Algebraic & Geometric Topology

Hochschild homology, Frobenius homomorphism and Mac Lane homology

Teimuraz Pirashvili

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We prove that Hi(A,Φ(A))=0, i>0. Here A is a commutative algebra over the prime field Fp of characteristic p>0 and Φ(A) is A considered as a bimodule, where the left multiplication is the usual one, while the right multiplication is given via Frobenius endomorphism and H denotes the Hochschild homology over Fp. This result has implications in Mac Lane homology theory. Among other results, we prove that HML(A,T)=0, provided A is an algebra over a field K of characteristic p>0 and T is a strict homogeneous polynomial functor of degree d with 1<d<Card(K).

Article information

Algebr. Geom. Topol., Volume 7, Number 2 (2007), 1071-1079.

Received: 14 March 2007
Accepted: 26 March 2007
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.) 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
Secondary: 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60] 55U10: Simplicial sets and complexes

Hochschild Homology Mac Lane homology


Pirashvili, Teimuraz. Hochschild homology, Frobenius homomorphism and Mac Lane homology. Algebr. Geom. Topol. 7 (2007), no. 2, 1071--1079. doi:10.2140/agt.2007.7.1071.

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  • M Bökstedt, The topological Hochschild homology of $\mathbb{Z}$ and $\mathbb{Z}/p$, unpublished manuscript
  • L Breen, Extensions du groupe additif, Inst. Hautes Études Sci. Publ. Math. 48 (1978) 39–125
  • L L Èsakia, Heyting algebras I: Duality theory (Russian), Metsniereba, Tbilisi (1985)
  • V Franjou, E M Friedlander, T Pirashvili, L Schwartz, Rational representations, the Steenrod algebra and functor homology, Panoramas et Synthèses 16, Société Mathématique de France, Paris (2003)
  • V Franjou, E M Friedlander, A Scorichenko, A Suslin, General linear and functor cohomology over finite fields, Ann. of Math. $(2)$ 150 (1999) 663–728
  • V Franjou, J Lannes, L Schwartz, Autour de la cohomologie de Mac Lane des corps finis, Invent. Math. 115 (1994) 513–538
  • E M Friedlander, A Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997) 209–270
  • M Jibladze, T Pirashvili, Cohomology of algebraic theories, J. Algebra 137 (1991) 253–296
  • J-L Loday, Cyclic homology, second edition, Grundlehren der Mathematischen Wissenschaften 301, Springer, Berlin (1998)
  • S Mac Lane, Homology, Grundlehren der Mathematischen Wissenschaften 114, Springer, Berlin (1963)
  • T I Pirashvili, Higher additivizations, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988) 44–54
  • T Pirashvili, Spectral sequence for Mac Lane homology, J. Algebra 170 (1994) 422–428
  • T Pirashvili, F Waldhausen, Mac Lane homology and topological Hochschild homology, J. Pure Appl. Algebra 82 (1992) 81–98
  • D Quillen, On the (co-) homology of commutative rings, from: “Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968)”, Amer. Math. Soc., Providence, R.I. (1970) 65–87
  • S Schwede, Stable homotopy of algebraic theories, Topology 40 (2001) 1–41