Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Third Mac Lane cohomology

HANS–JOACHIM BAUESa1, MAMUKA JIBLADZEa2 and TEIMURAZ PIRASHVILIa3

a1 Max-Planck-Institut für Mathematik, Vivatsgasse 7, Bonn 53111, Germany. e-mail: baues@mpim-bonn.mpg.de

a2 Razmadze Mathematical Institute, Alexidze st. 1, Tbilisi 0193, Georgia. e-mail: jib@rmi.acnet.ge

a3 Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH. e-mail: tp59-at-le.ac.uk

Abstract

MacLane cohomology is an algebraic version of the topological Hochschild cohomology. Based on the computation of the third author (see Appendix) we obtain an interpretation of the third Mac Lane cohomology of rings using certain kind of crossed extensions of rings in the quadratic world. Actually we obtain two such interpretations corresponding to the two monoidal structures on the category of square groups.

(Received April 26 2006)

(Revised March 05 2007)

(Online publication February 11 2008)