding the commutator quotient group 11/ [II, II] as the group of O-dimensional chains, with d[x]=x[n, TI]. The complex ^4°(I1) occurs in a topological problem in which IT plays the role of the fundamental group of a space. An analogous problem in which the fundamental group is replaced by a higher homotopy group has led us to believe that there exists for abelian groups a specific homology theory distinct from the one described above for…

## 36 Citations

Chapter 13. Mac Lane (co)homology

- Mathematics
- 1998

The second Hochschild cohomology group of rings (that is algebras over k = Z) classifies the extensions of a ring by a bimodule provided that the ex tensions are split as abelian groups. In order to…

MacLane homology and topological Hochschild homology

- Mathematics
- 1995

The aim of this paper is to show that the topological Hochschild homology of a discrete ring R in the sense of [2] and the MacLane homology of R (see [lo] or [S]) are isomorphic. The method is to…

On The Homology Theory Of Abelian Groups

- MathematicsCanadian Journal of Mathematics
- 1955

1. Introduction. In (1) we have introduced the notions of “construction” and “ generic acyclicity” in order to determine a homology theory for any class of multiplicative systems defined by…

Mac Lane (co)homology

- Mathematics
- 1998

The second Hochschild cohomology group of rings (that is algebras over k = Z) classifies the extensions of a ring by a bimodule provided that the ex-tensions are split as abelian groups. In order to…

Formal groups and stable homotopy of commutative rings

- Mathematics
- 2004

We explain a new relationship between formal group laws and ring spectra in stable homotopy theory. We study a ring spectrum denoted DB which depends on a commutative ring B and is closely related to…

Duality for local fields and sheaves on the category of fields

- Mathematics
- 2013

Duality for complete discrete valuation fields with perfect residue field with coefficients in (possibly p-torsion) finite flat group schemes was obtained by Begueri, Bester and Kato. In this paper,…

Comparison of MacLane, Shukla and Hochschild cohomologies

- Mathematics
- 2006

It is known that MacLane cohomology coincides with topological Hochschild cohomology ([28]) and coincides also with Baues-Wirsching cohomology of the category of finitely generated free A-modules…

Homological finiteness of functors and applications

- Mathematics
- 2021

We give sufficient conditions which ensure that a functor of finite length from an additive category to finite-dimensional vector spaces has a projective resolution whose terms are finitely…

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and…

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- MathematicsProceedings of the National Academy of Sciences of the United States of America
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