Limits and the Ext functor

Abstract

We show the identity Ext (lim← Xα, ℝ) = lim→ Ext(Xα;ℝ) for projective limits of quasi-Banach spaces X𝛼. The proof is derived from a pull-back lemma asserting that a topologically exact sequence 0→ℝ→E→ Z→ 0 of locally pseudoconvex spaces is the pull-back of an exact sequence of quasi-Banach spaces. Among the consequences we show that exact sequences 0→ℝ→E→Z→0 of locally pseudoconvex spaces come induced by quasi-linear maps, which extends a result of Kalton for Fréchet spaces; and that projective limits of 𝘒-spaces are 𝘒-spaces.