A homological approach to the splitting theory of PLSw spaces

J. Math. Anal. Appl. 433 (2016) 1305–1328

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A homological approach to the splitting theory of PLSw spaces Bernhard Dierolf a,∗ , Dennis Sieg b a b

Math.-Geogr. Fakultät, Katholische Universität Eichstätt-Ingolstadt, 85072 Eichstätt, Germany Faculty IV Mathematics, University of Trier, 54286 Trier, Germany

a r t i c l e

i n f o

Article history: Received 13 January 2015 Available online 17 August 2015 Submitted by J. Bonet Keywords: Splitting of short exact sequences Functors Extk Exact categories Maximal exact structure Space of Schwartz distributions

a b s t r a c t 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 Ext1PLSw 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)]. © 2015 Elsevier Inc. All rights reserved.

1. Introduction The aim of this paper is to give a purely homological approach to the splitting theory of PLSw spaces, i.e. 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. Applying the concept of exact structures we establish the same connection between splitting of short exact sequences of PLSw spaces, vanishing of the Yoneda Ext1 functor in PLSw and vanishing of the first derivative of the projective limit functor proj in a spectrum of operator spaces that we are used to from the Fréchet space context [40, (5.1.5)]. Furthermore, we apply the results to obtain a splitting theory for the space D  (Ω) of Schwartz Distributions that includes the higher Ext functors, thus extending the result about the vanishing of Ext1PLS (E, X) for closed subspaces * Corresponding author. E-mail address: [email protected] (B. Dierolf). http://dx.doi.org/10.1016/j.jmaa.2015.08.030 0022-247X/© 2015 Elsevier Inc. All rights reserved.

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E and complete Hausdorff quotients X of D  (Ω) due to Domański and Vogt [13] respectively Wengenroth [40, (5.3.8)]. In linear functional analysis, the so-called splitting theory is concerned with the following problem: Characterize the pairs (E, X) of locally convex spaces such that for each short topologically exact sequence of locally convex spaces and continuous linear maps f

g

0→X− →Y − →E→0

()

the map g admits a linear and continuous right inverse, in which case we say that the sequence () splits. The above sequence is called topologically exact if f is a topological embedding and g is an open surjection with f (X) = g −1 ({0}). The splitting problem for this sequence is easily translated into the question whether the functor Hom (E, _) transforms () into an exact sequence of vector spaces. This makes the splitting problem a natural candidate for the methods of homological algebra, especially derived functors, which measure to what degree an additive functor preserves exactness. The first one to use homological methods for the splitting problem in the context of functional analysis, and the one who introduced homological methods to functional analysis in general, was Palamodov in 1968 [25,26]. Using injective resolutions, he computed for a locally convex space E the right derived functors Extk (E, _) of the functor Hom (E, _) which associates to each locally convex space X the vector space Hom (E, X) := L (E, X) of linear and continuous operators. The splitting of each short topologically exact sequence () is then characterized by the vanishing of Ext1 (E, X). Using the standard homological toolbox of diagram lemmas, Palamodov connected the vanishing of the latter object to the vanishing of the first derived functor of the projective limit functor proj in a spectrum of operator spaces, thus making the vanishing of Ext1 accessible for concrete applications. In the context of Fréchet spaces splitting theory has been investigated with a strong emphasis on the functional analytic aspects by many authors, beginning with the famous (DN ) − (Ω) splitting theorem [37] due to Vogt and Wagner followed amongst others by [35,15,16,40,11] and can now be considered rather complete. Problems arise when considering classical non-metrizable, i.e. non-Fréchet, spaces of functional analysis as the space of Schwartz distributions D  (Ω) or the space A (Ω) of real analytic functions and their relatives from the theory of PDEs: We are unable to control the behavior of Ext1 (D  , D  ) in the category of locally convex spaces. A first step to the solution of this problem is to restrict the spaces Y we want to allow in the middle of the sequence () when E = X = D  to a suitable subcategory of the locally convex spaces (which later on will also contain X and E). This leads to the category of so-called PLS spaces, projective limits of strong duals of Fréchet Schwartz spaces, which is the smallest class of locally convex spaces which contains all duals of Fréchet Schwartz spaces and is closed with respect of taking countable products and closed subspaces. Furthermore it contains most of the important spaces which appear in analytic applications of linear functional analysis, like spaces of (ultra-)distributions, or spaces of real analytic or quasi-analytic functions as well as spaces of holomorphic or smooth functions. For more information on PLS spaces, we refer to [10]. Splitting theory in this category has been studied extensively by many authors, e.g. [12,13,20,40,2,3]. All those papers achieve their results using functional analytic arguments only. This lack of homological methods is due to the fact that Palamodov’s approach to the derived functors of Hom, cf. [26] respectively [40] for a survey, does not work in the category of PLS spaces, since it is neither semi-abelian, which we will prove in Section 3.1, nor is it known, whether it does have enough injective objects. Until rectified by Sieg [30] in 2010 this circumstance lead to an ad-hoc definition of the vanishing of Ext1PLS via the splitting of short exact sequences, cf. e.g. [40, (5.3)]. Hence the connection between the vanishing of Ext1PLS and proj1 – which allows for applicable characterizations of splitting – had to be implemented manually, cf. [2, Section 3] and [3, (3.4)]. In his PhD thesis [30] the second author has given a homological approach to the splitting theory of PLS spaces. Determining the maximal exact structure of PLS to be the class of short topologically exact sequences, he was able to use the diagram lemmas for exact categories [5] to establish the long exact

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sequence for the Yoneda Ext functors that are defined via equivalence classes of short exact sequences [41]. Furthermore he was able to link the vanishing of ExtLS – the Ext functors of the steps of the spectra giving rise to PLS spaces – to ExtPLS which in turn he linked to the first derivative of the projective limit functor in spectra of operator spaces. Finally he calculated the ExtkPLS functors for the space of distributions. A part of the PhD thesis of the first author is to extend the results of the second author to the larger category of PLSw space, that are defined as strongly reduced projective limits of inductive limits of reflexive Banach spaces. This category is not only of interest because it seems to be the more natural candidate but also because it contains several non-nuclear spaces of functional analysis as the Fréchet-Hilbert space DL2 , the Bloc 2,k (Ω) spaces due to Hörmander [18, Chap. 10] and Köthe sequence spaces of PLSw type that evaded proper investigation in the context of PLS spaces until now for the lack of compactness of the linking maps in the representing spectra. The paper is a joint presentation of the work of both: In the first part of Section 2 we give a short introduction into the setting of exact categories including Sieg and Wegner’s result about the maximal exact structure in pre-abelian categories. In the second part we introduce the Yoneda Ext functors in this context briefly, including the long exact sequence and a characterization of the vanishing of the Ext functors which is essentially due to Yoneda [41]. Knowledge of basic homological notions is of great use concerning Section 2 which is a summary of [30, Chapter 2 & 4]. For general concepts and unexplained notation of classic homological algebra we refer to [6,21,24,38]. Considering the concept of exact categories and exact structures we refer to [5]. Section 3 is the technical core of the paper. In the first part we study the categories of interest, PLSw and LSw (resp. LB), in detail. We show that they are pre-abelian and determine their maximal exact structure. As expected the category of LSw spaces turns out to be quasi-abelian. In contrast to that the maximal exact structure in PLSw turns out to consist of all short algebraic exact sequences where the right map (the cokernel) is open but the left map (the kernel) does not need to be open onto its range. Hence the class of short topologically exact sequences is a proper subclass of the maximal exact structure which in turn is a proper subclass of all kernel cokernel pairs. To our knowledge this is the first example of such a category. This is [8, Section 2.2]. Using the results of Sections 2 and 3.1 we establish the Yoneda Ext functor in PLSw as well as in LSw . We prove the connection of Ext 1LSw and ExtPLSw and show that Ext1PLSw coincides with proj1 in a spectrum of operator spaces under mild assumptions in complete analogy to the Fréchet case, cf. [40, (5.1.5)]. In the last part we prove that ExtkPLSw (E, X) vanishes for closed subspaces E and complete Hausdorff quotients X of D  (Ω) for all k ≥ 1. Note that we extend the result due to Domański and Vogt [13] respectively Wengenroth [39] from PLS to PLSw and (much more importantly) from k = 1 to k ≥ 1. Section 2 is [30, Chapter 5] adapted to the PLSw case, cf. [8, Section 2.3]. 2. The Yoneda Extk in pre-abelian categories The use of homological methods in functional analysis was started by Palamodov [25,26], re-invented by Vogt [35] and expanded by many others, we refer to the book of Wengenroth [40] for detailed references and concrete applications. What distinguishes the homological algebra in functional analysis from the classical one used in purely algebraic context (as presumed in [6,21,24,38]) is the fact that its categories are not abelian, i.e. the morphism induced between the coimage and the image of a morphism is not necessarily an isomorphism. Fortunately the classical categories of functional analysis, as the Banach spaces, Fréchet spaces or locally convex spaces share two other properties: They are quasi-abelian, i.e. kernels and cokernels holds certain stability properties in pullback and pushout constructions, and they have many injective objects. Palamodov took advantage of those to define for a given space Z the functors Extk (Z, _), k ≥ 1 as the right-derived functors of the Hom (Z, _) functor. Then the vanishing of the Ext1 functor characterizes the splitting of short exact sequences and the vanishing of the higher Ext functors give sufficient conditions for the vanishing of the Ext1 being passed to subspaces or quotients. Especially the connection Palamodov

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established between Ext1 and the first derivative of the projective limit functor in a spectrum of operator spaces proved to be very useful. A problem arises when considering classical non-metrizable spaces as the space of Schwartz Distributions  D (Ω) or the space of real analytic functions A (Ω) and their relatives, so-called PLSw spaces, which will be subject to investigation in the following sections. In those cases Palamodov’s approach reaches its limits, we do not know if Ext1 (D  , D  ) vanishes in the category of locally convex spaces and it remains unknown whether the categories of PLS and PLSw spaces posses many injective objects. Furthermore we will prove that PLSw is not even semi-abelian in (3.1.11)ii), hence is not quasi-abelian either. These structural deficits inhibited the use of homological methods in splitting theory for those spaces and led to ad hoc definitions of the vanishing of Ext1 (Z, X) via the splitting of short exact sequences, cf. [40,2,3]. Since Palamodov’s approach is not possible, we will exploit another access to Ext groups: In 1934, Baer [1] defined an addition on the class Ext1 (Z, X) of equivalence classes of extensions of abelian groups, named after him, in such a way that Ext1 (Z, X) becomes an abelian group. Cartan and Eilenberg   [6] introduced the higher Extk -groups as connected sequence of group valued bifunctors Extk (_ , _) , k≥0

which could be defined in any abelian category either having enough injective or projective objects. Yoneda [41] then showed that the groups Extk (Z, X) could also be defined without injectives or projectives in terms of equivalence classes of exact sequences of the form fk

f1

f0

0 −→ X −→ Yk−1 −→ . . . −→ Y1 −→ Y0 −→ Z −→ 0. With further work done by Schanuel and (independently) Buchsbaum [4], this led to the connected sequence of group valued bifunctors Extk (_ , _) for arbitrary abelian categories. This approach is not so well suited for direct calculations, hence it has not been used in the context of splitting theory. One possibility to conduct homological algebra for those functors all the same is to apply the concept of exact structures that originates from K theory, see e.g. [27]. To illustrate the main idea we recall the object under consideration, the short exact sequences f

g

0→X− →Y − →E→0

()

of locally convex spaces, we defined in the introduction. In the first step, we restricted the spaces X, Y and E to a subcategory of the locally convex spaces. Now in a second step we further restrict the kind of sequences () the splitting of which we want to consider, i.e. we specify the notion of exactness. To this end we regard general objects () as kernel-cokernel pairs, which makes sense as f is a kernel of g and g is a cokernel of f . Then we specify a certain subclass of all those kernel-cokernel pairs (f, g) in the category under consideration which has to satisfy a certain set of technical properties, the most prominent reminding of the axioms defining a quasi-abelian category. This subclass is then called an exact structure, its elements are the new short exact sequences and the category endowed with this class is called an exact category. Thanks to Bühler’s work in this area [5], we can establish almost the complete homological toolbox, in particular the long exact sequence also in merely exact categories, since we may substitute the basic homological diagram lemmas of exact categories, see [5], for their abelian counterparts, see e.g. [24]. We give a brief description of the construction and state a characterization of the vanishing of the higher Ext groups due to Yoneda [41] in the second part of this section. We start by a short introduction into the concept of exact structures and – to be able to apply the results of the second section – endow pre-abelian categories with a canonical exact structure, which exists (and is even maximal) by Sieg and Wegner’s result [31, (3.3)]. 2.1. Exact categories – notions and auxiliary results Definition and Remark 2.1.1. Our vantage point is a pre-abelian category C, i.e. an additive category C where every morphism has a kernel and a cokernel. A pair (f, g) of composable morphisms in C is called a

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kernel-cokernel pair if f is the kernel of g and g is the cokernel of f . If a class E of kernel-cokernel pairs in C is fixed, then a morphism f : X −→ Y is called an admissible kernel if there is a morphism g : Y −→ Z in C such that (f, g) is in E. Dually a morphism g : Y −→ Z in C is called an admissible cokernel if there is a morphism f : X −→ Y in C such that (f, g) is in E. A class E of kernel-cokernel pairs in C is an exact structure if it is closed under isomorphisms and the following hold: (E0) op (E0) (E1) op (E1)

idX is an admissible cokernel for all X ∈ C. idX is an admissible kernel for all X ∈ C. If allowed, the composition of two admissible cokernels is an admissible cokernel. If allowed, the composition of two admissible kernels is an admissible kernel.

(E2)

For all X − →Y − → Z in E the pushout

f

g

f

X

exists for all morphisms s and qS is an

Y qY

s

S

Q

qS

admissible kernel. (E2)

op

f

g

For all X − →Y − → Z in E the pullback

P

pS

exists for all morphisms s and pS is an

S

pY

s

Y

g

Z

admissible cokernel. An exact category (C, E) is an additive category C together with an exact structure E. The kernel-cokernel pairs in E are called short exact sequences. The definition of an exact category is self-dual, i.e. (C, E) is an exact category if and only if (C op , E op ) is an exact category. Given a pre-abelian category C we always have a minimal exact structure, i.e. an exact structure that is contained in any other exact structure in C, the class of short exact sequences that are split-exact, i.e. the class of kernel-cokernel pairs (f, g) in C such that g has a right inverse (see [5, Lemma (2.7) & Remark (2.8)]). C In [31] Sieg and Wegner proved that every pre-abelian category C admits an exact structure Emax that is maximal in the sense that it contains every other exact structure in C. A pre-abelian category C is then quasi-abelian if and only if the maximal exact structure coincides with the class of all kernel-cokernel pairs (see [31, (3.3)]). We fix some notation to formulate their result: Definition 2.1.2. Let C be a pre-abelian category. i) A cokernel g : Y −→ Z in C is said to be semi-stable, if for every morphism s : S −→ Z in C the morphism pS in the pullback diagram

pS

P

is also a cokernel in C.

S

pY

s

Y

Z

g

ii) A kernel f : X −→ Y in C is said to be semi-stable if for every morphism s : X −→ S in C the morphism qS in the pushout diagram

X

f

is also a kernel in C.

Y qY

s

S

qS

Q

C Theorem 2.1.3. (See [31, (3.3)].) If C is a pre-abelian category, then the class Emax , which consists of all kernel-cokernel pairs (f, g) in C such that f is a semi-stable kernel and g is a semi-stable cokernel, is an exact structure in C. Moreover, it is maximal in the sense that it contains any other exact structure in C.

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Theorem (2.1.3) allows us to fix a distinguished exact structure in every pre-abelian category: C obtaining Definition 2.1.4. We endow every pre-abelian category C with its maximal exact structure Emax   C C the exact category C, Emax . We call an element (f, g) of Emax a short exact sequence of C objects or a sequence exact in C.

2.2. Yoneda Ext functors in exact categories As mentioned in the introduction of this section, the Yoneda Ext groups are defined using equivalence classes of exact sequences. We start by giving the definition of exactness of sequences that are longer than short exact sequences and the equivalence relation under consideration. Definition and Remark 2.2.1. Let (C, E) be an exact category. i) A sequence A:

fk

f1

f0

0 −→ X −→ Yk−1 −→ . . . −→ Y1 −→ Y0 −→ Z −→ 0

in C is called exact (in C, with left end X and right end Z), if every morphism fl factorizes as fl = ml ◦el for an admissible cokernel el : Yl −→ Il and an admissible kernel ml : Il −→ Yl−1 , such that (ml , el−1 ) is in E for 1 ≤ l ≤ k. The integer k is called the length of the sequence. ii) Let A :

 0 −→ X  −→ Yk−1 −→ . . . −→ Y1 −→ Y0 −→ Z  −→ 0

be another exact sequence of length k in C. A morphism with fixed ends Φ : A −→ A between the exact sequences A and A of length k is a commutative diagram A: 0

X

Yk−1

Yk−2

Φk−1

A : 0

X

 Yk−1

...

Y1

Φk−2  Yk−2

Φ1

...

Y1

Y0

Z

0

Z

0.

Φ0

Y0

iii) For Z and X in C we denote by EkC (Z, X) the exact sequences of length k with right end Z and left end X. On EkC (Z, X) we define the following equivalence relation: Two elements A and A of EkC (Z, X) are said to be equivalent, A ∼ A , if there is a sequence A = A0 , A1 , . . . , Al−1 , Al = A of elements of EkC (Z, X), such that for every 0 ≤ i ≤ l − 1 there is a morphism with fixed ends either from Ai to Ai+1 or from Ai+1 to Ai . We define  ExtkC (Z, X) = EkC (Z, X) ∼ and denote by [A]C the equivalence class of A in ExtkC (Z, X). iv) Using the usual pullback and pushout constructions with kernel-cokernel pairs, cf. [30, (2.1.2)], it is possible to endow the equivalence classes ExtkC (Z, X) with an abelian group structure via the Baer sum and assign each morphism λ : X −→ X  in C a homomorphism of abelian groups ExtkC (Z, λ) : ExtkC (Z, X) −→ ExtkC (Z, X  ), cf. [30, (4.1.5)].

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Then we have: Theorem 2.2.2. Let (C, E) be an exact category and let E be an object of C. For each k ≥ 1 the assignment ExtkC (E, _) : C −→ (AB) , X → ExtkC (E, X) λ → ExtkC (E, λ) is a covariant, additive, abelian-group-valued functor, which induces for every short exact sequence 0 −→ X −→ Y −→ Z −→ 0 a long exact sequence δ

0 0 −→ Hom (E, X) −→ Hom (E, Y ) −→ Hom (E, Z) −→

δ

δ

0 1 −→ Ext1C (E, X) −→ Ext1C (E, Y ) −→ Ext1C (E, Z) −→

δ

1 −→ Ext2C (E, X) −→

δk−1

−→ Extk−1 (E, Z) −→ C

...

δk−1

δ

k −→ ExtkC (E, X) −→ ExtkC (E, Y ) −→ ExtkC (E, Z) −→

δ

k −→ Extk+1 (E, X) −→ C

...

.

Proof. See [6, Chapter 5, §4] for the abelian case and [5] for the basic diagram lemmas in exact categories respectively [30, Theorem (4.1.1)] for a more detailed explanation. 2 Remark 2.2.3. Applying the concept of universal δ-functors it is possible to show that if an exact category (C, E) has enough injective objects, for all objects E in C the ExtkC (E, _) groups are isomorphic to the derived functors of HomC (E, _) in the sense of Palamodov, cf. [26] and [30, (2.1.4) & p. 36]. In the following, we connect the vanishing of Ext1 to the splitting of short exact sequences and give the announced characterizations of the vanishing of the (higher) Ext groups, which is due to Yoneda [41], for detailed proofs see [30, (4.2.1), (4.2.2), (4.2.4)]: Proposition 2.2.4. Let (C, E) be an exact category. i) For any short exact sequence f

g

0 −→ X −→ G −→ E −→ 0 in C a) b) c)

the following are equivalent: f has a left inverse. g has a right inverse. There is a commutative diagram ⎛

⎞

⎝1⎠

0

0

X

X ⊕E

(0,1)

E

0

E

0

β

0 such that β is an isomorphism.

X

f

G

g

B. Dierolf, D. Sieg / J. Math. Anal. Appl. 433 (2016) 1305–1328

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ii) For two objects X and E in C the following are equivalent: a) Ext1C (E, X) = 0 f

g

b) Every exact sequence 0 −→ X −→ G −→ E −→ 0 in C is split exact. iii) For every exact sequence A: 0

fn

X

Yn−1

fn−1

Yn−2

...

Y1

f1

f0

Y0

Z

0

in C with n > 1 the following are equivalent: a) [A] = 0 b) There is a commutative diagram A: 0

X

fn

Yn−1

fn−1

φn−1 

A: 0

X

gn

Vn−1

gn−1

Yn−2

...

φn−2

Y1

f1

Y0

φ1

Vn−2

...

V1

Yn−2

...

Y1

f0

Z

0

φ0 g1

V0

0

in C with exact rows. c) There is a commutative diagram A: 0

X

fn

Yn−1

fn−1

φn−1

A : 0

X

gn

Vn−1

gn−1

φn−2

f1

Y0

f0

Z

0

φ1

Vn−2

...

V1

Vn−2

...

V1

g1

Y0

0

in C with exact rows. d) There is a commutative diagram A :

0

Vn−1

gn−1

φn−1

A: 0

X

fn

Yn−1

fn−1

φn−2

g1

V0

φ1

Yn−2

...

Y1

Vn−2

...

V1

g0

Z

0

Z

0

Z

0

Z

0

φ0

f1

Y0

f0

in C with exact rows. e) There is a commutative diagram A :

0

Yn−1

gn

φn−2

A: 0

X

fn

Yn−1

fn−1

Yn−2

g1

V0

φ1

...

Y1

g0

φ0

f1

Y0

f0

in C with exact rows. We conclude the section by the following result for later use, cf. [30, (4.2.6)]: Proposition 2.2.5. Let (C, E) be an exact category and let X be an object of C such that Ext2C (E, X) vanishes for all objects E of C. Then ExtkC (E, X) vanishes for all objects E of C and all k ≥ 2.

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Proof. We prove the assertion by induction. Let k ≥ 3, A: 0

X

fk

fk−1

Yk−1

Yk−2

...

Y1

f1

Y0

f0

E

0

be an exact sequence of length k and m1 ◦ e1 the canonical factorization of f1 . Then we have an exact sequence of length k − 1 A : 0

X

Yk−1

Yk−2

...

Y1

I1

0,

which induces a commutative diagram with exact rows by (2.2.4)iii), since Extk−1 (I1 , X) vanishes: C A : 0

X

Yk−1

Yk−2

...

Y1

I1

0

A : 0

X

Vk−1

Vk−2

...

V1

0

.

Then we also have a commutative diagram with exact rows A: 0

X

Yk−1

Yk−2

...

Y1

Y0

E

0

A : 0

X

Vk−1

Vk−2

...

V1

0

0

,

which shows [A] = 0 by (2.2.4)iii) and thus ExtkC (E, X) = 0.

2

3. Extk in the category of PLSw spaces In this chapter we will apply the tools presented in the preceding section to the splitting problem in a category that became of interest in the nineties of the last century. As mentioned in the introduction of Section 2 the Palamodov approach to calculate Ext groups as right derivatives of the Hom functor does not work for classical spaces as the space of Schwartz distributions D  (Ω) or the space of real analytic functions A (Ω) as the category of Fréchet spaces is obviously too small and the category of locally convex spaces seems too big as we don’t know whether Ext1LCS (D  , D  ) vanishes. Thus we are in need of a category in between. The category we choose to investigate is suggested by the topological structure of the spaces above. N Valdivia [32] and independently Vogt [34] have proven the isomorphy of D  (Ω) and (s ) where s denotes as usual the space of rapidly decreasing sequences. Thus the space of distributions is a projective limit of strong duals of nuclear Fréchet spaces. If we call those spaces LN spaces we arrive at a category that seems to be too small for our purpose since those spaces are even strongly nuclear, hence the category PLN does not even contain s. This lead to the introduction of the so called PLS spaces – strongly reduced projective limits of strong duals of Fréchet-Schwartz spaces. This category contains all Fréchet Schwartz spaces and most of the non-metrizable examples as well, hence it seems large enough. Furthermore it has the great advantage that the compactness of the embeddings in the inductive spectra giving rise to the steps of the projective spectra guarantees that the category PLS is stable with respect to closed subspaces. Those types of spaces were investigated by the second author during the course of his thesis. We will consider the natural extension of the category of PLS spaces, the so-called PLSw spaces – strongly reduced projective limits of inductive limits of reflexive Banach spaces. Losing the rather artificial appearing asymmetric compactness assumption on the embeddings does not only have idealistic reasons. Apart from illustrating the assumptions needed much more clearly, the extension includes natural non-nuclear spaces of functional analysis as the Fréchet-Hilbert

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space DL2 , the Bloc 2,k (Ω) spaces due to Hörmander [18, Chap. 10] and Köthe sequence spaces of PLSw type that evaded proper investigation in the context of PLS spaces until now for the lack of compactness of the linking maps in the representing spectra. This lack of compactness for PLSw spaces results in the loss of several stability properties, e.g. subspace stability, that poses technical difficulties, when determining the maximal exact structure, which will be dealt with in the first part. In the second part we are going to establish the classical connection between Ext1PLSw and proj1 that makes the vanishing of the functors ExtkPLSw (k ∈ N) accessible to concrete investigation, cf. [8, Chapter 3] for results at least for the hilbertized version PLH of PLSw . In the last part we are going to investigate splitting for the space D  (Ω) of Schwartz distributions in the category of PLSw spaces. We prove that Ext1PLSw (E, X) vanishes for subspaces E and complete Hausdorff quotients X of D  (Ω), which slightly extends the splitting theorem due to Wengenroth [39] for the space of distributions, which was itself an improvement of a result due to Domański and Vogt [13, (2.19) & (3.1)]. The results of the first part are part of the PhD thesis of the first author [8, Chapter 2], whereas the results of part 2 and 3 are minor adaptions of results in the PLS context of the PhD thesis of the second author [30, Chapter 5]. 3.1. The maximal exact structure in PLSw We begin by relating the results to the classical notion of exactness: Palamodov [26] calls a short sequence f

g

0→X− →Y − →Z→0

()

of objects and morphisms in a semi-abelian category C exact at Y if the image of f is a kernel of g and the coimage of g is a cokernel of f and both morphisms are homomorphisms. In quasi-abelian categories this notion coincides with our notion of exactness, cf. (2.2.1)i), as in those categories the maximal exact structure coincides with the class of all kernel-cokernel pairs by definition. Thus we call the sequence () of vector spaces and linear maps a short algebraic exact sequence if it is exact in the category VS of vector spaces, i.e. if f is injective, g is surjective and f (X) = g −1 ({0}). A sequence () of locally convex spaces (lcs) and linear and continuous maps is called topologically exact if it is exact in the category LCS of lcs and linear and continuous maps, i.e. if it is algebraically exact and all the maps involved are continuous and open onto their ranges. In the following we will start by investigating the category LB of inductive limits of embedding spectra of Banach spaces that harbors the spaces of the steps of the projective spectra giving rise to our spaces of interest. We will prove that the maximal exact structure in LB coincides with the class of all kernel-cokernel pairs where the kernel is weak isomorphism onto its range and the cokernel is surjective, which contains the short topologically exact sequences as a proper subclass and is a proper subclass of the short algebraically exact sequences. By considering the subcategory LS, inductive limits of compact embedding spectra, we will arrive at a category where all the maximal exact structures coincides with both classical ones. In contrast to that it coincides with the short algebraically exact sequences in LSw , the inductive limits of reflexive Banach spaces, but differs from the short topologically exact sequences. Note, that the result about LS is classic, cf. [40, (5.3)]. Then by applying a dual variant of Grothendieck’s factorization theorem for LF spaces [23, (24.33)] due to Vogt [36] we will use the knowledge about the categories of the steps to determine the maximal exact structure in PLSw . We will show that it consists of all kernel-cokernel pairs in PLSw , where the cokernel is surjective, which again contains the short topologically exact sequences of PLSw spaces as a proper subclass. The short topologically exact sequences indeed constituting an exact structure in the categories under investigation is proved in [9] where also various examples of subcategories of LCS where the class of short topologically exact sequences is not even an exact structure are given. We can easily see that the category LB is pre-abelian:

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Remark 3.1.1. Let t : E −→ F be a linear and continuous map between the LB spaces E = indn∈N En and F = indn∈N Fn . As LB is stable under Hausdorff quotients, the cokernel of t in C is given by the quotient map    F −→ F t (E) = ind Fn t (E) ∩ F , n n∈N i.e. LB inherits cokernels from the category (LCS) of Hausdorff lcs. With the closed graph theorem we obtain that the kernel of t in C is given by the continuous inclusion of the algebraic kernel of t endowed with its associated ultrabornological topology:  −1   ub t ({0}) = ind t−1 ({0}) ∩ En → E. n∈N

In LB the product of E and F endowed with the product topology and the four standard maps πE , πF , ωE , ωF of projections and inclusions is a biproduct of E and F , as E ×F = indn∈N (En × Fn ) holds topologically. Thus the claim is valid. Furthermore, it is easy to verify that t is a kernel in C iff it is injective with closed range and a cokernel in C iff it is surjective. Now we can determine the maximal exact structure of LB: LB Proposition 3.1.2. The maximal exact structure Emax in the category LB consists of all kernel-cokernel LB pairs (f, g) such that f is a weak isomorphism onto its range. In other words: Emax consists of all short f

g

algebraically exact sequences 0 → X − →Y − → Z → 0 of LB spaces such that f is a weak isomorphism onto its range. Proof. With (2.1.3) it is sufficient to prove that a kernel in LB is semi-stable iff it is a weak isomorphism onto its range and each cokernel is semi-stable. Let f : X −→ Y be a kernel in LB, s : X −→ S be a linear and continuous map from X into an LB space S and

X

f

be their pushout diagram in LB. The map qS is a kernel in LB iff it is injective

Y qY

s

S

qS

Q

with closed range. Since f has closed range, so does qS . Again by mere calculation the injectivity of qS is equivalent to the graph of s being closed in the product of X endowed with the topology of Y via f and S. As S is an arbitrary LB space, a standard application of the Hahn-Banach theorem yields that this is equivalent to f being weak isomorphism onto. As a morphism in LB is a cokernel iff it is surjective, in every pullback diagram

P

pS

in LB

S

pY

s

Y

g

Z

the morphism pS is a cokernel if g is a cokernel. Hence all cokernels in LB are semi-stable and the proof is complete. 2 We finish by considering the mentioned subcategories of LB: Corollary 3.1.3. i) Let E = indn∈N En be an LB space. We say that E is an LSw space if the Banach spaces En , n ∈ N, can be chosen to be reflexive. It is an LS space if the inclusions EN → En+1 , n ∈ N, can be chosen

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compact. As every weakly compact operator between lcs factorizes over a Banach space by [7] the category LS of LS spaces is a proper subcategory of the category LSw of LSw spaces. Both categories inherit products, kernels and cokernels from LB, hence the maximal exact structure as well. As LS spaces are subspacestable we arrive here at the class of short topologically exact sequences, whereas LSw spaces are not by [23, (27.23)c)]. LSw spaces are weakly acyclic by [26], from which we can deduce with [28] that closed subspaces of LSw are at least well-located, hence with the open mapping theorem for webbed spaces [23, (24.30)] we arrive at the class of all short algebraically exact sequences. ii) The perfect dualities between LS and its dual category FS of Fréchet-Schwartz spaces and between LSw and its dual category FSw of Fréchet spaces that are countable projective limits of reflexive Banach spaces extend to the exact structures. This allows us to control splitting in both inductive categories by dualizing to the subcategories of the Fréchet spaces where splitting theory is well established, cf. [40, Chapter 5.2] for a short survey. The preparations being complete, we can start with the investigation of PLSw : A Hausdorff locally convex   N space E is called a PLSw space if it is the projective limit of a strongly reduced spectrum EN , EM of LSw spaces EN (N ∈ N). Recall that such a projective spectrum is called strongly reduced if for every N ∈ N N N N (EM ) is a subset of the closure of E∞ (E) in EN , where E∞ denotes the there is an M ≥ N such that EM restriction of the canonical projection on the nth component to E. Furthermore E is called a PLS space if all EN are even LS spaces. The latter category has been under investigation in the context of splitting theory by several authors for over a decade now. For a short survey of examples and results cf. [10,40]. The main difference between the classic category of PLS spaces and the PLSw spaces is the loss of subspacestability: As closed subspaces of LS spaces equipped with the relative topology are again LS spaces [23, (26.17)] the same is true for PLS spaces. As the same does not hold for LSw by (3.1.3)i), it is not true for PLSw either. This leads to problems constructing kernels in PLSw . To deal with those we need the announced dual variant of Grothendieck’s factorization theorem due to Vogt [36]:     N N and FN , FM be two strongly reduced projective spectra of complete Proposition 3.1.4. Let EN , EM     N N separated LB spaces and t : proj EN , EM −→ proj FN , FM be a linear and continuous map. Then   there is a subsequence (K (N ))N ∈N and a morphism of spectra tK(N ) N ∈N , called localization of t, from       K(N ) N such that t = proj tK(N ) N ∈N . EK(N ) , EK(M ) to FN , FM Now we can construct the kernel and cokernel in PLSw : Proposition 3.1.5. Let t : E −→ F be a linear and continuous map between PLSw spaces (E, T ) = projN ∈N indn∈N EN,n and (F, S) = projN ∈N indn∈N FN,n . i) The kernel of t in PLSw is given by the continuous inclusion of the algebraic kernel of t, endowed with   its associated PLSw topology, into E. We denote this object by t−1 ({0}) PLS . By definition, it is the w PLSw space arising from the PLSw spectrum consisting of the closures of the projections of the algebraic kernel of t to EN (N ∈ N) endowed with their associated ultrabornological topology – the LSw spaces  ub E N N (t−1 ({0})) N E∞ – and the corresponding restrictions of the linking maps EM . ii) As in the category of PLS spaces, the cokernel of t in the category of PLSw spaces is given by the  Hausdorff completion of the quotient F t (E), which arises as the projective limit of the strongly reduced  spectrum consisting of the LSw spaces FN F N (t (E))FN together with the maps induced by the linking N maps FM .

∞

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Proof. i) By the closed graph theorem and considering the fact that closed subspaces of LSw spaces are well-located by (3.1.3)i), the above defined kernel is again a PLSw space that coincides algebraically with the classic kernel of t. By construction it carries a topology that is finer than the relative topology induced by E, which is induced by the projective limit of closures of the projections of the algebraic kernel of t to EN (N ∈ N) endowed with the relative topology of EN and the corresponding restrictions of the linking N maps EM . We obtain our spectrum by passing to the associated LSw topologies and we have the     continuous identity map from t−1 ({0}) PLS to t−1 ({0}) , T ∩ t−1 ({0}) . Let now s : S −→ E be a w   N linear and continuous map from a PLSw space S = proj SN , SN t ◦ s = 0. Then we +1 into E with  −1  −1 have the unique linear and continuous map λ : S −→ t ({0}) , T ∩ t ({0}) induced by the locally   convex kernel of t and it remains to show that λ : S −→ t−1 ({0}) PLS is continuous. As λ : S −→ E is w continuous, by passing to a subsequence, we may assume by (3.1.4) that it arises as a morphism of spectra     N N −1 (λN )N ∈N : SN , SN ({0}), we obtain by construction that λN (SN ) +1 −→ EN , E N +1 . As λ (S) ⊂ t  N −1 is a subset of the closure of E∞ t ({0}) in EN . Thus, again with the closed graph theorem, λN remains continuous when passing to the associated ultrabornological topologies and λ = proj (λN )N ∈N   is continuous even from S to t−1 ({0}) PLS . w ii) Here we can essentially follow the proof for the category of PLS spaces in [30, (3.1.3)]: For N ∈ N N (t (E)) in FN , here endowed with the relative topology, and AN we define AN as the closure of F∞ N +1 N as the restriction of FN +1 to AN +1 and the topological inclusion iN of AN into FN . Furthermore, we  define YN as the topological quotient FN AN , YNN+1 as the map from YN +1 to YN induced by FNN+1   and the quotient map qN from FN to YN (N ∈ N). Then AN , AN is a strongly reduced spectrum N +1   N of locally convex spaces, YN , YN +1 is a strongly reduced spectrum of LSw spaces and we have a short topologically exact sequence of projective spectra   (iN )N ∈N   (qN )N ∈N   0 −→ AN , AN −−−−−→ FN , FNN+1 −−−−−−→ YN , YNN+1 −→ 0. N +1 −   The map proj (qN )N ∈N is open onto its range by [40, (3.3.1)], as the spectrum AN , AN N +1 is strongly reduced, hence the map     N  F j : Y t (E)F −→ proj YN , YNN+1 , y + t (E) → Y∞ (y) + AN N ∈N N ∈N

  is open onto its range as well. The space projN ∈N YN , YNN+1 is a complete Hausdorff space, since each YN (N ∈ N) is complete and Hausdorff. As j has dense range, it is a version of the Hausdorff completion  of Y t (E)F . Thus it is an easy exercise to check that it satisfies the universal property of the cokernel of f in PLSw . 2 Corollary 3.1.6. The category PLSw is pre-abelian. Proof. As the product of two PLSw spaces is again a PLSw space, (3.1.5) yields the assertion. 2 Before we can determine the maximal exact structure of PLSw , we need to apprehend the possibilities of localization and its limits: Remark 3.1.7. Let t : E −→ F be linear and continuous operator between PLSw spaces E =     N N projN ∈N EN , EM and F = projN ∈N FN , FM with localization (tN )n∈N given by (3.1.4). Denoting by

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ϕN the kernel of tN in LSw (N ∈ N), we arrive at the morphism of spectra (ϕN )N ∈N that relates to the so-called spectrum of local kernels, which always contains the spectrum giving rise to the PLSw kernel continuously. Of course, if the spectrum of local kernels is strongly reduced, this inclusion is an isomorphism of spectra but since we do not know whether the spectrum of local kernels holds this property we cannot localize kernels in general. In contrast to that a localization of the cokernel is very well possible. Since we may   N N (and will) assume that for a given PLS space X = proj XN , XM the canonical projections E∞ : E −→ EN have dense range, the closures of the images of tN coincide with the closures of the projection of the image of t to FN for all N ∈ N, i.e. the spectrum of local cokernels is a localization of the PLSw cokernel of t. However bad the situation concerning the localization of kernels may be, in complete analogy to the PLS case, cf. [13, p. 64], a localization of kernel-cokernel pairs is possible in PLSw , as the category of the steps, LSw , is quasi-abelian by (3.1.3)i): Proposition 3.1.8. Let (f, g) be a kernel-cokernel pair in PLSw . Then there are strongly reduced spectra         N N N X , X N , Y N , YM , ZN , ZM of LSw spaces as well as morphisms of spectra (fN )N ∈N : XN , XM −→  N NM     N N YN , YM and (gN )N ∈N : YN , YM −→ ZN , ZM such that each (fN , gN ) is a kernel-cokernel pair in LSw , LSw i.e. (fN , gN ) ∈ Emax (N ∈ N), and (f, g) arises as the projective limit of the sequence of spectra       N (fN )N∈N N (gN )N∈N N 0 −→ XN , XM −→ YN , YM −→ ZN , ZM −→ 0.       ˜N , X ˜ N , Y˜N , Y˜ N and Z˜N , Z˜ N are strongly reduced spectra of LSw spaces giving rise to Moreover, if X M M M         N N N N X, Y and Z then we can either take YN , YM = Y˜N , Y˜M or XN , XM and ZN , ZM as subsequences     N N ˜ ˜ ˜ ˜ of XN , XM and ZN , ZM . Now we can determine the semi-stable kernels and cokernels in PLSw , which leads to the maximal exact structure of PLSw by (2.1.3): Proposition 3.1.9. i) In PLSw every kernel is semi-stable. ii) In PLSw a cokernel is semi-stable if and only if it is surjective. Proof. Let (f, g) be a kernel-cokernel pair in PLSw with corresponding localization (3.1.8):       N (fN )N∈N N (gN )N∈N N 0 −→ XN , XM −→ YN , YM −→ ZN , ZM −→ 0.   N i) Let s : X −→ S be a linear and continuous map from X = proj XN , XM into any PLSw space S =   N proj SN , SM . By passing to a subsequence we may assume by (3.1.4) that there is a morphism of spectra       N N N (sN )N ∈N from XN , XM to SN , SM such that s = proj (sN )N ∈N . We denote Y := proj YN , YM . Since the pushout is a cokernel, cf. [40, (5.1.2)], [30, (2.1.1)], (3.1.7) yields that the pushout diagram X

f

in PLSw arises as the projective limit of the local pushout diagrams

Y qY

s

S

qS

Q

(N ∈ N) in LSw . Then it is straightforward to check, that each qSN is the kernel of

XN

fN

qYN

sN

SN

YN

qSN

QN

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  ψN : QN = YN × SN (fN , −sN ) (XN ) −→ ZN = YN fN (XN ), (yN , rN ) + (fN , −sN ) (XN ) −→ yN + fN (XN ) . Thus qS = proj (qSN )N ∈N is the projective limit of the spectrum of the local kernels (qSN )N ∈N of the   N map ψ := proj (ψN )N ∈N . As the spectrum SN , SM is strongly reduced, qS is indeed the PLSw kernel of ψ by (3.1.7), hence f is a semi-stable kernel. ii) If g is semi-stable, it has to be surjective by [30, (2.2.3)]. Now let g be surjective and s : S −→ Z be a     N N linear and continuous map from a PLSw space S = proj SN , SM to Z = proj ZN , ZM . By passing to a subsequence we may assume by (3.1.4) that s arises as the projective limit of a morphism of spectra     pS N N (sN )N ∈N from SN , SM to ZN , ZM . We show that the pullback diagram P S in PLSw pY

s

Y arises as the projective limit of the local pullback diagrams

PN

pSN

pYN

YN

SN

g

Z

(N ∈ N). By (3.1.7) we

sN

gN

ZN

have to show that the spectrum of local pullbacks is strongly reduced. Then pS arises as the projective limit of the local cokernels (pSN )N ∈N which makes it a cokernel by (3.1.7). ub As pullbacks are kernels, cf. [40, (5.1.2)], [30, (2.1.1)], we have PN = ({(yN , rN ) : gN (yN ) = sN (rN )}) and pYN and pSN are the restrictions of the canonical projections to PN . We show that for each N ∈ N N the restriction of the projection P∞ : P −→ PN even has dense range: Let (yN , rN ) be an element of PN and let UN and VN be 0-neighborhoods in YN respectively SN .    ˜N , which is a 0-neighborhood in SN as gN is a ˜N := 1 UN and V˜N := VN ∩ s−1 gN U We define U N 2 N quotient map, hence open. As we may assume that S∞ has dense range, we can choose an element N ˜ r in S with S∞ (r) − rN ∈ VN . As g is surjective, there is an element y in Y with g (y) = s (r).  N   N    ˜N holds. Thus Then we have by construction that gN Y∞ (y) − yN = sN S∞ (r) − rN ∈ gN U    −1 −1 N ˜N ˜N + g ({0}) = U ˜N + XN . Since by construction we have Y∞ (y) − yN is in gN gN U = U N YN N N N N (X) ˜N , there is an x ∈ X with Y∞ ˜N = UN . Now XN = Y∞ ⊂ Y∞ (X) + U (y) − yN − Y∞ (x) ∈ 2U we define z := y − x and obtain that (z, r) is an element of P , because g (z) = g (y) − g (x) = s (r) as  N  N N x ∈ X = g −1 ({0}). Furthermore, we calculate P∞ ((z, r)) − (yN , rN ) = Y∞ (z) − yN , S∞ (r) − rN =   N N N (x) − yN , S∞ (r) − rN ∈ UN × VN . Thus the spectrum of local pullbacks is indeed Y∞ (y) − Y∞ strongly reduced and the proof is complete. 2

PLSw Theorem 3.1.10. The class Emax , which consists of all kernel-cokernel pairs (f, g) in PLSw such that g is surjective, is the maximal exact structure in PLSw .

Proof. Sieg and Wegner’s characterization (2.1.3) of the maximal exact structure of a pre-abelian category and the determination of semi-stable kernels and cokernels (3.1.9) yield the assertion. 2 We conclude this part with some general remarks about the categories under consideration: Remark 3.1.11. i) When analyzing if the vanishing of Ext groups is passed down to subcategories of the locally convex spaces, three-space properties and the relation of the respective maximal exact structures in categories

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and their subcategories are important aspects to consider. We have proven in (3.1.3)i) and in the previous theorem (3.1.10) that neither LSw nor PLSw are exact subcategories of the locally convex spaces since the restriction of the short topologically exact sequences to both categories is a proper subclass of the respective maximal exact structures. Note that the second author proved in [30] that both LS and PLS are exact subcategories of LCS. Unfortunately we do not know, whether being one of the four types of spaces is a three space property in LCS, i.e. whether C is a fully exact subcategory of LCS for C ∈ {LS, LSw , PLS, PLSw }, meaning that we do not now if for every short exact sequence 0 −→ X −→ Y −→ Z −→ 0 of lcs, X and Z being spaces in C already implies Y being an element of C as well. We only do know that this is not true for the hilbertized versions LH and PLH of LSw and PLSw – we replace the reflexive Banach spaces in the definitions by Hilbert spaces. This is an easy consequence of Enflo, Lindenstrauss and Pisier’s counterexample for Hilbert spaces [14], cf. [8, (2.2.18)i)a)]. Furthermore as an easy consequence of the Five Lemma for exact categories [5, (3.2)] we obtain that being an LSw (resp. an LS) space is a three space property in PLSw (resp. PLS), cf. [30, (5.13)]. ii) We show that neither PLS nor PLSw is quasi-abelian or even semi-abelian, cf. [30, ((3.1.5))]: Any non-surjective linear partial differential operator with constant coefficients P (D) : D  (Ω) −→ D  (Ω) on the space of Schwartz distributions over an open subset Ω ⊂ Rd that is surjective as an operator P (D) : C ∞ (Ω) −→ C ∞ (Ω) yields a kernel NP (Ω) in D  (Ω) with proj1 NP (Ω) = 0 by [40, (3.4.5)]. Hence the cokernel of the embedding of Np (Ω) into D  (Ω), which is given by the Hausdorff completion  of the quotient D  (Ω) NP (Ω) by (3.1.5)ii), is not surjective by [13, (1.4)]. Classical results of Malgrange [22, Chapitre 1, Théorème 4] and Hörmander [17] reduce this problem to finding an open subset Ω ⊂ Rd which is P -convex but not strongly P -convex. For d ≥ 3 such pairs (P, Ω) are easy to find, see e.g. [10, (3.5) a)]. Note that Kalmes showed in [19] that this is not possible for d = 2, thus solving an old conjecture of Trève in the affirmative. Using the above example to construct a morphism f in PLS such that the induced morphism f˜ from the coimage of f into its image in the canonical factorization the same way as in [30, (3.1.6)], we obtain that neither PLS nor PLSw is even semi-abelian. 3.2. Extk functors for PLSw spaces The splitting theory of PLSw spaces is concerned with the following problem: Characterize the pairs (Z, X) of PLSw spaces Z and X such that every short exact sequence 0 −→ X −→ Y −→ Z −→ 0

()

of PLSw spaces is split exact. This problem is of special interest if one of the spaces X or Z is a classical PLSw space as the space D  (Ω) of Schwartz distributions or the space A (Ω) of real analytic functions. The splitting theory has been investigated in the context of PLS spaces by different authors [13,40,3,20] but the theory is far from being complete. Up to the thesis of the second author, the splitting problem has been investigated using an ad-hoc definition of Ext1PLS (Z, X) = 0, i.e. the splitting characterization (2.2.4)ii). The second author proved in his thesis that the abstract tools of the second chapter applied to the exact category PLS yield, in complete analogy to the Fréchet spaces and locally convex spaces, abelian group valued functors ExtkPLS such that every exact sequence (1) of PLS spaces splits if and only if the group Ext1PLS (Z, X) is trivial. We prove that with the same methodology one can achieve the same results for the even larger and more general category of PLSw spaces. In the following we will investigate the functors ExtkPLSw more closely. We will establish a connection between ExtkPLSw and ExtkLSw for LSw spaces, which is

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already suggested by (3.1.8) and (3.1.11)i), analyze the vanishing of Ext1PLSw (Z, X) when X is a countable product of LSw spaces and apply these results to the canonical resolution 0 −→ X −→

N ∈N

ψX

XN −−→



XN

N ∈N

of a PLSw space X to arrive at an analogue of a result for Fréchet spaces, which connects the functors Ext1 and proj1 and also gives a sufficient condition for the vanishing of the higher Extk (see [40, (5.1.5)]) We start by fixing the basic facts about ExtPLSw resp. ExtLSw : Remark 3.2.1. Let C be either the category of PLSw spaces or the category of LSw spaces. Since C is pre-abelian by (3.1.6) respectively (3.1.1) and (3.1.3)i), it is an exact category when endowed with the canonical maximal exact structure, cf. (2.1.3) resp. (2.1.4). Hence we obtain by (2.2.2) for any PLSw or LSw space E and each k ≥ 1 a covariant, additive, abelian-group-valued functor ExtkC (E, _) : C −→ (AB), and a contravariant, additive, abelian-group-valued functor ExtkC (_, E) : C −→ (AB), which both induce for every short exact sequence of PLSw respectively LSw spaces the covariant and contravariant long exact sequences. An immediate consequence of these long exact sequences is that the vanishing of ExtkPLSw (resp. ExtkLSw ) is a three space property: Proposition 3.2.2. Let 0 −→ X −→ Y −→ Z −→ 0 be a short exact sequence of PLSw spaces and let E be a PLSw space. If for any position in one of the long exact sequences the spaces on the left and on the right are trivial, then the middle space is trivial as well. In particular we have that the vanishing of Ext1PLSw (E, Y ) inherits to the complete Hausdorff quotient Z of Y if Ext2PLSw (E, X) = 0 and dually that the vanishing of Ext1PLSw (Y, E) inherits to the ‘closed subspace’ X of Y if Ext2PLSw (Z, E) = 0. The same holds for LSw instead of PLSw . Note that (3.2.2) is an improvement of a result of Domański and Vogt who showed in [13, (1.12)] that in the above situation Ext1PLS (E, X) = 0 and Ext1PLS (E, Z) = 0 implies Ext1PLS (E, Y ) = 0 if all the spaces are PLS and E is ultrabornological. We proceed by analyzing the connection of ExtkPLSw and ExtkLSw , cf. [30, (5.1.6)]. In principle, the result is a consequence of LSw being an exact subcategory of PLSw , (3.1.11)i), and the fact that each short exact sequence of PLS spaces arises as projective limit of a sequence of short exact sequence of LSw spaces, (3.1.8). Nonetheless the proof is not trivial. Proposition 3.2.3. If X, Z are LS-spaces then there is an isomorphism of abelian groups ExtkPLSw (Z, X) ∼ = ExtkLSw (Z, X) for all k ∈ N. Proof. Given two LSw spaces X and Z and k ∈ N, it is an easy task to check that the map Φk : ExtkLSw (Z, X) −→ ExtkPLSw (Z, X) , [E]LSw −→ [E]PLSw is a well-defined group morphism. The injectivity of Φk is yielded by the characterization of [E] = 0, (2.2.4)iii) and the following lemma (3.2.4) that also plays a crucial role for the proof of surjectivity: Given an exact sequence E of length k of LSw spaces with [E]PLSw = 0, (2.2.4)iii) yields that we may append an

B. Dierolf, D. Sieg / J. Math. Anal. Appl. 433 (2016) 1305–1328

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exact sequence of PLSw spaces of length k − 1. This appended sequence can be reduced to an exact sequence of length k − 1 of LSw spaces by the next lemma (3.2.4), hence again (2.2.4)iii) yields [E]LSw = 0. The proof of surjectivity is exactly the statement of the next lemma, cf. [30, (5.1.4)]. 2 N Lemma 3.2.4. Let X be an LSw -space, Z = proj(ZN , ZM ) a PLSw -space and

A: 0

fn

X

Yn−1

fn−1

Yn−2

...

Y1

f1

Y0

f0

Z

0

an exact sequence of PLSw -spaces. Then there exists N0 ∈ N and an exact sequence H: 0

X

gn

Wn−1

gn−1

Wn−2

...

W1

g1

W0

g0

ZN0

0

N0 N0 of LSw -spaces with [A] = [H]Z∞ , i.e. A is contained in [H]Z∞ , which is defined as the equivalence class of the upper row of the commutative diagram

P

pZ

Z

0

i

I1

pW0

PB

N0 Z∞

e1 m1

H: 0

X

gn

Wn−1

gn−1

...

g1

W1

W0

g0

ZN0

0,

where e1 and m1 are given by the exactness of the lower row and the square PB is a pullback. Proof. The proof of the above will also make use of the following factorization lemma for morphisms of short exact sequences, cf. [5, (3.1)]: Lemma 3.2.5. Let (C, E) be an exact category and let X

f

Y

a

X

g

Z c

b

Y

f

g

Z

be a commutative diagram in C with exact rows. Then there is a commutative diagram in C with exact rows X a

X

f

Y

D

m

f

Z

b

(1)

b

X

g

e

(2)

Y

g

Z c

Z,

such that the diagrams (1) and (2) are pullback as well as pushout squares and b ◦ b = b.

B. Dierolf, D. Sieg / J. Math. Anal. Appl. 433 (2016) 1305–1328

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First, we decompose the exact sequence A into short exact sequences of PLSw spaces (1) 0

X

(2) 0

Ik

(3) 0

I1

fn mk m1

Yn−1 Yk−1 Y0

en−1 ek−1

f0

In−1

0,

Ik−1

0 for 2 ≤ k ≤ n − 1,

Z

0.

By 3.1.8 there exist strongly reduced spectra X , Yn−1 , In−1 of LSw spaces and a sequence X

0

Yn−1

In−1

(I)

0

of short exact sequences of LSw spaces such that (1) is the projective limit of (I). Furthermore, we can take X to be the constant spectrum X = (X, idX )N ∈N . The fact that (1) arises as the projective limit of (I) implies that we have a commutative diagram 0

fn

X

en−1

Yn−1

In−1

1 Yn−1,∞

0

X

fn,1

Yn−1,1

0

1 In−1,∞

en−1,1

In−1,1

0

with exact rows, where the lower row is the first step of the sequence (I) of projective spectra of LSw -spaces and where the vertical arrows are the canonical morphisms from the projective limits to the respective steps. Again, use 3.1.8 to find strongly reduced spectra J n−1 , Yn−2 , In−2 of LSw spaces and a sequence J n−1

0

Yn−2

In−2

(II)

0

of exact sequences of LSw spaces such that (2), in the case k = n − 1, is the projective limit of (II) and choose J n−1 as a subsequence of Jn−1 . As before, we find a commutative diagram 0

In−1

mn−1

m

m

0 In−1,∞

0

en−2

Yn−2

0 Yn−2,∞

In−1,m0 m

n−1,m0

Yn−2,m0

en−2,m0

In−2

0

m

0 In−2,∞

In−2,m0

0

with exact rows, where the lower row is the m0 -th step of the sequence (II) for an m0 ≥ 1. Let 1 (Sn−2 , sYn−2 , sIn−1 ) be the pushout of mn−1,m0 and the connecting morphism In−1,m of the spectrum 0 In−1 , then we get a commutative diagram of LSw spaces with exact rows 0

In−1,m0 1 In−1,m 0

0

In−1,1

mn−1,m0

PO sIn−1

Yn−2,m0

en−2,m0

In−2,m0

0

In−2,m0

0.

sYn−2

Sn−2

cn−2

Define Wn−1 := Yn−1,1 , Wn−2 := Sn−2 , gn := fn,1 and gn−1 := sIn−1 ◦ en−1,1 , then the extended diagram

B. Dierolf, D. Sieg / J. Math. Anal. Appl. 433 (2016) 1305–1328

1324

0

X

fn

fn−1

Yn−1

Yn−2

en−1

en−2

In−2

0

mn−1

In−1 m

0 sYn−2 ◦Yn−2,∞

1 In−1,∞

1 Yn−1,∞

m

0 In−2,∞

In−1,1 sIn−1

en−1,1

0

X

gn

Wn−1

Wn−2

gn−1

cn−2

In−2,m0

0

is commutative with exact rows and its lower row is an exact sequence of LSw spaces. Proceeding inductively in this way, we get a commutative diagram A: 0

X

fn

Yn−1

fn−1

...

Y1

f1

f0

Y0

Z

0 N0 Z∞

H: 0

X

gn

Wn−1

gn−1

...

W1

g1

W0

g0

ZN0

0

with exact rows for an N0 ∈ N, whose lower row consists of LSw spaces. Let then 0

I0

Y0

Z

0 N0 Z∞

0

I0, k0

W0

ZN0

0

be the right end of this commutative diagram. Then Lemma 3.2.5 yields a commutative diagram 0

I0

Y0

Z

0

0

I0, k0

D

Z

0

PB 0

I0, k0

W0

N0 Z∞

ZN0

0

with exact rows such that the lower-right square is a pullback. This in turn shows that the diagram with exact rows A: 0

X

A : 0

X

fn

gn

Yn−1

Wn−1

fn−1

gn−1

...

Y1

...

W1

f1

Y0

f0

D PB

H: 0

X

gn

Wn−1

gn−1

...

N0 is commutative, which implies [A] = [A ] = [H]Z∞ .

W1 2

g1

W0

g0

Z

0

Z

0 N0 Z∞

ZN0

0

B. Dierolf, D. Sieg / J. Math. Anal. Appl. 433 (2016) 1305–1328

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Lemma (3.2.4) is a useful tool to analyze the vanishing of ExtkPLSw (E, X) for PLSw spaces E =     N N proj EN , EM and X = proj XN , XM under the assumption of local vanishing, i.e. vanishing of ExtkLSw (EN , XM ), M, N ∈ N. A first result we can deduce is the following lemma, cf. [30, (5.1.7)]:   N be a PLSw space and k ∈ N. Then the vanishing Corollary 3.2.6. Let X be an LSw space, E = proj EN , EM of all ExtkLSw (EN , X) (N ∈ N) implies the vanishing of ExtkPLSw (E, X). Proof. This is Lemma (3.2.4) and the fact that for PLSw spaces E and X and natural numbers k and N0 N0 the map ExtkPLSw (E, X) −→ ExtkLSw (E, XN0 ) , [A] −→ [A]X∞ from (3.2.4) is a group homomorphism. 2 Another consequence is the following proposition, cf. [30, (5.1.8)]:   N . Proposition 3.2.7. Let (XN )N ∈N be a sequence of LSw spaces and E a PLSw space with E = proj EN , EM    Then the vanishing of all ExtkLSw (EN , XM ) (M, N ∈ N) implies the vanishing of ExtkPLSw E, N ∈N XN .    Proof. Let H be an element of ExtkPLSw E, N ∈N XN and L ∈ N. Using the pushout construction dual to the pullback construction for [H]γ from (3.2.4), we arrive at a commutative diagram with exact rows H: 0



fk

XN

fk−1

Yk−1

N ∈N

ΠL ∞

ek−1

sYk−1

PO

Yk−2

...

f0

Z

0

mk−1

Ik−1 c

HL : 0

XL

SL

sXL

,

where ΠL ∞ is the canonical projection, PO is a pushout diagram and the pair (ek−1 , mk−1 ) is induced by the exactness of the first row. Since ExtkLSw (XL , Z) vanishes, (3.2.3) and the characterization of the vanishing of Extk (2.2.4) yield a commutative diagram with exact rows HL : 0

XL

SL

Yk−2

...

Y1

HL : 0

XL

VLk−1

VLk−2

...

VL1

f1

Y0

f0

VL0

Z

0

0

.

Since a countable product of exact sequences of PLSw spaces is again exact in PLSw we obtain a commutative diagram with exact rows H: 0



XN

fk

YK−1

fk−1

...

Y0

f0

Z

0

0

,

N ∈N

0

N ∈N

XN



VNk−1



...

N ∈N

   hence (2.2.4) yields the vanishing of ExtkPLSw E, N ∈N XN .

N ∈N

2

VN0

B. Dierolf, D. Sieg / J. Math. Anal. Appl. 433 (2016) 1305–1328

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By applying the long exact sequence (2.2.2) to the canonical resolution of a PLS space X, Sieg established the same connection between Ext1PLS and proj1 in [30, (5.2.1) & (5.2.3)], which we are used to from the Fréchet setting (see e.g. [40, (5.1.5)]), in [30, (5.2)]. The same proofs yield the same results in the category of PLSw spaces:     N N Theorem 3.2.8. Let X = proj XN , XM and E = proj EN , EM be two PLSw spaces. Then we have:     N N ∗ = 0 and Ext1PLSw (E, X) = 0, then proj1 L (E, XN ) , XM = 0. i) If proj1 XN , XM ii) If   N a) proj1 XN , XM = 0 and b) ExtkLSw (EN , XM ) = 0 for all M, N ∈ N and all 1 ≤ k ≤ k0 for a k0 ∈ N, then   N ∗ α) Ext1PLSw (E, X) = proj1 L (E, XN ) , XM and β) ExtkPLSw (E, X) = 0 for all 2 ≤ k ≤ k0 . N Here XM

∗

  N N denotes as usual HomPLSw E, XM ◦ T. : L (E, XM ) −→ L (E, XN ) , T −→ XM

Proof. As we want to apply the long exact sequence of ExtPLSw to the canonical resolution of X, we just   N have to check, that under the correct assumption, i.e. proj1 XN , XM = 0, it is a short exact sequence in PLSw . Hence we recall:   N we define the map Given a PLSw space X = proj XN , XM

ΨX :

XN −→

N ∈N

N ∈N

 N  XN , (xN )N ∈N −→ XN +1 xN +1 − xN N ∈N .

  N As X is the kernel of ΨX in PLSw and XN , XM is strongly reduced, [40, (3.3.1)] yields that ΨX is open onto its range. Since it has dense range, it is the cokernel of its kernel. Thus the canonical resolution of X 0 −→ X →



Ψ

X XN −→

N ∈N



XN

N ∈N

    PLSw N is an element of Emax if and only if ΨX is surjective. Since proj1 XN , XM = N ∈N XN im (ΨX ) by [40,   N (3.1.4)], this is equivalent to proj1 XN , XM = 0. Thus the long exact sequence (3.2.1) yields an exact sequence 0 → L (E, X) →



L (E, XN ) →

N ∈N

 Extk−1 PLSw

E,





L (E, XN ) → ExtPLSw (E, X) → . . . 1

N ∈N

 XN

 →

ExtkPLSw

N ∈N

(E, X) →

ExtkPLSw

E,



 XN

→ ...,

N ∈N

which immediately implies i). To obtain ii) we observe that (3.2.7) yields the vanishing of   XN 0 for 1 ≤ k ≤ k0 , which implies ExtkPLSw (E, X) = 0 for 2 ≤ k ≤ k0 . FurtherExtkPLSw E, N ∈N

more it follows from the exactness of the sequence

B. Dierolf, D. Sieg / J. Math. Anal. Appl. 433 (2016) 1305–1328

0 → L (E, X) →



ΨL

L (E, XN ) −−−→

N ∈N



1327

L (E, XN ) → ExtPLSw (E, X) → 0, 1

N ∈N

  N ∗ , that Ext1PLSw (E, X) is isomorphic to a L (E, XN ) , XM cokernel of ΨL , which is just the space proj1 (L ). 2 where L denotes the projective spectrum

3.3. ExtPLSw for the space of distributions Domański and Vogt [13] respectively Wengenroth [39] have proven with functional analytic methods that the space of Schwartz distributions D  (Ω) plays the same role in the splitting theory of PLS spaces as the space of rapidly decreasing sequences s does in the splitting theory for Fréchet spaces, i.e. if E is isomorphic to a closed subspace of D  (Ω) and X is isomorphic to a complete Hausdorff quotient of D  (Ω), then Ext1PLS (E, X) = 0. Especially keeping in mind the remark after (3.2.2) about the significance of the vanishing of Ext2 it makes sense to determine the higher Ext groups in the above situation. The second author has given a purely homological proof in [30, (5.3.6)] for the fact that ExtkPLS (E, X) vanishes for all k ≥ 1 for closed subspaces E of D  (Ω) and complete Hausdorff quotients F of D  (Ω), that we can transfer with minor adaptions to the category PLSw : Theorem 3.3.1. If E and X are PLSw spaces such that E is isomorphic to a closed subspace of D  (Ω) and X is isomorphic to a complete Hausdorff quotient of D  (Ω), then ExtkPLSw (E, X) = 0 for all k ≥ 1. N Proof. We work on the right hand side of the sequence space representation D  (Ω) ∼ to Valdivia = (s ) due   1 N [32] and (independently) Vogt [34]. To apply (3.2.8)ii) and (3.2.7) we need to prove that proj XN , XM =0 as well as the vanishing of ExtkLSw (EN , XM ) for all M, N, k ≥ 1. The first statement is a consequence of [13, (1.1)], since X is ultrabornological as a quotient of D  (Ω). For the second statement we have to prove that ExtkLSw (F, Y ) vanishes for closed subspaces F of s and complete Hausdorff quotients Y of s . This is equivalent to the dual statement, the vanishing of ExtkFSw (Y, F ) for closed subspaces Y of s and complete Hausdorff quotients F of s, since the exact structures of LSw and FSw are dual to each other and (2.2.4) yields the equivalence of the vanishing of ExtkLSw (F, Y ) and ExtkFSw (Y  , F  ) for all LSw spaces F and Y and all k ≥ 1. By [35, (1.2) & (1.3)] we know that ExtkF (Y, F ) vanishes for Fréchet spaces Y and F if F is nuclear. Since being an FSw space is three space property by [29, (4.3)] the characterization of the vanishing of the higher Ext (2.2.4)iii) yields that at least Ext2FSw (Y, F ) = 0 if F is nuclear. Since Y is arbitrary, (2.2.5) yields that ExtkFSw (Y, F ) vanishes for all k ≥ 2. Furthermore, again the three space property of being an FSw space yields that the classic splitting result for subspaces and quotient spaces of s due to Vogt and Wagner [33,37] also holds in the category FSw . All in all we have proven ExtkLSw (EN , XM ) = 0 for all M, N, k ≥ 1. Thus, (3.2.8)ii) yields the assertion for k ≥ 2. To see the case k = 1 we observe that (3.2.7) yields Ext1PLSw (E, D  (Ω)) = 0, hence we obtain Ext1PLSw (E, X) = 0 by applying the long exact sequence to the (even) topologically exact sequence

0 −→ D  (Ω) −→ D  (Ω) −→ X −→ 0, which exists by [13, (2.1)]. 2 Acknowledgments The authors thank L. Frerick and J. Wengenroth for many fruitful discussions on the subjects of this article during the supervision of the PhD theses. Furthermore the financial support of the “Stipendienstiftung Rheinland-Pfalz” for both PhD projects is acknowledged.

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