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Izv. RAN. Ser. Mat., 2012, Volume 76, Issue 4, Pages 65–124 (Mi izv6792)  

This article is cited in 3 scientific papers (total in 3 papers)

Homological dimensions and Van den Bergh isomorphisms for nuclear Fréchet algebras

A. Yu. Pirkovskii

National Research University "Higher School of Economics"

Abstract: We prove the equation $\operatorname{w{.}dg} A=\operatorname{w{.}db} A$ for every nuclear Fréchet–Arens–Michael algebra $A$ of finite weak bidimension, where $\operatorname{w{.}dg} A$ is the weak global dimension and $\operatorname{w{.}db} A$ the weak bidimension of $A$. Assuming that $A$ has a projective bimodule resolution of finite type, we establish the estimate $\operatorname{db}A\le\operatorname{dg}A+1$, where $\operatorname{dg} A$ is the global dimension and $\operatorname{db} A$ the bidimension of $A$. We also prove that $\operatorname{dg}A=\operatorname{db}A=\operatorname{w{.}dg}A= \operatorname{w{.}db} A=n$ for all nuclear Fréchet–Arens–Michael algebras satisfying the Van den Bergh conditions $\operatorname{VdB}(n)$. As an application, we calculate the homological dimensions of smooth and complex-analytic quantum tori.

Keywords: nuclear Fréchet algebra, global dimension, bidimension, Van den Bergh isomorphisms, Hochschild homology.

DOI: https://doi.org/10.4213/im6792

Full text: PDF file (1074 kB)
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English version:
Izvestiya: Mathematics, 2012, 76:4, 702–759

Bibliographic databases:

UDC: 517.986.2+512.664.2
MSC: 46M18, 46A04, 46H05, 18G20
Received: 19.01.2011
Revised: 20.04.2011

Citation: A. Yu. Pirkovskii, “Homological dimensions and Van den Bergh isomorphisms for nuclear Fréchet algebras”, Izv. RAN. Ser. Mat., 76:4 (2012), 65–124; Izv. Math., 76:4 (2012), 702–759

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. Yu. Pirkovskii, “Homological dimensions of modules of holomorphic functions on submanifolds of Stein manifolds”, J. Funct. Anal., 266:12 (2014), 6663–6683  crossref  mathscinet  zmath  isi  scopus
    2. Pirkovskii A.Yu., “Holomorphically Finitely Generated Algebras”, J. Noncommutative Geom., 9:1 (2015), 215–264  crossref  mathscinet  zmath  isi  elib  scopus
    3. St. Petersburg Math. J., 31:4 (2020), 607–656  mathnet  crossref  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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