Projective Limit Functor Research Articles

Theorems A and B for dagger quasi-Stein spaces

Abstract In this article, we use the homological methods of the theory of quasi-abelian categories and results from functional analysis to prove Theorems A and B for (a broad sub-class of) dagger quasi-Stein spaces. In particular, we show how to deduce these theorems from the vanishing, under certain hypothesis, of the higher derived functors of the projective limit functor. Our strategy of the proof generalizes and puts in a more formal framework Kiehl’s proof for rigid quasi-Stein spaces.

A homological approach to the splitting theory of [formula omitted] spaces

We give a homological approach to the splitting theory of PLSw spaces, that is strongly reduced projective limits of inductive limits of reflexive Banach spaces – a category that contains the PLS spaces that have been considered up to now. In particular we connect the problem under which conditions for given PLSw spaces E and X each short exact sequence(⋆)0→X→Y→E→0 of PLSw spaces splits to the vanishing of the Yoneda ExtPLSw1 functor in the category of PLSw spaces. Using the concept of exact categories this in turn is connected to the vanishing of the first derivative of the projective limit functor in a spectrum of operator spaces, thus generalizing results for special cases due to Bonet and Domański [2,3]. Furthermore, we apply the results to obtain a splitting theory for the space of Schwartz Distributions that includes the higher Ext functors, thus extending the result due to Domański and Vogt [13] respectively Wengenroth [40, (5.3.8)].

A new proof for the bornologicity of the space of slowly increasing functions

A. Grothendieck proved at the end of his thesis that the space of slowly increasing functions and the space of rapidly decreasing distributions are bornological. Grothendieck's proof relies on the isomorphy of these spaces to a sequence space and we present the first proof that does not utilize this fact by using homological methods and, in particular, the derived projective limit functor.

Projective limits of weighted (LB)-spaces of continuous functions

Countable projective spectra of countable inductive limits, called (PLB)-spaces, of weighted Banach spaces of continuous functions are investigated. It is characterized when the derived projective limit functor vanishes in terms of the sequences of the weights defining the spaces. The locally convex properties of the corresponding projective limits are analyzed, too.

The exactness of the projective limit functor on the category of quotients of Frechet spaces

We give conditions under which the functor projective limit is exact on the category of quotients of Fréchet spaces of L.Waelbroeck [18].

The projective limit functor for spectra of webbed spaces

The projective limit functor for spectra of webbed spaces

Derived Projective Limits of Topological Abelian Groups

In this paper, we prove that the category TAbof topological Abelian groups is quasi-Abelian. Using results about derived projective limits in quasi-Abelian categories, we study exactness properties of the projective limit functor in TAb. IfXis a projective system of TAbindexed by a filtering ordered set, we give a necessary and sufficient condition for the derived projective limit ofXto be strict. We also characterize the countable projective systems of complete metrizable Abelian groups which are [equation] -acyclic in TAb.

A new characterization of 𝑃𝑟𝑜𝑗¹𝒳=0 for countable spectra of (LB)-spaces

The derived projective limit functor Proj 1 ^1 is a very useful tool for investigating surjectivity problems in various parts of analysis (e.g. solvability of partial differential equations). We provide a new characterization for vanishing Proj 1 ^1 on projective spectra of (LB)-spaces which improves a classical result of V. P. Palamodov and V. S. Retakh.

A sufficient condition for vanishing of the derived projective limit functor

A sufficient condition for vanishing of the derived projective limit functor

Note on the Projective Limit on Small Categories

Let X be a small category and let ${\mathbf {A}}{{\mathbf {b}}^X}$ be the category of covariant functors on X with values in Ab. Consider the projective limit functor proj ${\lim _X}:{\mathbf {A}}{{\mathbf {b}}^X} \to {\mathbf {Ab}}$. The categories X for which proj ${\lim _X}$ is exact are characterized, proving a conjecture of Oberst.

Note on the projective limit on small categories

Let X be a small category and let A b X {\mathbf {A}}{{\mathbf {b}}^X} be the category of covariant functors on X with values in Ab. Consider the projective limit functor proj lim X : A b X → A b {\lim _X}:{\mathbf {A}}{{\mathbf {b}}^X} \to {\mathbf {Ab}} . The categories X for which proj lim X {\lim _X} is exact are characterized, proving a conjecture of Oberst.

Derived functors of the projective limit functor

Derived functors of the projective limit functor