A Künneth formula in topological homology and its applications to the simplicial cohomology of $ℓ¹(ℤ₊^{k})$

• Autorzy: F. Gourdeau , Z. A. Lykova , M. C. White
• Języki publikacji: EN

Abstrakty

EN

We establish a Künneth formula for some chain complexes in the categories of Fréchet and Banach spaces. We consider a complex 𝓧 of Banach spaces and continuous boundary maps dₙ with closed ranges and prove that Hⁿ(𝓧') ≅ Hₙ(𝓧)', where Hₙ(𝓧)' is the dual space of the homology group of 𝓧 and Hⁿ(𝓧') is the cohomology group of the dual complex 𝓧'. A Künneth formula for chain complexes of nuclear Fréchet spaces and continuous boundary maps with closed ranges is also obtained. This enables us to describe explicitly the simplicial cohomology groups $ℋⁿ(ℓ¹(ℤ₊^{k}),ℓ¹(ℤ₊^{k})')$ and homology groups $ℋₙ(ℓ¹(ℤ₊^{k}),ℓ¹(ℤ₊^{k}))$ of the semigroup algebra $ℓ¹(ℤ₊^{k})$.

Kategorie tematyczne

• 46J40: Structure, classification of commutative topological algebras
• 46H20: Structure, classification of topological algebras
• 43A20: L 1 -algebras on groups, semigroups, etc.
• 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
• 22E41: Continuous cohomology

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2005
• Tom: 166
• Numer: 1
• Strony: 29-54

Daty

wydano

2005

Twórcy

autor

F. Gourdeau

• Département de Mathématiques, Université Laval, Cité Universitaire (Québec), G1K 7P4, Canada

autor

Z. A. Lykova

• School of Mathematics and Statistics, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, UK

autor

M. C. White

• School of Mathematics and Statistics, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, UK

Identyfikatory

DOI

10.4064/sm166-1-3