Standard exact projective resolutions relative to a countable class of Fréchet spaces

P. Domański; J. Krone; D. Vogt

Studia Mathematica (1997)

  • Volume: 123, Issue: 3, page 275-290
  • ISSN: 0039-3223

Abstract

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We will show that for each sequence of quasinormable Fréchet spaces ( E n ) ℕ there is a Köthe space λ such that E x t 1 ( λ ( A ) , λ ( A ) = E x t 1 ( λ ( A ) , E n ) = 0 and there are exact sequences of the form . . . → λ ( A ) → λ ( A ) → λ ( A ) → λ ( A ) → E n → 0 . If, for a fixed ℕ, E n is nuclear or a Köthe sequence space, the resolution above may be reduced to a short exact sequence of the form 0 → λ ( A ) → λ ( A ) → E n → 0 . The result has some applications in the theory of the functor E x t 1 in various categories of Fréchet spaces by providing a substitute for non-existing projective resolutions.

How to cite

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Domański, P., Krone, J., and Vogt, D.. "Standard exact projective resolutions relative to a countable class of Fréchet spaces." Studia Mathematica 123.3 (1997): 275-290. <http://eudml.org/doc/216393>.

@article{Domański1997,
abstract = {We will show that for each sequence of quasinormable Fréchet spaces $(E_n)_ℕ$ there is a Köthe space λ such that $Ext^1(λ(A), λ(A) = Ext^1 (λ(A), E_n)=0$ and there are exact sequences of the form $... → λ(A) → λ(A) → λ(A) → λ(A) → \{E_n\} → 0$. If, for a fixed ℕ, $E_n$ is nuclear or a Köthe sequence space, the resolution above may be reduced to a short exact sequence of the form $0 → λ(A) → λ(A) → \{E_n\} → 0$. The result has some applications in the theory of the functor $Ext^1$ in various categories of Fréchet spaces by providing a substitute for non-existing projective resolutions.},
author = {Domański, P., Krone, J., Vogt, D.},
journal = {Studia Mathematica},
keywords = {Fréchet spaces; Köthe sequence spaces; splitting of short exact sequences; nuclear spaces; Schwartz spaces; quasinormable spaces; functor $Ext^1$; projective spaces; projective resolution; quasinormable Fréchet spaces; Köthe space; exact sequences; functor ; substitute for nonexisting projective resolutions},
language = {eng},
number = {3},
pages = {275-290},
title = {Standard exact projective resolutions relative to a countable class of Fréchet spaces},
url = {http://eudml.org/doc/216393},
volume = {123},
year = {1997},
}

TY - JOUR
AU - Domański, P.
AU - Krone, J.
AU - Vogt, D.
TI - Standard exact projective resolutions relative to a countable class of Fréchet spaces
JO - Studia Mathematica
PY - 1997
VL - 123
IS - 3
SP - 275
EP - 290
AB - We will show that for each sequence of quasinormable Fréchet spaces $(E_n)_ℕ$ there is a Köthe space λ such that $Ext^1(λ(A), λ(A) = Ext^1 (λ(A), E_n)=0$ and there are exact sequences of the form $... → λ(A) → λ(A) → λ(A) → λ(A) → {E_n} → 0$. If, for a fixed ℕ, $E_n$ is nuclear or a Köthe sequence space, the resolution above may be reduced to a short exact sequence of the form $0 → λ(A) → λ(A) → {E_n} → 0$. The result has some applications in the theory of the functor $Ext^1$ in various categories of Fréchet spaces by providing a substitute for non-existing projective resolutions.
LA - eng
KW - Fréchet spaces; Köthe sequence spaces; splitting of short exact sequences; nuclear spaces; Schwartz spaces; quasinormable spaces; functor $Ext^1$; projective spaces; projective resolution; quasinormable Fréchet spaces; Köthe space; exact sequences; functor ; substitute for nonexisting projective resolutions
UR - http://eudml.org/doc/216393
ER -

References

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