A Sufficient Condition for Proj~1 X = 0

摘要

In [7], Palamodov established a homological theory for projective spectra of topological vector spaces. In applications of this theory, it is crucial to decide whether, for a given projective spectrum X of (DFS) spaces, a certain vector space Proj~1 X is trivial. A topological characterization of Proj~1 X = 0 has been given by Retakh. In practical cases, its evaluation is hard. In Vogt, more tractable conditions were given, which were motivated from the structure theory of nuclear Frechet spaces. There is a sufficient as well as a necessary condition, but these are probably different. In the case of sequence spaces, it is shown in [9] that the necessary condition is also sufficient. Recently, Wengenroth has proved the sufficiency of the necessary condition also for (DFM) spectra. His proof is based on the investigation of topological properties of the dual inductive spectrum. In the present paper, we give a direct proof of Wengenroth's result for the case of (DFS) spectra. It grew out of a third condition, the sufficiency of which was shown in Braun.