ACYCLIC INDUCTIVE SPECTRA OF FRECHET SPACES

We provide new characterizations of acyclic inductive spectra of Frech et spaces which improve the classical theorem of Palamodov and Retakh. It turns out that acyclicity, sequential retractivity (defined by Flo ret) and further strong regularity conditions (introduced e.g. by Bier stedt and Meise) are all equivalent. This solves a problem that was fo lklore since around 1970. For inductive limits of Frechet-Montel space s we obtain even stronger results, in particular, Grothendieck's probl em whether regular (LF)-spaces are complete has a positive solution in this case and we show that even the weakest regularity conditions alr eady imply acyclicity. One of the main benefits from our results is an improvement in the theory of projective spectra of (DFM)-spaces. We p rove the missing implication in a theorem of Vogt and thus obtain eval uable conditions for vanishing of the derived projective limit functor which have direct applications to classical problems of analysis like surjectivity of partial differential operators on various classes of ultradifferentiable functions (as was explained e.g. by Braun, Meise a nd Vogt).