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

P. Domański; J. Krone; D. Vogt

ArticleOriginal scientific text

Title

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

Authors ¹, ², ³

Affiliations

Faculty of Mathematics and Computer Science, A. Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland
Fachbereich Physikalische Technik, Märkische Fachhochschule Iserlohn, Frauenstuhlweg 31, D-58644 Iserlohn, Germany
Bergische Universität, Gesamthochschule Wuppertal, FB Mathematik, Gaussstr. 20, D-42097 Wuppertal, Germany

Abstract

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

... \to λ (A) \to λ (A) \to λ (A) \to λ (A) \to {E_{n}} \to 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 \to λ (A) \to λ (A) \to {E_{n}} \to 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.

Keywords

ENG

Fréchet spaces, Köthe sequence spaces, splitting of short exact sequences, nuclear spaces, Schwartz spaces, quasinormable spaces, functor

E x t^{1}

, projective spaces, projective resolution

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Title

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

Affiliations

Abstract

Keywords

Bibliography