STANDARD EXACT PROJECTIVE-RESOLUTIONS RELATIVE TO A COUNTABLE CLASS OF FRECHET SPACES

We will show that for each sequence of quasinormable Frechet spaces (E -n)(n epsilon M) there is a Kothe space lambda(A) such that Ext(1)(lam bda(A),lambda(A)) = Ext(1)(lambda(A),E-n) = 0 and there are exact sequ ences of the form ... --> lambda(A) --> lambda(A) --> lambda(A) --> la mbda(A) --> E-n --> 0. If, for a fixed n epsilon N, E-n is nuclear or a Kothe sequence space, the resolution above may be reduced to a short exact sequence of the form 0 --> lambda(A) --> lambda(A) --> E-n --> 0. The result has some applications in the theory of the functor Ext(1 ) in various categories of Frechet spaces by providing a substitute fo r non-existing projective resolutions.