Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-30T19:34:52.936Z Has data issue: false hasContentIssue false
  • English
  • Français

Third Mac Lane cohomology

Published online by Cambridge University Press:  01 March 2008

HANS–JOACHIM BAUES ,
MAMUKA JIBLADZE  and
TEIMURAZ PIRASHVILI
HANS–JOACHIM BAUES
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, Bonn 53111, Germany. e-mail: [email protected]
MAMUKA JIBLADZE
Affiliation:
Razmadze Mathematical Institute, Alexidze st. 1, Tbilisi 0193, Georgia. e-mail: [email protected]
TEIMURAZ PIRASHVILI
Affiliation:
Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH. e-mail: tp59-at-le.ac.uk
Get access Rights & Permissions [Opens in a new window]

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 2 , March 2008 , pp. 337 - 367
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Baues, H.-J.. The algebra of secondary cohomology operations. Progr. Math. 297 (2006), 483 pp.Google Scholar
[2]Baues, H.-J.. The homotopy category of simply connected 4-manifolds. London Math. Soc. Lecture Note Series 297 (Cambridge University Press, 2003), xii+184 pp.Google Scholar
[3]Baues, H.-J.. Combinatorial homotopy and 4-dimensional complexes. De Gruyter Expositions in Math 2 (de Gruyter, 1991), 380 pp.Google Scholar
[4]Baues, H.-J. and Dreckmann, W.. The cohomology of homotopy categories and the general linear group. K-theory 3 (1989), 307–338.CrossRefGoogle Scholar
[5]Baues, H.-J.Hartl, M. and Pirashvili, T.. Quadratic categories and square rings. J. Pure Appl. Algebra 122 (1997), 1–40.Google Scholar
[6]Baues, H.-J. and Iwase, N.. Square rings associated to elements in homotopy groups of spheres. Contemp. Math. 274 (2001), 57–78.Google Scholar
[7]Baues, H.-J.Jibladze, M. and Pirashvili, T.. Quadratic algebra of square groups. Preprint MPIM2006-9.Google Scholar
[8]Baues, H.-J. and Minian, E. C.. Crossed extensions of algebras and Hochschild cohomology. Homology, Homotopy Appl. 4 (2002), 63–82.Google Scholar
[9]Baues, H.-J. and Pirashvili, T.. Quadratic endofunctors of the category of groups. Adv. Math. 141 (1999), 167–206.Google Scholar
[10]Baues, H.-J. and Pirashvili, T.. A universal coefficient theorem for quadratic functors. J. Pure Appl. Algebra 148 (2000), 1–15.Google Scholar
[11]Baues, H.-J. and Pirashvili, T.. Comparison of Mac Lane, Shukla and Hochschild cohomologies. J. Reine Angew. Math. 598 (2006), 25–69.Google Scholar
[12]Baues, H.-J. and Wirsching, G.. Cohomology of small categories. J. Pure Appl. Algebra 38 (1985), 187–211.CrossRefGoogle Scholar
[13]Eilenberg, S. and Lane, S. Mac. On the groups H(π,n), II. Ann. Math. 60 (1954), 49–139.Google Scholar
[14]Hochschild, G.On the cohomology groups of an associative algebra. Ann. of Math. (2) 46 (1945), 58–67.Google Scholar
[15]Janelidze, G.Internal crossed modules. Georgian Math. J. 10 (2003), 99–114.Google Scholar
[16]Jibladze, M. and Pirashvili, T.. Some linear extensions of a category of finitely generated free modules (russian). Soobshch. Akad. Nauk Gruzin. SSR 123 (1986), 481–484.Google Scholar
[17]Jibladze, M. and Pirashvili, T.. Cohomology of algebraic theories. J. Algebra 137 (1991), 253–296.Google Scholar
[18]Lazarev, A.Homotopy theory of A∞ ring spectra and applications to MU-modules. K-theory 24 (2001), 243–281.Google Scholar
[19]Loday, J.-L.. Spaces with finite many nontrivial homotopy groups. J. Pure Appl. Algebra 24 (1982), 179–202.Google Scholar
[20]Loday, J.-L.. Cyclic Homology (Second edition. Grundlehren der Mathematischen Wissenschaften) 301 (Springer, 1998), xx+513 pp.Google Scholar
[21]Mac Lane, S. Homologie des anneaux et des modules. Coll. Topologie Algebrique (Louvain, 1956), 55–80.Google Scholar
[22]Mac Lane, S.Extensions and obstructions for rings. Ill. J. Math. 2 (1958), 316–345.Google Scholar
[23]Mac Lane, S. and Whitehead, J. H. C.. On the 3-type of a complex. Proc. Nat. Acad. Sci. USA 36 (1950), 41–48.CrossRefGoogle Scholar
[24]Pirashvili, T.Higher additivizations (Russian). Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988), 44–54.Google Scholar
[25]Pirashvili, T.Models for the homotopy theory and cohomology of small categories (russian). Soobshch. Akad. Nauk Gruzin. SSR 129 (1988), 261–264.Google Scholar
[26]Pirashvili, T. Cohomology of small categories in homotopical algebra. In: K-theory and homological algebra (Tbilisi, 1987–88) Lecture Notes in Math. 1437 (Springer, 1990), 268–302.Google Scholar
[27]Pirashvili, T.. Polynomial approximation of and groups in functor categories. Comm. Algebra 21 (1993), 1705–1719.Google Scholar
[28]Pirashvili, T.. On the topological Hochschild homology of ℤ/p kℤ. Comm. Algebra 23 (1995), no. 4, 1545–1549.Google Scholar
[29]Pirashvili, T. On the cohomology of the category NIL. Appendix to [2].Google Scholar
[30]Pirashvili, T. and Waldhausen, F.. Mac Lane homology and topological Hochschild homology. J. Pure Appl. Algebra 82 (1992), 81–98.Google Scholar
[31]Quillen, D. G.Homotopical algebra. Lecture Notes in Math. 43 (Springer, 1967), iv+156 pp.Google Scholar
[32]Shukla, U.Cohomologie des algébres associatives. Ann. Sci. École Norm. Sup. (3) 78 (1961), 163–209.CrossRefGoogle Scholar
[33]Schwede, S.Stable homotopy of algebraic theories. Topology 40 (2001), 1–41.Google Scholar
[34]Shipley, B. HZ-algebra spectra are differential graded algebras. To appear in Amer. J. Math.Google Scholar