Analytic sheaves in Banach spaces

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4^esérie, t. 40, 2007, p. 453 à 486.

ANALYTIC SHEAVES IN BANACH SPACES

^✩

B

Y

L

ÁSZLÓ

LEMPERT

AND

I

MRE

PATYI

ABSTRACT. – We introduce a class of analytic sheaves in a Banach space X, that we shall call cohesive sheaves. Cohesion is meant to generalize the notion of coherence from finite dimensional analysis.

Accordingly, we prove the analog of Cartan’s Theorems A and B for cohesive sheaves on pseudoconvex open subsetsΩ⊂X, providedXhas an unconditional basis. We also give a few applications.

RÉSUMÉ. – On introduit une classe de faisceaux analytiques dans un espace de Banach complexeX, que nous appelons faisceaux cohésifs. La cohésion généralise l’idée de cohérence en dimension finie.

On démontre l’analogue des Théorèmes A et B de Cartan pour un faisceau cohésif sur un ouvert pseudoconvexeΩ⊂X, en supposant queX admette une base inconditionnelle. Finalement, on propose quelques applications.

0. Introduction

In finite dimensional complex analysis and geometry coherent analytic sheaves play a central role, for the following four reasons:

(i) Most sheaves that occur in the subject are coherent.

(ii) Over pseudoconvex subsets ofCⁿtheir higher cohomology groups vanish.

(iii) The class of coherent sheaves is closed under natural operations.

(iv) Whether a sheaf is coherent can be decided locally.

The purpose of this paper is to introduce a comparable class of sheaves, that we shall call cohesive, in Banach spaces. This notion is different from coherence, which formally makes sense in infinite dimensions as well. However, coherence is not relevant for infinite dimensional geometry, since it has to do with finite generation, while in infinite dimensional spaces one frequently encounters sheaves, such as tangent sheaves and ideal sheaves of points, that are not finitely generated over the structure sheaf. The structure sheaf itself is not known to be coherent in any infinite dimensional Banach space, either.

We shall define cohesive sheaves in general Banach spaces (always over C). However, we are able to prove meaningful results about cohesive sheaves only in some Banach spaces, e.g., in those that have an unconditional basis. (Our main results do not generalize to certain nonseparable spaces, and we do not know whether they hold in all separable spaces, or at least in those that have a Schauder basis.—For the notion of Schauder and unconditional bases, see Section 1.)

✩Research partially supported by NSF grant DMS 0203072 and a Research Initiation Grant from Georgia State University.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

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Cohesive sheaves are sheaves of modules with an extra structure and a special property. We shall arrive at their definition in four steps. Given Banach spacesX, E, and an openΩ⊂X, the sheafOÊ=OÊ_Ω of germs of holomorphic functionsU →E,U ⊂Ωopen, will be called a plain sheaf. It is regarded as a sheaf of modules over O=O^C. If U ⊂Ω is open, F is another Banach space andHom(E, F)denotes the Banach space of bounded linear operators, then any holomorphic ϕ:U →Hom(E, F)induces a homomorphism OÊ|U → O^F|U. Such homomorphisms will be called plain.

Next one defines when a sheaf A of O-modules over Ω is analytic. In traditional finite dimensional terminology analytic sheaves are the same as sheaves ofO-modules. However, the latter notion is adequate only if one is satisfied with studyingO-modules of finite type; already in finite dimensional spaces the sheaf of Banach (space or bundle) valued holomorphic germs has a richer structure than a mereO-module. For example, certain infinite sums of sections make sense, which cannot be explained in terms of theO-module structure. In the finite dimensional context Leiterer in [13] proposed to capture this richness by introducing a Fréchet space structure onΓ(U,A),U⊂Ωopen; in this way he obtained the notion of an analytic Fréchet sheaf. For infinite dimensionalΩthe corresponding notion would not be practical, though, and instead our definition will be inspired by a suggestion of Douady [7]. To generalize the notion of a complex analytic space, he proposed that the infinite dimensional analog of a ringed space should be a

“functored space”. We shall say that a sheafAofO-modules overΩis endowed with an analytic structure if for every plain sheafEa submoduleHom(E,A)⊂Hom_O(E,A)is specified. The correspondenceE →Hom(E,A)should satisfy two natural conditions; then we also say that Ais an analytic sheaf. AnO-homomorphismE → Awill be called analytic if its germs are in Hom(E,A). Any plain sheafAhas a natural analytic structure, namelyHom(E,A)will consist of germs of plain homomorphisms.

Our notion of analyticity slightly conflicts with the traditional terminology: while every sheaf ofO-modules admits an analytic structure, this structure is not unique, see 3.7.

Now consider an infinite sequence

· · · → F2→ F1→ A →0 (∗)

of analytic sheaves and homomorphisms overΩ, with eachFnplain. We shall say that (∗) is a complete resolution ofAif for each pseudoconvexU⊂Ωthe induced sequence on sections

· · · →Γ(U,F2)→Γ(U,F1)→Γ(U,A)→0 (∗∗)

is exact and, moreover, the same is true if in (∗∗) eachFnis replaced byHom(E,Fn)andAby Hom(E,A), for any plain sheafE.

Finally, we shall call an analytic sheafAoverΩcohesive ifΩcan be covered by open sets over whichAhas a complete resolution.

Analytic sheaves locally isomorphic to plain sheaves obviously are cohesive, but at first glance the notion of a complete resolution is quite formidable, and it is not clear how such a resolution can be constructed beyond trivial situations. An exact sequence like (∗) is perhaps doable, but how will one ensure the exactness of (∗∗)beforecohomology vanishing is known? An answer is that in the spaces we consider it suffices to check the exactness of (∗∗) forU in an appropriate subclass of pseudoconvex sets, see Section 6. Using this device it is possible to show that certain important ideal sheaves are cohesive (see Section 10):

THEOREM1. –Suppose a Banach spaceX has an unconditional basis,Ω⊂X is open, and M ⊂Ωis a direct submanifold (see Section1). IfE is another Banach space, then the sheaf J ⊂ O^E_Ωof germs that vanish onM is cohesive.

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The main result of this paper is the following generalization of Cartan’s Theorems A and B (see Section 9):

THEOREM2. –If a Banach spaceXhas an unconditional basis,Ω⊂Xis pseudoconvex, and Ais a cohesive sheaf overΩ, then

(a) Aadmits a complete resolution overΩ;and (b) H^q(Ω,A) = 0forq1.

If coherence was useful because of the four reasons listed earlier, it would make sense to test the notion of cohesion against (i), . . . ,(iv). As to (iv) and (ii), being a cohesive sheaf is a local property, and at least in spaces with unconditional bases, higher cohomology groups of cohesive sheaves do vanish. As to (i), so far two types of sheaves have occurred in infinite dimensional analysis and geometry: the sheaf of sections of a holomorphic Banach bundle, and the ideal sheaf of a submanifold. The former is always cohesive, and, as said, the ideal sheaf of a direct submanifold is also, provided one works in a Banach space that has an unconditional basis. For this reason, the theory of cohesive sheaves allows one to study geometry and analysis on direct submanifolds.

In the Appendix we show that the ideal sheaf of certain analytic subsets is not cohesive; but, since those analytic subsets are “pathological” anyway, this result counts in favor of the notion of cohesion. Indeed, it suggests that intuitively bizarre analytic subsets can be eliminated from complex geometry by testing whether their ideal sheaf is cohesive. We shall briefly introduce and study analytic subvarieties along these lines in Section 11.

Finally, item (iii) above. The class of cohesive sheaves is closed under certain sheaf theoretical operations, but not under all operations that coherent sheaves admit; we give various examples as we develop the theory. This seems to be a feature one has to learn to live with, and is due to peculiarities that Banach space valued functions can exhibit, even those of finitely many variables.

The reader may nevertheless ask why do sheaf theory in infinite dimensional spaces? Are there applications? There certainly are many, within infinite dimensional geometry, and in Section 12 we give a sample. These applications follow from our Theorems 1 and 2 in the same way as many geometric consequences follow from Cartan’s analogous theorems; and there does not seem to be any other way to obtain them. In fact, as in finite dimensions, to understand the analysis and geometry of complex submanifolds, one must develop sheaf theory.

The present paper is the fruit of research independently done by the two authors. Since our main results were essentially equivalent, and the proofs very similar, we decided to publish them jointly. (However, Sections 11 and 13 are solely due to the first author, and the manuscript [29]

contains some further applications not discussed here.)

A few words about the history of analytic cohomology vanishing in vector spaces. Cartan’s Theorems A and B were generalized by Bishop and Bungart, and then by Leiterer to certain, so called Banach coherent analytic Fréchet sheaves over finite dimensional Stein spaces. Douady proved the vanishing of higher cohomology groups of certain sheaves over compact subsets of Banach spaces; see [1,3,4,7,13]. More recently the first author considered pseudoconvex setsΩ in Banach spaces that have an unconditional basis, and proved in [17] that for a trivial Banach (or even Fréchet) bundle overΩhigher cohomology groups vanish. This was generalized in [23, 24], and finally in [19] and [27] to arbitrary locally trivial holomorphic Banach bundles. Further results on sheaf cohomology were obtained by the second author in [25,26,28]. The present paper borrows ideas from most of these works.

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1. Glossary

1.1. For matters of sheaf theory we refer to [32] or [2], for complex analysis in Banach spaces to [21], and for basic notions of infinite dimensional complex geometry to [14, Sections 1–2].

In this glossary we shall nevertheless spell out a few basic definitions of infinite dimensional complex analysis and geometry. LetX andE be Banach spaces, always overC, and Ω⊂X open. We denote the space of continuous linear operatorsX→EbyHom(X, E), endowed with the operator norm.

1.2. DEFINITION. – A function f: Ω→E is holomorphic if for each x∈Ω there is an L∈Hom(X, E)such that

f(y)−f(x) =L(y−x) + o

y−xX

, Ωy→x.

This leads to the notion of what is called in [14] a rectifiable complex Banach manifold: it is a Hausdorff space, sewn together from open subsets of Banach spaces (charts), with holomorphic sewing functions. Using the charts one can define holomorphic maps between rectifiable complex Banach manifolds.

1.3. LetM be a rectifiable complex Banach manifold.

DEFINITION. – (a) A closed subsetN ⊂M is a submanifold if for eachx∈N there are a neighborhoodU⊂M, an open subsetOof a Banach spaceX, a closed subspaceY ⊂X, and a biholomorphic mapU→Othat mapsU∩N ontoO∩Y.

(b)N is a direct submanifold if, in addition,Y above has a closed complement.

The definition implies that the submanifoldNitself is a rectifiable complex Banach manifold.

DEFINITION. – A locally trivial holomorphic Banach bundle overM is a holomorphic map π:L→M, whereLis a rectifiable complex Banach manifold, and each fiberπ⁻¹x,x∈M, is endowed with the structure of a complex vector space. It is required that for everyx∈M there be a neighborhoodU ⊂M, a Banach spaceE, and a biholomorphic mapπ⁻¹U→E×U that maps each fiberπ⁻¹y,y∈U, linearly onE× {y} ≈E.

1.4. DEFINITION. – An upper semicontinuous function u: Ω→[−∞,∞) is plurisubhar- monic if for every pairx, y∈X the functionCλ→u(x+λy)is subharmonic where defined.

DEFINITION. – The open setΩis pseudoconvex if for each finite dimensional subspaceY ⊂X the setΩ∩Y is pseudoconvex inY.

It follows from the characterization of pseudoconvexity in [21, 37.5 Theorem (e) or (f)] that if Ωis pseudoconvex, then so is anyΩ⊂Xbiholomorphic to it.

1.5. DEFINITION. – In a Banach spaceX a sequencee₁, e₂, . . .is a Schauder basis if every x∈Xcan be uniquely represented as a norm convergent sum

x= ∞ n=1

λnen, λn∈C.

The basis is unconditional if, in addition, the above series converge after arbitrary rearrange- ments.

The spacesl^p,L^p[0,1]for1< p <∞,l¹, and the spacec0of sequences converging to zero all have unconditional bases, butL¹[0,1]andC[0,1]have Schauder bases only, see [33].

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2. Plain sheaves

2.1. Fix a Banach space(X, )and an openΩ⊂X. This notation will be used throughout the paper. As explained in the Introduction, ifE is another Banach space, the sheafOÊ=OÊ_Ω of germs of holomorphic functionsU →E,U ⊂Ωopen, is called a plain sheaf. In particular, O=O^Cis a sheaf of rings and eachOÊis in a natural way a sheaf ofO-modules. In general, a sheaf ofO-modules will simply be called anO-module. The sheaf of homomorphisms between O-modulesAandBwill be denotedHom_O(A,B), itself anO-module.

If F is also a Banach space and U ⊂Ω is open, any holomorphic ϕ:U →Hom(E, F) defines a homomorphism O^E|U e→ϕe∈ O^F|U. Such homomorphisms are called plain homomorphisms. The following is obvious:

PROPOSITION. –If a plain homomorphism induced byϕ:U→Hom(E, F)annihilates germs of constant functions, thenϕ= 0.

It follows that germs of plain homomorphismsO^E|U→ O^F|U form anO-module Hom_plain

O^E,O^F

⊂Hom_O

O^E,O^F ,

isomorphic toO^Hom(E,F⁾; its sections over any openU⊂Ωare in one-to-one correspondence with the plain homomorphismsO^E|U→ O^F|U.

3. Analytic sheaves

3.1. If Ais an O-module over Ω, an analytic structure on Ais the choice, for each plain sheafE, of a submoduleHom(E,A)⊂Hom_O(E,A), subject to

(i) ifEandFare plain sheaves,x∈Ω, andϕ∈Homplain(E,F)x, then ϕ^∗Hom(F,A)x⊂Hom(E,A)x;

(ii) Hom(O,A) =Hom_O(O,A).

AnO-module endowed with an analytic structure is called an analytic sheaf. The restriction of an analytic sheaf to an openU⊂Ωinherits an analytic structure. Given analytic sheavesA,B over Ω and an openU ⊂Ω, a homomorphism ϕ:A|U → B|U is called analytic if for every plainEthe induced homomorphism

ϕ_∗:Hom_O(E|U,A|U)→Hom_O(E|U,B|U) mapsHom(E|U,A|U)inHom(E|U,B|U).

Any plain sheafFcan be endowed with an analytic structureHom(E,F) =Hom_plain(E,F).

In what follows, we shall automatically use this analytic structure on plain sheaves. More generally, the sheafSof holomorphic sections of a locally trivial holomorphic Banach bundleS→Ω has a canonical analytic structure: if over an openU⊂Ωthe restrictionS|U is isomorphic to a trivial bundleF×U→U, this isomorphism induces an isomorphism ofO-modules

Hom_O

O^E,O^FU−→^∼ Hom_O

O^E,SU,

and we defineHom(O^E,S)so that its restriction to suchU is the image ofHom(O^E,O^F)|U. Analytic homomorphisms between plain sheaves are just the plain homomorphisms, and, more generally, a homomorphism between plain, resp. analytic, sheaves E andBis analytic if its germs are in Hom(E,B). We shall writeHom(A,B) for the sheaf of germs of analytic homomorphisms between analytic sheaves A and B; if A is plain, this is consistent with

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notation already in use. Further, we shall write Hom(A,B)for the Abelian group of analytic homomorphisms A → B; it is canonically isomorphic to the group Γ(Ω,Hom(A,B)) of sections.

It is possible to define an analytic structure on the sheaf Hom(A,B) itself, but in this generality the structure will not have useful properties. To keep the discussion simple we shall therefore not considerHom(A,B)as an analytic sheaf.

3.2. The notion of an analytic structure is naturally expressed in the language of category theory—even in more than one way—, of which we shall avail ourselves only very sparingly.

Consider the category S of O-modules andO-homomorphisms overΩ (a monoidal category with unitO). NowO-modules overΩalso form a categoryenrichedoverS, or anS-category, see [12], that we shall denote byO. This means that with any pairA,B ∈ObO, i.e.,O-modules, anO-moduleHom_O(A,B)∈Sis associated (the “hom-object”); further, homomorphisms

Hom_O(A,B)⊗Hom_O(B,C)→Hom_O(A,C)

are specified, satisfying certain axioms. There is also theS-subcategoryPofO, whose objects are plain sheaves and the hom-object associated withE,F ∈ObPisHomplain(E,F).

AnyA ∈ObOdetermines a contravariantS-functorHom_O(·,A)fromPtoO. This functor associates withE,F ∈ObPthe homomorphism

Homplain(E,F)→Hom_O

Hom_O(F,A),Hom_O(E,A) ,

induced by composition. In this language an analytic structure onA ∈ObOis anS-subfunctor Hom(·,A)ofHom_O(·,A)satisfyingHom(O,A) =Hom_O(O,A); subfunctor meaning that Hom(E,A)⊂Hom_O(E,A)is a submodule for everyE ∈ObP.

3.3. The sumA ⊕ Bof analytic sheaves has a natural analytic structure: one simply says that ϕ∈Hom_O(E,A ⊕ B)is analytic ifpr_Aϕandpr_Bϕare both analytic. IfCis a further analytic sheaf, anO-homomorphismθ:C|U→ A ⊕ B|Uwill be analytic precisely whenα= pr_Aθand β= pr_Bθare. Nowθis uniquely determined byαandβ, and is denoted byα⊕β. Conversely, an O-homomorphism ψ:A ⊕ B|U → C|U will be analytic if its compositions γ, δ with the embeddingsι_A:A → A ⊕ B,ι_B:B → A ⊕ Bare; in this case we writeψ=γ+δ.

Clearly,O^E⊕ O^Fis analytically isomorphic toO^E⊕F.

3.4. IfAis an analytic sheaf, an analytic structure can be defined on anyO-submoduleS ⊂ A by letting

Hom(E,S) =Hom(E,A)∩Hom_O(E,S).

It is straightforward that (i) and (ii) in 3.1 are satisfied. Similarly, there is an analytic structure onA/S. With the projectionπ:A → A/Sone lets

Hom(E,A/S) =π_∗Hom(E,A)⊂Hom_O(E,A/S).

(3.1)

Again, (i) in 3.1 is straightforward to check. As for (ii), let τ ∈Hom_O(O,A/S)x. Over a neighborhood U of x there is a section a∈Γ(U,A) such that τ1 =πa(x). If an analytic homomorphismα:O|U→ A|U is defined byαf=fa, its germαatxwill satisfy

τ=π_∗α∈π_∗Hom(O,A) =Hom(O,A/S).

In view of (3.1) therefore indeedHom(O,A/S) =Hom_O(O,A/S).

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3.5. In particular, if ϕ:A → B is an analytic homomorphism,Kerϕ⊂ A,Imϕ⊂ B, and Cokerϕ=B/Imϕall have natural analytic structures. Also,ϕfactors through an isomorphism ofO-modules

ψ:A/Kerϕ→Imϕ, (3.2)

which is easily seen to be analytic. In spite of this,ψis not necessarily an isomorphism of analytic sheaves, as the following example shows.

LetΩ =X={0}, and consider a closed subspaceKof a Banach spaceE. The epimorphism E →E/K =F induces an analytic epimorphism ϕ:OÊ → O^F over Ω. In this case (3.2) becomes ψ:OÊ/O^K→ O^F, with each plain sheafOÊ,O^F,O^K endowed with its canonical analytic structure. Now the image of the induced map

ψ_∗:Hom

O^F,O^E/O^K

→Hom

O^F,O^F

containsid_O^F only ifKis complemented. Otherwiseψ_∗ is not surjective, and the inverse ofψ is not analytic.

3.6. Another way to express the same is that given an exact sequence 0→ A−→ B^β −→ C →^γ 0

of analytic homomorphisms, it is not necessarily the case that A ≈Imβ andB/Imβ≈ C as analytic sheaves. These isomorphisms hold precisely when for all plainEthe induced sequence

0→Hom(E,A)−−→^β^∗ Hom(E,B)−−→^γ^∗ Hom(E,C)→0

is also exact. For this reason, whenever one considers analytic homomorphisms and their diagrams, one should also consider the diagrams obtained by applying the functorsHom(E,·).

This, at least partly, motivates the definitions in Section 4 to be presented.

3.7. Any sheaf A of O-modules can be endowed with an analytic structure by setting Hom(E,A) =Hom_O(E,A). In addition to this maximal structure there is also a minimal analytic structure defined as follows. IfU⊂Ωis open, let us say that a homomorphismα:E|U→ A|U is of finite type if there are a finite dimensional Banach spaceF, a plain homomorphism ϕ:E|U → O^F|U, and a homomorphism β:O^F|U → A|U such that α=βϕ. Germs of finite type homomorphisms form a sheaf Hommin(E,A), and the choice Hom(E,A) = Hommin(E,A)defines an analytic structure. Clearly, any analytic structure onAsatisfies

Hommin(E,A)⊂Hom(E,A)⊂Hom_O(E,A).

Neither of the two extreme structures will be of any importance in the sequel, except whenA is locally finitely generated. In this case it is often useful to endow it with the minimal analytic structureHom_min(E,A).

3.8. Further examples of analytic sheaves come from analytic Fréchet sheaves over a finite dimensionalΩ, as defined in [13]. IfAis such a sheaf, the sheavesHom(O^E,A)of so called AF-homomorphisms endowAwith an analytic structure in the sense of this paper.

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4. Resolutions

4.1. DEFINITION. – A sequence A → B → C of analytic homomorphisms over Ωis called completely exact if for each pseudoconvexU ⊂Ωand each plain sheafE over Ωthe induced sequence on sections

Γ

U,Hom(E,A)

→Γ

U,Hom(E,B)

→Γ

U,Hom(E,C) or, equivalently,

Hom(E|U,A|U)→Hom(E|U,B|U)→Hom(E|U,C|U),

is exact. In general, a sequence of analytic homomorphisms is completely exact if its subsequences of length three are all completely exact.

Completely exact sequences are clearly exact, and an exact sequence0→ A−→ B → C^β is completely exact precisely when β is an isomorphism between the analytic sheaves A and β(A)⊂ B.

PROPOSITION. –IfB−→ C →^γ 0is a completely exact sequence over a pseudoconvex Ω,F is a plain sheaf overΩ, andϕ:F → C is an analytic homomorphism, then there is an analytic homomorphismψ:F → Bsuch thatϕ=γψ.

Proof. –Chooseψin the preimage ofϕunder the surjective homomorphism γ_∗: Hom(F,B)→Hom(F,C).

4.2. DEFINITION. – An infinite completely exact sequence

· · ·−−→ F^ϕ² 2 ϕ1

−−→ F1 ϕ0

−−→ A →0 (4.1)

is called a complete resolution ofAif eachFnis plain.

We shall denote (4.1) byF_•→ A →0, or evenA_•→0, withA0=A,An=Fnforn1.

PROPOSITION. –In a complete resolution (4.1) each Kerϕ_n−1 = Imϕ_n has a complete resolution.

Proof. –Indeed,· · ·−−−→ F^ϕⁿ⁺¹ n+1−−→ϕn Imϕn→0is a complete resolution.

4.3. A homomorphismβ_•= (βn)of analytic complexesA•andB•is called analytic if each β_n:An→ Bnis analytic.

PROPOSITION. –Given a complete resolution A• →0 and a completely exact sequence B_•→0 over a pseudoconvexΩ, any analytic homomorphism ϕ0:A0→ B0 can be extended to an analytic homomorphismA_•→ B_•.

Proof. –Letαn:An+1→ Anandβn:Bn+1→ Bn denote the differentials of the complexes, α₋₁= 0,β₋₁= 0. Suppose for0jnanalytic homomorphismsϕ_jhave been constructed so thatϕ_j−1α_j−1=β_j−1ϕ_j. Thenβ_n−1(ϕ_nα_n) =ϕ_n−1α_n−1α_n= 0; sinceHom(An+1,B•)→0 is exact, there is a

ϕn+1∈Hom(An+1,Bn+1) withϕnαn=βnϕn+1. Continuing in the same way we obtain the desired homomorphismϕ_•:A•→ B•.

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4.4. THEOREM. –Let · · ·−−→ A^α¹ 1 −−→ Aα0 0 →0 be a completely exact sequence over a pseudoconvexΩ. IfAnhas a complete resolution forn1, then so doesA0.

Proof. –First we construct a completely exact sequenceB_•→ A0→0, where eachBn has a complete resolution, andB1is plain. Let· · · → F_n → Fn−−→ Aϕn n→0be a complete resolution ofAn, n1. By 4.3 there are analytic homomorphismsψnsuch that the diagram

0 0 0

· · · A3 α2

A2 α1

A1 α0

A0 0

· · · F3 ψ2

ϕ3

F2 ψ1

ϕ2

F1 ϕ1

(4.2)

commutes. We claim that the sequence

· · ·−−→ F^β³ 3⊕Kerϕ2−−→ Fβ2 2⊕Kerϕ1−−→ Fβ1 1−−→ Aβ0 0→0 (4.3)

is completely exact, where we write ιn for the inclusion Kerϕn → Fn and, with notation introduced in 3.3,

β0=α0ϕ1, β1=ψ1−ι1, βn= (ιn⊕ψn−1)(ψn−ιn), n2.

One checks that theβ_n’s do map into the sheaves indicated in (4.3).

With a pseudoconvex U ⊂Ωand a plainE, applyΓ(U,Hom(E,·))to both diagrams (4.2) and (4.3). We obtain diagrams of Abelian groups

· · · â³ A₃ â² A₂ â¹ A₁ â⁰ A₀ 0

· · · ^p³ F3 p2

f3

F2 p1

f2

F1 f1

(4.4)

and

· · ·−→^b³ F3⊕Kerf2−→b2 F2⊕Kerf1−→b1 F1−→b0 A0→0;

(4.5)

the first is commutative, its top row is exact, and eachfnis surjective. Here b₀=a₀f₁, b₁=p₁−i₁, b_n= (i_n⊕p_n−1)(p_n−i_n), n2,

where in = Kerfn →Fn is the inclusion. It is straightforward computation that (4.5) is a complex. To prove (4.3) is completely exact we have to show (4.5) is exact.

First,b₀is surjective because botha₀andf₁are. Second, ifξ∈Kerb₀then f₁ξ∈Kera₀= Ima₁= Ima₁f₂= Imf₁p₁;

we used the fact that f2 is surjective. Let f1ξ=f1p1ζ and ω=p1ζ−ξ∈Kerf1, so that ξ=b₁(ζ, ω)∈Imb1. Third, if n1 and(ξ, η)∈Kerbn= Ker(pn−in), thenη=pnξ and 0 =f_np_nξ=a_nf_n+1ξ. Hence, as before

fn+1ξ∈Iman+1= Iman+1fn+2= Imfn+1pn+1.

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Chooseζso thatfn+1ξ=fn+1pn+1ζ; thenω=pn+1ζ−ξ∈Kerfn+1. We conclude(ξ, η) = bn+1(ζ, ω)∈Imbn+1. Thus (4.5) is exact and (4.3) is completely exact. It follows from the Proposition in 4.2 that eachFn⊕Kerϕ_n−1has a complete resolution; hence indeed there is a completely exact sequence

· · · → B3−−→ Bβ2 2−−→ Eβ1 1−→ Aε0 0→0, whereE1=F1is plain and eachBnhas a complete resolution.

Consider the completely exact sequence

· · · → B3 β2

−−→ B2 β1

−−→Kerε0→0.

By what we have proved there are completely exact sequences

· · · → C4→ C3→ E2−→ε1 Kerε₀→0, and so

· · · → C4→ C3→ E2−→ Eε1 1−→ Aε0 0→0,

withE2 plain and eachCn having a complete resolution. Continuing in this way we obtain a complete resolutionE•→ A0→0.

4.5. THEOREM (“Three lemma”). –Suppose0→ A−→ B^β −→ C →^γ 0 is a completely exact sequence over a pseudoconvexΩ. If two amongA,B, andChave a complete resolution, then so does the third.

Proof. –We can assume thatA ⊂ Bandβis the inclusion map.

(a) IfAandBhave a complete resolution, then 4.4 implies thatCalso has one.

In both remaining casesCis known to have a complete resolution· · · → F−→ C →^ϕ 0. By the Proposition in 4.1 there is a commutative diagram

0

0 A ^β B ^γ C 0

F

ψ

F,

ϕ

(4.6)

whereψ:F → Bis an analytic homomorphism. We letι: Kerϕ→ Fdenote the inclusion, and claim that the sequence

0→Kerϕ−−−→ A ⊕ F^ψ⊕ι −−−→ B →^β−ψ 0 (4.7)

is completely exact.

With a pseudoconvexU⊂Ωand a plainEapplyΓ(U,Hom(E,·))to (4.6) and (4.7) to obtain diagrams of Abelian groups

0 A ^b B ^c C 0

F

p

F

f

e ◦

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and

0→Kerf−−→^p⊕i A⊕F−−−→^b−p B→0.

(4.8)

The first is commutative, its top row is exact, andf is surjective. The latter is clearly a complex and exact atKerf; we have to check it is exact at the next two terms. If(ξ, η)∈Ker(b−p)then pη=bξ=ξ, whence0 =cpη=f η. Thus η∈Kerf and(ξ, η) = (p⊕i)η∈Imp⊕i. On the other hand, ifζ∈Bthen with someη∈F

−cζ=f η=cpη, i.e., ζ+pη=ξ∈Kerc=A.

Thusζ=ξ−pη∈Im(b−p). We conclude that (4.8) is exact and so (4.7) is completely exact.

Note thatKerϕhas a complete resolution by 4.2.

(b) Now supposeAtoo has a complete resolution. ThenA ⊕ F also has one and by part (a) of this proof, (4.7) implies that so doesB.

(c) IfB, rather thanA, is known to have a complete resolution, then by part (b) (4.7) implies thatA ⊕ Fhas a complete resolution. In view of the completely exact sequence

0→ F → A ⊕ F → A →0, part (a) lets us concludeAhas a complete resolution.

4.6. There is a clear parallel between properties of coherent sheaves and sheaves that have complete resolutions. However, one should bear in mind that if A and B have complete resolutions andϕ:A → B is an analytic homomorphism, thenKerϕ⊂ Awill not necessarily have a complete resolution, not even locally; nor willImϕ⊂ B, see 5.3 below. Because of this, the proof of the Three lemma is more complicated than the corresponding proof for coherent sheaves.

5. Cohesive sheaves

5.1. DEFINITION. – An analytic sheaf A over Ω is called cohesive if each x∈Ω has a neighborhood over whichAadmits a complete resolution.

An analytic sheafAthat is locally isomorphic to plain sheaves (in other words, the sheaf of sections of a locally trivial holomorphic Banach bundle) is cohesive. Indeed, ifEis a plain sheaf over an openU⊂Ωsuch thatE ≈ A|U, then· · · →0→ E−→ A|U^≈ →0is a complete resolution.

Over a finite dimensionalΩthere are many more examples of cohesive sheaves. It can be shown that coherent sheaves with their minimal analytic structure discussed in 3.7 are cohesive. We do not know whether Leiterer’s Banach coherent analytic Fréchet sheaves, with their analytic structure defined in 3.8, are cohesive or not.—The theory developed in this paper nevertheless can be generalized so that it includes the sheaves considered by Leiterer. In the definition of plain sheavesO^E one can restrict to Banach spacesE in some full subcategory of all Banach spaces, and base the notion of cohesive sheaves on this restricted class of plain sheaves. As long as the subcategory of Banach spaces we choose is closed under (an appropriate completion of) countable direct sums, all our results up to Section 9 carry over. If the subcategory consists ofL¹ spaces of discrete measure spaces, also known asl¹spaces, the resulting cohesive sheaves over finite dimensionalΩwill be the same as Banach coherent analytic Fréchet sheaves. This follows from [13, 1.3. Proposition, Theorems 2.2, 2.3, and Lemma 3.4].

Further examples of cohesive sheaves can be constructed by the following theorem, an immediate consequence of 4.5.

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THEOREM. –Suppose0→ A → B → C →0is a completely exact sequence overΩ. If two amongA,B, andCare cohesive, then so is the third one.

5.2. IfXis another Banach space,Ω⊂Xis open, andf: Ω→Ωis biholomorphic, sheaves ofOΩ-modules can be pulled back byfto sheaves ofOΩ-modules. The pullback of a plain sheaf will be plain, and the pullback of an analytic sheaf will carry a natural structure of an analytic sheaf; finally, the pullback of a cohesive sheaf will be cohesive.

5.3. However, not all properties of coherent sheaves carry over to cohesive ones. The following is an adaptation of an example of Leiterer [13, p. 94].

Example. – OverCthere is an analytic endomorphismϕ:F → F of a plain sheaf such that the sequence

0→Kerϕ→ F →Imϕ→0 is completely exact, but neitherKerϕnorImϕis cohesive.

LetF=l², F=O^F_C, and withζ∈Cconsiderψ(ζ)∈Hom(F, F),

ψ(ζ)(y0, y1, . . .) = (y1−ζy0, y2−ζy1, . . .), y= (yn)∈F.

In fact,ψ:C→Hom(F, F)is holomorphic, and induces an analytic endomorphismϕ of F.

The kernelKofϕis supported on the discD={ζ∈C: |ζ|<1}, where it is generated by the functionh(ζ) = (ζⁿ)^∞_n=0.

As toJ = Imϕ, overDit agrees withF. Indeed,ψ(ζ)has a right inverse

(yn)^∞₀ → _n

ν=1

ζ^ν−1y_n−ν _∞

0

whenζ∈D. It follows that overD ϕhas an analytic right inverseF → F, whenceJ |D=F|D;

also for any plainE=O^Eand openU⊂Dthe sequence 0→Γ

U,Hom(E,K)

→Γ

U,Hom(E,F) ϕ_∗

−−→Γ

U,Hom(E,J)

→0 (5.1)

is exact.

In fact, (5.1) is exact for all openU⊂C. To prove this we shall need two auxiliary results.

PROPOSITION 1. –Let W ⊂C be open and σ:W →Hom(E, F). If for each e∈E the functionσ(·)e:W→F is holomorphic, thenσitself is holomorphic.

Proof. –This is the content of [21, Exercise 8E], whose solution rests on Cauchy’s formula and the principle of uniform boundedness.

PROPOSITION 2. –Let c ∈∂D, let U ⊂C be a neighborhood of c, and σ:U\D → Hom(E, F)holomorphic. If for each e∈E the function σ(·)e analytically continues across c, then so doesσ.

Proof. –Fixε >0so thatcis the unique point on∂(U\D)at distanceεto(1 +ε)c=c. WithA∈(0,∞)andB∈(0,1/ε)consider

EAB=

e∈E: σ^(k)(c)e

FAB^kk!, k= 0,1, . . . .

e ◦

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These are closed subsets of E, and the assumption, combined with Cauchy estimates, implies that their union is all ofE. By Baire’s category theorem someEABhas an interior point, whence 0∈intEABfor some (perhaps other)A, B; and in fact

σ^(k)(c)e

FAB^kk!eE, e∈E

(with yet anotherA, B). But then this implies that the Taylor series ofσaboutchas radius of convergence1/B > ε, and represents the analytic continuation ofσacrossc.

Now we return to the analysis of the example, and show that for every plain E=O^E the sequence

0→Hom(E,K)→Hom(E,F)−−→^ϕ^∗ Hom(E,J)→0 (5.2)

is exact; exactness overD we already know. As always, the issue is whetherϕ_∗is surjective.

To verify this, considerc∈C\Dand a germι∈Hom(E,J)crepresented by a homomorphism E → J over some connected neighborhoodU⊂Cofc. Let this homomorphism be induced by a holomorphic functionθ:U→Hom(E, F), and write

θ(ζ)e=

θn(ζ)e_∞

n=0∈F, ζ∈U, e∈E.

For eache∈E thenθ(·)einduces a section ofJ |U. SinceK|C\D= 0, there is a uniqueF valued holomorphic functiong= (gn)on a neighborhood ofU\Dsuch that

θ(ζ)e=ψ(ζ)g(ζ), i.e.,θν(ζ)e=gν+1(ζ)−ζgν(ζ) forν= 0,1,2, . . .andζ∈U\D.

One computes

−

nν<N

ζ^n−ν−1θν(ζ)e=gn(ζ)−ζ^n−NgN(ζ) forNn, hence

N→∞lim

−

nν<N

ζⁿ⁻^ν⁻¹θ_ν(ζ)e _∞

n=0

=g(ζ).

(5.3)

For fixedζ∈U \D, if we vary e, the N-th term on the left-hand side of (5.3) represents a continuous linear operatorSN:E→F. According to (5.3) theSN converge pointwise, so that by the principle of uniform boundednessσ(ζ) = limS_N is a continuous linear operator. Hence we have a functionσ:U\D→Hom(E, F),

σ(ζ)e=

−^∞

ν=n

ζ^n−ν−1θν(ζ)e _∞

n=0

∈F, e∈E,

to which Proposition 1 applies, cf. (5.3). We conclude thatσis holomorphic; alsoψσ=θover U\D.

Now ifc /∈∂D thenU\D is a neighborhood ofc. Ifc∈∂D then by (5.3)σ(·)econtinues acrosscfor eache∈E. Therefore Proposition 2 implies thatσitself continues acrossc. In either case there are a neighborhoodV ofcand a holomorphicσ:V →Hom(E, F)satisfyingψσ=θ overV. Passing to germs atcwe obtainι∈Imϕ_∗, and (5.2) is indeed exact.

Note that for any openU⊂C H¹

U,Hom(E,K)

=H¹

U∩D,Hom(E,O)

=H¹

U∩D,O^E^∗

= 0,

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see [1, Theorem 4], [3, p. 331], [13, Theorem 2.3], or 9.1 below. Hence the cohomology sequence of (5.2) gives that (5.1) is exact.

Thus we proved that0→ K → F → J →0 is completely exact. However, K= Kerϕ is not cohesive. Indeed, on a connected neighborhoodU of 1∈C any analytic homomorphism E → K|U⊂ F|U of a plain sheafEmust be 0 onU\D, hence on all ofU. ThereforeJ = Imϕ cannot be cohesive, either, by virtue of the theorem in 5.1.

5.4. Similar constructions lead to two more noteworthy examples.

Example. – There is a cohesive sheafAoverCwhose support has interior points; yet it is not all ofC.

We take F =l², F=O^F as in the previous example, but now we define a holomorphic ψ:C→Hom(F, F)by

ψ(ζ)(y0, y1, . . .) = (y0, y1−ζy0, y2−ζy1, . . .).

The analytic endomorphismϕ:F → F induced by ψ has kernel 0 this time. As in 5.3, one shows that the sequence0→ F−→^ϕ Imϕ→0is completely exact. ThusImϕ≈ F and—by 9.2 below—A=F/Imϕare cohesive; and the support of this latter isC\D.

5.5. Example. – There are a plain sheaf F over Cand two cohesive subsheavesA,B ⊂ F such thatA ∩ Bis not cohesive.

Again letF=l²,F=O^F and consider holomorphic mapsψ1, ψ2:C→Hom(F, F) ψ₁(ζ)(y₀, y₁, . . .) = (y₀, ζy₀, y₁, ζy₁, y₂, ζy₂, . . .),

ψ2(ζ)(y0, y1, . . .) = (y0, y1, ζy1, y2, ζy2, y3, . . .),

and the induced analytic endomorphismsϕ1, ϕ2ofF. The left inverses ofψ1(ζ),ψ2(ζ)given by y→(y0, y2, y4, . . .), resp.y→(y0, y1, y3, y5, . . .)

induce analytic left inverses ofϕ1, ϕ2, so thatϕjare in fact analytic isomorphisms on their im- ages. In particular, A= Imϕ1 and B= Imϕ2 are cohesive. However,A ∩ B coincides with Kerϕof the Example in 5.3, and is not cohesive.

6. Resolutions and cohomology

6.1. Now we turn to the following two questions. First, how can cohesive sheaves be constructed in infinite dimensional spaces, beyond those that are locally isomorphic to plain sheaves? Second, do higher cohomology groups of cohesive sheaves over pseudoconvex sets vanish? Eventually we shall answer the second question in the affirmative in rather general Banach spaces, and will obtain a useful if quite particular answer to the first. In this section and in the next we shall concern ourselves with a special case of the questions: how to recognize when a sequence of analytic homomorphisms is a complete resolution; and, once a complete resolution E•→ A →0 over a pseudoconvexΩ is granted, how to prove H^q(Ω,A) = 0 for q1? Of course, one hopes to exploitH^q(Ω,En) = 0,q1, which is known in a large class of Banach spaces. It turns out that the two issues are related, and are best treated in generality greater than that of Banach spaces, at least initially.

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6.2. We start with a simple result on the continuity of ˇCech cohomology groups, versions of which have been well understood and widely used in complex analysis for a long time.

LEMMA. –Let K be a sheaf of Abelian groups over a topological space Ω, and q2 an integer. ForN= 1,2, . . . ,letU_N,V_N, andW_N be families of open subsets ofΩ, each finer than the previous one. Assume the sequences{U_N}^∞_N=1,{V_N}^∞_N=1, and{W_N}^∞_N₌₁are increasing, and the refinement homomorphisms

H^q(U_N,K)→H^q(V_N,K), H^q⁻¹(V_N,K)→H^q⁻¹(W_N,K) are zero. Then, writing U=_∞

1 U_N and W= _∞

1 W_N, the refinement homomorphism H^q(U,K)→H^q(W,K)is also zero.

Proof. –We shall write V=_∞

1 V_N. Let us introduce the following notation. If S is a family of open subsets ofΩ,(C^•(S), δ)denotes the ˇCech complex ofS, with values inK, and Z^•(S) = Kerδthe complex of cocycles. Given a finer familyTand a refinement mapT→S, we denote the induced homomorphism C^•(S)→C^•(T) by f →f|T. Among the families in the Lemma there are various refinement maps: first of all, the inclusions U_N →U_M →U, etc. forNM; then refinement mapsW_N →V_N →U_N, that we choose to commute with the inclusions; and finally,W→U, the union of the mapsW_N →U_N. We fix these maps and the induced homomorphisms on the groups of cochains.

To prove the Lemma, let f ∈Z^q(U)represent a cohomology class[f]∈H^q(U,K). By the assumption, for each N there is agN ∈C^q−1(V)such thatf|V_N =δgN|V_N. It follows that (gN −gN+1)|V_N ∈Z^q−1(VN), hence, again by the assumption, there is anhN ∈C^q−2(W) such that(gN−gN+1)|W_N=δhN|W_N. If we define

kN =gN +δ

N−1 1

hn∈C^q−1(W),

6.3. LetPbe a basis of open sets in a topological spaceΩ. (In subsequent applications,Ωwill be an open subset of a Banach space andPthe family of pseudoconvex subsets ofΩ.)

DEFINITION. – We shall say thatPis exhaustive if, given anyP∈Pand any open coverU ofP, there is an increasing sequence ofP_N∈P, each covered by finitely many elements ofU, such that_∞

1 PN=P.

6.4. THEOREM. –LetΩbe a topological space,Pan exhaustive basis of open sets inΩ, with each U ∈P paracompact, and let O⊂Pbe another basis of open sets. Both P and O are assumed to be closed under finite intersections. Consider an infinite sequence

· · · → A2−−→ Aα1 1−−→ Aα0 0→0 (6.1)

of sheaves of Abelian groups overΩ. Assume (i) the induced sequence on sections

· · · →Γ(U,A2)→Γ(U,A1)→Γ(U,A0)→0 (6.2)

is exact for allU∈O;and

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(ii) H^q(U,Ar) = 0forq, r1,U∈P.

Then

(a) (6.2)is exact for allU∈P;and (b) H^q(U,A0) = 0forq1,U∈P.

If (6.2) is exact then clearly so is (6.1). Now given an exact sequence (6.1) and a fixed U ⊂Ω, the sole hypothesisH^q(U,Ar) = 0 for all q, r1 already implies H^q(U,A0) = 0, q1, provided that eitherAr= 0for somer, orΩis finite dimensional. In the absence of these finiteness conditions it is not hard to construct examples whereH^q(U,Ar) = 0forr1but not forr= 0. Since the resolutions we are working with in this paper are over infinite dimensionalΩ, and typically allAr= 0, we are forced to make the stronger assumptions (i) and (ii). The proof will still rest on a finiteness property, the one in the definition of exhaustivity. When it comes to applying the Theorem in complex analysis, the hardest is precisely to prove that the family of pseudoconvex sets is exhaustive.

Proof of (b). –Since for anyU∈Pthe familyP|U={P∈P: P⊂U}is an exhaustive basis of open sets inU, it will suffice to verify (b) forU = Ω, under the assumption that Ωitself is inP. This is what we shall do.

We start by introducingKr= Kerαr−1= Imαr⊂ Ar andK0=A0, that fit in short exact sequences

0→ Kr+1→ Ar+1−−→ Kαr r→0, r0.

(6.3)

By (i) the associated sequences

0→Γ(U,Kr+1)→Γ(U,Ar+1)→Γ(U,Kr)→0 (6.4)

are also exact forU∈O.

The heart of the matter is the following: given an increasing sequence of familiesU_N ⊂O such thatU=_∞

1 U_N coversΩ, there is an increasing sequence of familiesV_N ⊂O, eachV_N finer thanU_N, such that_∞

1 V_N also coversΩ, and the refinement homomorphism H^q(UN,Kr)→H^q(VN,Kr)is zero forq1.

(6.5)

To verify this, choose an increasing sequence ofP_N ∈P, each covered by a finiteT_N ⊂U, such that_∞

1 PN= Ω. At the price of introducing empty sets at the start and repeating some of thePN’s, we can assume thatT_N ⊂U_N∩T_N₊₁, and hasN elements. Consider the covering

V_N ={V ∈O: V ⊂P_N∩T with someT∈T_N} ofPN. ThusV_N is finer thanU_N.

(6.4) induces exact sequences

0 C^•(U_N,Kr+1) C^•(U_N,Ar+1) C^•(U_N,Kr) 0

0 C^•(V_N,Kr+1) C^•(V_N,Ar+1) C^•(V_N,Kr) 0.

With vertical arrows determined by a refinement mapV_N →U_N, this diagram is commutative.

The bottom row induces an exact sequence

e ◦