On the derived tensor product functors for (DF)- and Fréchet spaces

Abstract

Abstract. For a (DF)-space E and a tensor norm α we investigate the derivatives Tor l α (E, ·) of the tensor product functor E ⊗ α · : F S → LS from the category of Fréchet spaces to the category of linear spaces. Necessary and sufficient conditions for the vanishing of Tor 1 α (E, F ), which is strongly related to the exactness of tensored sequences, are presented and characterizations in the nuclear and (co-)echelon cases are given.

A decomposition lemma for elementary tensors

“…We will need some homological definitions. For details and results on this subject we refer to [19] and [18].…”

“…Instead of recalling the definition of the derivatives we remark: If the above tensor product functor is left exact, for instance this is true if E is a nuclear (DF)-space or a coechelon space of type one, then Tor 1 π (E, F ) = 0 if and only if the tensor product functor is exact on all short exact sequences of Fréchet spaces starting with F (c.f. [19,Remark 2.2]). This easy remark shows the importance of the first derivative and its computation.…”

“…The assertion follows from Lemma 2.4,[19, Proposition 3.22] and an application of the long exact cohomology sequence for the projective limit functor, see[19, Theorem 2.1]. Note that the complete projective tensor product commutes with reduced projective limits (c.f.…”

“…The starting point of our considerations is the following problem, which inspired different authors to further studies, compare [5], [7], [8], [18], [19], [24]: Let E be a complete (DF)-and F a Fréchet space. If 0 → F → G q → H → 0 is an exact sequence of Fréchet spaces the question arises whether the tensored sequence 0 → E ⊗ π F −→ E ⊗ π G id⊗q −→ E ⊗ π H → 0 is again exact, where π denotes the projective tensor product.…”

“…The aim of this paper is to present a sufficient condition for the solvability of the above question and to complement therewith the theory of derived tensor product functors for (DF)-and Fréchet spaces, which was investigated in detail by the author in [19] and has some relations to A. Grothendieck's research on topological properties of locally convex tensor products (see [5, II, §4, n • 3] and [18]), by a result corresponding to the well-known (DN)-(Ω) splitting theorems Vol. 90 (2008) A decomposition lemma for elementary tensors 247 of the Ext theory ( [20] and [12]).…”