## Algebraic & Geometric Topology

### Hochschild homology, Frobenius homomorphism and Mac Lane homology

Teimuraz Pirashvili

#### Abstract

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.

#### Article information

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

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

https://projecteuclid.org/euclid.agt/1513796714

Digital Object Identifier
doi:10.2140/agt.2007.7.1071

Mathematical Reviews number (MathSciNet)
MR2336249

Zentralblatt MATH identifier
1137.19001

#### Citation

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. https://projecteuclid.org/euclid.agt/1513796714

#### References

• M Bökstedt, The topological Hochschild homology of $\mathbb{Z}$ and $\mathbb{Z}/p$, unpublished manuscript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