Open embeddings and pseudoflat epimorphisms
aa r X i v : . [ m a t h . F A ] N ov OPEN EMBEDDINGS AND PSEUDOFLAT EPIMORPHISMS
O. YU. ARISTOV AND A. YU. PIRKOVSKII
Dedicated to Professor Alexander Ya. Helemskii on the occasion of his 75th birthday
Abstract.
We characterize open embeddings of Stein spaces and of C ∞ -manifolds interms of certain flatness-type conditions on the respective homomorphisms of functionalgebras. Introduction
Our main motivation comes from the following fact in algebraic geometry. If ( X, O X )and ( Y, O Y ) are affine schemes, then a morphism f : ( Y, O Y ) → ( X, O X ) is an open embed-ding if and only if the respective homomorphism f • : O ( X ) → O ( Y ) is a flat epimorphismof finite presentation [26, 17.9.1]. We are interested in complex analytic and smooth ver-sions of this result. Specifically, given a morphism f : ( Y, O Y ) → ( X, O X ) of Stein spaces,we are looking for a condition on f • : O ( X ) → O ( Y ) that is necessary and sufficientfor f to be an open embedding. A similar question makes sense for C ∞ -manifolds. Toget a reasonable answer, we equip the algebras of holomorphic and smooth functionswith their canonical Fr´echet space topologies and consider them as functional analyticobjects [29, 30].It is easy to see that the above-mentioned algebraic result does not extend verbatim tothe complex analytic case. Indeed, if U is an open subset of a Stein space ( X, O X ), then O ( U ) is normally not flat as a Fr´echet O ( X )-module. This observation is essentially dueto M. Putinar [44] (see also [16]) and is closely related to the spectral theory of linearoperators on Banach spaces. Actually, if O ( U ) were flat over O ( C ) for every open subset U ⊂ C , then each Banach space operator would possess Bishop’s property ( β ), which isnot the case [6]. For a direct proof of the fact that O ( D ) is not flat over O ( C ) (where D ⊂ C is the open unit disc), see [38].A reasonable substitute for the flatness property was introduced by J. L. Taylor [52].Given a continuous homomorphism ϕ : A → B of Fr´echet algebras, he says that ϕ isa localization if, for each Fr´echet B -bimodule M , the induced map of the continuousHochschild homology H • ( A, M ) → H • ( B, M ) is an isomorphism. Taylor also proved that,if A and B are nuclear, then the above condition means precisely that (i) Tor Ai ( B, B ) = 0for all i ≥
1, and (ii) Tor A ( B, B ) ∼ = B canonically. Homomorphisms satisfying (i) and (ii)were rediscovered several times under different names [4, 12, 20, 33, 35], both in the purelyalgebraic and in the functional analytic contexts (see Remark 3.16 for historical details).We adopt the terminology of [20] and call such maps homological epimorphisms. To bemore precise, there are two types of homological epimorphisms in the functional analyticsetting, weak and strong homological epimorphisms. For nuclear Fr´echet algebras, weakhomological epimorphisms are the same as Taylor’s localizations, while strong homologicalepimorphisms are the same as Taylor’s absolute localizations. See Section 3 for details. Mathematics Subject Classification.
The fundamental (and chronologically the first) example of a weak homological epi-morphism that is not necessarily flat is the restriction map O ( C n ) → O ( U ), where U isa Stein open subset of C n (i.e., a domain of holomorphy). This fact was proved by Tay-lor [52, Prop. 4.3] and was the main motivation for him to introduce weak homologicalepimorphisms. The second author [38, Theorem 3.1] observed that the same result holdsif we replace C n by an arbitrary Stein manifold. Recently, F. Bambozzi, O. Ben-Bassatand K. Kremnizer [2], working in the setting of bornological algebras, proved that theabove property actually characterizes open embeddings of Stein spaces (not only over C ).Other examples of homological epimorphisms in the functional analytic context can befound in [13–15, 40, 41, 52, 53].In the present paper, we introduce a wider class of Fr´echet algebra homomorphisms A → B that we call n -pseudoflat epimorphisms (where n is a fixed nonnegative integer).Such homomorphisms are defined by the conditions that Tor Ai ( B, B ) = 0 for all 1 ≤ i ≤ n and Tor A ( B, B ) ∼ = B canonically. For n = 1, pseudoflat epimorphisms were introducedby G. M. Bergman and W. Dicks [5] in the purely algebraic setting. They also appearnaturally in [1, 3, 49], for example. As far as we know, pseudoflat epimorphisms were notconsidered before in the functional analytic framework. Our main results are Theorems 4.2and 5.3, which characterize open embeddings of Stein spaces and of smooth manifolds interms of pseudoflat epimorphisms.The paper is organized as follows. Section 2 contains some preliminaries from homolog-ical algebra in categories of Fr´echet modules. Our main reference is [29]; some facts thatare missing in [29] can be found in [16, 41, 51]. In Section 3, we introduce n -pseudoflatepimorphisms of Fr´echet algebras, give some examples, and characterize epimorphisms,0-pseudoflat epimorphisms, and 1-pseudoflat epimorphisms in terms of noncommutativedifferential forms. In particular, we show that not every Fr´echet algebra epimorphism is0-pseudoflat (in contrast to the purely algebraic case). Our main results are contained inSections 4 and 5. In Section 4, we show that a map f : ( Y, O Y ) → ( X, O X ) of Stein spacesis an open embedding if and only if the respective homomorphism f • : O ( X ) → O ( Y ) isa 1-pseudoflat epimorphism. Some other equivalent homological conditions on f • are alsogiven. This is a partial generalization of the main result of [2]. However, in contrast to [2],we work only over C , and we deal with topological (rather than bornological) algebras. InSection 5, we show that a similar result holds for the algebras of C ∞ -functions on smoothreal manifolds. Section 6 contains some remarks and open questions related to functionalgebras on Stein spaces and on C ∞ -differentiable spaces.2. Preliminaries
Throughout, all vector spaces and algebras are assumed to be over the field C of complexnumbers. All algebras are assumed to be associative and unital. By a Fr´echet algebra wemean an algebra A equipped with a complete, metrizable locally convex topology (i.e., A is an algebra and a Fr´echet space simultaneously) such that the multiplication A × A → A is continuous. A left Fr´echet A -module is a left A -module M equipped with a complete,metrizable locally convex topology in such a way that the action A × M → M is continuous.We always assume that 1 A · x = x for all x ∈ M , where 1 A is the identity of A . Left Fr´echet A -modules and their continuous morphisms form a category denoted by A - mod . Thecategories mod - A and A - mod - A of right Fr´echet A -modules and of Fr´echet A -bimodulesare defined similarly. Note that A - mod - A ∼ = A e - mod ∼ = mod - A e , where A e = A b ⊗ A op , andwhere A op stands for the algebra opposite to A . The space of morphisms from M to N in PEN EMBEDDINGS AND PSEUDOFLAT EPIMORPHISMS 3 A - mod (respectively, in mod - A , in A - mod - A ) will be denoted by A h ( M, N ) (respectively, h A ( M, N ), A h A ( M, N )). Given Fr´echet algebras A and B , we denote by Hom( A, B ) theset of all continuous algebra homomorphisms from A to B .If M is a right Fr´echet A -module and N is a left Fr´echet A -module, then their A -moduletensor product M b ⊗ A N is defined to be the quotient ( M b ⊗ N ) /L , where L ⊂ M b ⊗ N isthe closed linear span of all elements of the form x · a ⊗ y − x ⊗ a · y ( x ∈ M , y ∈ N , a ∈ A ). As in pure algebra, the A -module tensor product can be characterized by theuniversal property that, for each Fr´echet space E , there is a natural bijection betweenthe set of all continuous A -balanced bilinear maps from M × N to E and the set of allcontinuous linear maps from M b ⊗ A N to E .A chain complex C • = ( C n , d n ) n ∈ Z of Fr´echet A -modules is admissible if it splits inthe category of topological vector spaces, i.e., if it has a contracting homotopy consistingof continuous linear maps. Geometrically, this means that C • is exact, and Ker d n is acomplemented subspace of C n for each n . A left Fr´echet A -module P is projective if thefunctor A h ( P, − ) : A - mod → Vect (where
Vect is the category of vector spaces and linearmaps) is exact is the sense that it takes admissible sequences of Fr´echet A -modules toexact sequences of vector spaces. Similarly, a left Fr´echet A -module F is flat if the tensorproduct functor ( − ) b ⊗ A F : mod - A → Vect is exact in the same sense as above. It isknown that every projective Fr´echet module is flat.A projective resolution of M ∈ A - mod is a pair ( P • , ε ) consisting of a nonnegative chaincomplex P • = ( P n , d n ) n ≥ in A - mod and a morphism ε : P → M such that the sequence0 ← M ε ←− P • is an admissible complex and such that all the modules P n ( n ≥
0) areprojective. It is a standard fact that A - mod has enough projectives , i.e., each left Fr´echet A -module has a projective resolution. The same is true of mod - A and A - mod - A . Inparticular, the (unnormalized) bimodule bar resolution of A [29, Section III.2.3] looks asfollows: 0 ← A µ A ←− A b ⊗ A d ←− A b ⊗ A b ⊗ A ← · · · ← A b ⊗ n ← · · · (2.1)Here µ A is the multiplication map, and d : A b ⊗ → A b ⊗ is given by d ( a ⊗ b ⊗ c ) = ab ⊗ c − a ⊗ bc ( a, b, c ∈ A ) . (2.2)The explicit formula for the higher differentials A b ⊗ ( n +1) → A b ⊗ n is similar [loc. cit.]; we donot need it here. The augmented complex (2.1) is a projective resolution of A in A - mod - A .If M ∈ mod - A and N ∈ A - mod , then the space Tor An ( M, N ) is defined to be the n thhomology of the complex P • b ⊗ A N , where P • is a projective resolution of M . Equivalently,Tor An ( M, N ) is the n th homology of the complex M b ⊗ A Q • , where Q • is a projectiveresolution of N . The spaces Tor An ( M, N ) do not depend on the particular choices of P • and Q • and have the usual functorial properties (see [29, Section III.4.4] for details).Note that Tor An ( M, N ) is not necessarily Hausdorff, but the associated Hausdorff space(i.e., the quotient of Tor An ( M, N ) modulo the closure of zero) is a Fr´echet space. If M ∈ A - mod - A , then the n th Hochschild homology of A with coefficients in M is definedby H n ( A, M ) = Tor A e n ( M, A ).In contrast to the purely algebraic case, Tor A ( M, N ) is not the same as M b ⊗ A N .Nevertheless, there is a natural continuous open linear surjection α M,N : Tor A ( M, N ) → M b ⊗ A N, (2.3) Some authors (see, e.g., [16, 32, 45, 51]) define M b ⊗ A N in a different way. Actually, their M b ⊗ A N isour Tor A ( M, N ) (see below). We adopt the definition given by M. A. Rieffel [47] (see also [8,10,29,30,48]).
O. YU. ARISTOV AND A. YU. PIRKOVSKII whose kernel is the closure of zero in Tor A ( M, N ) [29, III.4.27]. In other words, M b ⊗ A N is isomorphic to the Hausdorff space associated to Tor A ( M, N ). Hence the followingequivalences hold:Tor A ( M, N ) is Hausdorff ⇐⇒ α M,N is injective ⇐⇒ α M,N is bijective ⇐⇒ α M,N is a topological isomorphism . (2.4)Under some nuclearity assumptions, the derived functor Tor can be calculated with thehelp of exact (not necessarily admissible) sequences of projective modules. The followingresult is an easy modification of [16, Corollary 3.1.13] (which, in turn, goes back to [51,Proposition 4.5]). Proposition 2.1.
Let A be a Fr´echet algebra, M ∈ mod - A , and N ∈ A - mod . Supposethat ← M ← P ← P ← · · · ← P n ← P n +1 is an exact sequence in mod - A such that P , . . . , P n are projective. Assume that one ofthe following conditions holds: (i) P , . . . , P n +1 are nuclear; (ii) A and N are nuclear.Then for each m = 0 , . . . , n the space Tor Am ( M, N ) is topologically isomorphic to the m thhomology of the complex P • b ⊗ A N . In particular, if either M or N is flat, then the tensoredsequence ← M b ⊗ A N ← P b ⊗ A N ← · · · ← P n +1 b ⊗ A N is exact. Given a Fr´echet algebra A and a Fr´echet A -bimodule M , we let Der( A, M ) denote thespace of all continuous derivations of A with values in M . The bimodule of noncommuta-tive differential -forms over A is a Fr´echet A -bimodule Ω A together with a derivation d A : A → Ω A such that for each Fr´echet A -bimodule M and each derivation D : A → M there exists a unique A -bimodule morphism Ω A → M making the following diagramcommute: Ω A / / MA d A O O D < < ②②②②②②②②② In other words, we have a natural isomorphism A h A (Ω A, M ) ∼ = Der( A, M ) ( M ∈ A - mod - A ) . It is a standard fact (see, e.g., [9, 41]) that Ω A exists and is isomorphic to the kernelof the multiplication map µ A : A b ⊗ A → A . Under the above identification, the universalderivation d A : A → Ω A acts by the rule d A ( a ) = 1 ⊗ a − a ⊗ a ∈ A ). Thus we havean exact sequence 0 → Ω A j A −→ A b ⊗ A µ A −→ A → A - mod - A , where j A is uniquely determined by j A ( d A ( a )) = 1 ⊗ a − a ⊗ a ∈ A ).Note that (2.5) splits in A - mod and in mod - A ( [41], cf. also [9]). In particular, (2.5) isadmissible. PEN EMBEDDINGS AND PSEUDOFLAT EPIMORPHISMS 5 Pseudoflat epimorphisms
We begin this section with the following “truncated” version of the transversality rela-tion ⊥ A introduced in [32] (see also [11, 16, 45]). Proposition 3.1.
Let A be a Fr´echet algebra, M ∈ mod - A , N ∈ A - mod , and n ∈ Z + .Then the following conditions are equivalent: (i) Tor Am ( M, N ) = 0 for ≤ m ≤ n , and Tor A ( M, N ) is Hausdorff; (ii) for some (or, equivalently, for each) projective resolution ← M ← P • in mod - A the sequence ← M b ⊗ A N ← P b ⊗ A N ← · · · ← P n +1 b ⊗ A N (3.1) is exact; (iii) for some (or, equivalently, for each) projective resolution ← N ← Q • in A - mod the sequence ← M b ⊗ A N ← M b ⊗ A Q ← · · · ← M b ⊗ A Q n +1 is exact; (iv) for some (or, equivalently, for each) projective resolution ← A ← L • in A - mod - A the sequence ← M b ⊗ A N ← M b ⊗ A L b ⊗ A N ← · · · ← M b ⊗ A L n +1 b ⊗ A N (3.2) is exact.Proof. The equivalences between “for some” and “for each” in (ii)–(iv) are immediatefrom the fact that all projective resolutions of a module are homotopy equivalent.(i) ⇐⇒ (ii). Since b ⊗ A preserves surjections [29, II.4.12], (3.1) is always exact at M b ⊗ A N . If 1 ≤ m ≤ n , then (3.1) is exact at P m b ⊗ A N if and only if Tor Am ( M, N ) = 0.On the other hand, (3.1) is exact at P b ⊗ A N if and only ifKer( P b ⊗ A N → Tor A ( M, N )) = Ker( P b ⊗ A N → M b ⊗ A N ) , i.e., if and only if the canonical map Tor A ( M, N ) → M b ⊗ A N is injective. By (2.4), thelatter condition holds if and only if Tor A ( M, N ) is Hausdorff.(i) ⇐⇒ (iii). This is similar to (i) ⇐⇒ (ii).(iii) ⇐⇒ (iv). If 0 ← A ← L • is a projective resolution of A in A - mod - A , then0 ← N ← L • b ⊗ A N is a projective resolution of N in A - mod . The rest is clear. (cid:3) Definition 3.2.
Let A be a Fr´echet algebra, M ∈ mod - A , N ∈ A - mod , and n ∈ Z + .We say that M and N are n -transversal over A (and write M ⊥ nA N ) if the (equivalent)conditions of Proposition 3.1 are satisfied. If M ⊥ nA N for all n ∈ Z + , then M and N aresaid to be transversal [32] (see also [11, 16, 45]). In this case, we write M ⊥ A N . Corollary 3.3.
Let A be a Fr´echet algebra, M ∈ mod - A , N ∈ A - mod , and n ∈ Z + . Thenthe following conditions are equivalent: (i) M ⊥ nA N ; (ii) ( N b ⊗ M ) ⊥ nA e A ; (iii) H m ( A, N b ⊗ M ) = 0 for ≤ m ≤ n , and H ( A, N b ⊗ M ) is Hausdorff.Proof. This is immediate from Proposition 3.1 and from the isomorphisms Tor Am ( M, N ) ∼ = H m ( A, N b ⊗ M ) [29, III.4.25]. (cid:3) O. YU. ARISTOV AND A. YU. PIRKOVSKII
Here is our main definition.
Definition 3.4.
Let ϕ : A → B be a Fr´echet algebra homomorphism, and let n ∈ Z + .We say that ϕ is n -pseudoflat if B ⊥ nA B .We are mostly interested in those pseudoflat homomorphisms which are epimorphisms(in the category-theoretic sense). Recall that a morphism ϕ : A → B in a category C isan epimorphism if for each pair ψ, ψ ′ : B → C of morphisms in C the equality ψϕ = ψ ′ ϕ implies that ψ = ψ ′ . Equivalently, this means that, for each object C of C , the mapHom C ( B, C ) → Hom C ( A, C ) induced by ϕ is injective.For the reader’s convenience, let us recall the following result (see, e.g., [50, Prop. XI.1.2],[41, Prop. 6.1]). Proposition 3.5.
Let ϕ : A → B be a homomorphism of Fr´echet algebras. Then thefollowing conditions are equivalent: (i) ϕ is an epimorphism in the category of Fr´echet algebras; (ii) the multiplication map µ B,A : B b ⊗ A B → B is a topological isomorphism; (iii) for each M ∈ mod - B and each N ∈ B - mod , the canonical map M b ⊗ A N → M b ⊗ B N is a topological isomorphism. We also need the following well-known fact (see, e.g., [29, Chap. 0, Corollary 4.2]).
Proposition 3.6.
Let A be a Fr´echet algebra, and let ϕ : M → N be a morphism of leftFr´echet A -modules. Then ϕ is an epimorphism in A - mod if and only if ϕ ( M ) is dense in N . The same result holds for the categories mod - A and A - mod - A . Given a Fr´echet algebra homomorphism ϕ : A → B , we define ¯ µ B,A : Tor A ( B, B ) → B to be the composition of the canonical map α = α B,B : Tor A ( B, B ) → B b ⊗ A B (see (2.3))and the multiplication map µ B,A : B b ⊗ A B → B . Lemma 3.7.
The following conditions are equivalent: (i) ϕ is a -pseudoflat epimorphism; (ii) ¯ µ B,A : Tor A ( B, B ) → B is bijective; (iii) ¯ µ B,A : Tor A ( B, B ) → B is a topological isomorphism.Proof. (i) = ⇒ (iii). The fact that ϕ is 0-pseudoflat means precisely that Tor A ( B, B )is Hausdorff, which happens if and only if α : Tor A ( B, B ) → B b ⊗ A B is a topologicalisomorphism (see (2.4)). On the other hand, the fact that ϕ is an epimorphism meansprecisely that µ B,A : B b ⊗ A B → B is a topological isomorphism (see Proposition 3.5).Hence so is ¯ µ B,A = µ B,A ◦ α .(iii) = ⇒ (ii). This is clear.(ii) = ⇒ (i). Since ¯ µ B,A is continuous and bijective, we conclude that Tor A ( B, B )is Hausdorff (i.e., ϕ is 0-pseudoflat). Hence α is bijective by (2.4), and so µ B,A is atopological isomorphism by the Open Mapping Theorem. Applying Proposition 3.5, wesee that ϕ is an epimorphism. (cid:3) Let ϕ : A → B be a Fr´echet algebra homomorphism, and let 0 ← A ← L • be a projectiveresolution in A - mod - A . Applying B b ⊗ A ( − ) b ⊗ A B to L → A and composing with themultiplication µ B,A : B b ⊗ A B → B , we get a B -bimodule morphism ε L : B b ⊗ A L b ⊗ A B → B . Similarly, if 0 ← B ← P • and 0 ← B ← Q • are projective resolutions of B in mod - A and in A - mod , respectively, then we have morphisms ε P : P b ⊗ A B → B in mod - B and ε Q : B b ⊗ A Q → B in B - mod . PEN EMBEDDINGS AND PSEUDOFLAT EPIMORPHISMS 7
Proposition 3.8.
Let ϕ : A → B be a Fr´echet algebra homomorphism. The followingconditions are equivalent: (i) ϕ is an n -pseudoflat epimorphism; (ii) for some (or, equivalently, for each) projective resolution ← B ← P • in mod - A thesequence ← B ε P ←− P b ⊗ A B ← · · · ← P n +1 b ⊗ A B (3.3) is exact; (iii) for some (or, equivalently, for each) projective resolution ← B ← Q • in A - mod the sequence ← B ε Q ←− B b ⊗ A Q ← · · · ← B b ⊗ A Q n +1 is exact; (iv) for some (or, equivalently, for each) projective resolution ← A ← L • in A - mod - A the sequence ← B ε L ←− B b ⊗ A L b ⊗ A B ← · · · ← B b ⊗ A L n +1 b ⊗ A B (3.4) is exact.Proof. (i) ⇐⇒ (ii). Clearly, if m ≥
1, then (3.3) is exact at P m b ⊗ A B if and only ifTor Am ( B, B ) = 0. Since the 0th homology of P • b ⊗ A B is precisely Tor A ( B, B ), we see that(3.3) is exact at P b ⊗ A B if and only if ¯ µ B,A : Tor A ( B, B ) → B is bijective. Now the resultfollows from Lemma 3.7.Equivalences (i) ⇐⇒ (iii) and (i) ⇐⇒ (iv) are proved similarly. (cid:3) Corollary 3.9.
Let ϕ : A → B be a Fr´echet algebra homomorphism. Define d ϕ : B b ⊗ A b ⊗ B → B b ⊗ B, b ⊗ a ⊗ c bϕ ( a ) ⊗ c − b ⊗ ϕ ( a ) c ( b, c ∈ B, a ∈ A ) . Then ϕ is a -pseudoflat epimorphism if and only if the sequence ← B µ B ←− B b ⊗ B d ϕ ←− B b ⊗ A b ⊗ B (3.5) is exact.Proof. If 0 ← A ← L • is the bimodule bar resolution of A (see (2.1)), then (3.4) for n = 0is precisely (3.5). (cid:3) Corollary 3.10.
A surjective Fr´echet algebra homomorphism is a -pseudoflat epimor-phism.Proof. If we replace A by B in (3.5), then we get an exact sequence (in fact, this is thelow-dimensional segment of the bimodule bar resolution for B ). Since ϕ : A → B is onto,we conclude that (3.5) is exact as well. (cid:3) Remark . As we shall see below (Example 3.23), a Fr´echet algebra homomorphismwith dense image, while being an epimorphism for an obvious reason, is not necessarily0-pseudoflat.The next proposition (which is a Fr´echet algebra version of [5, (87)]) emphasizes thedifference between 0-pseudoflat and 1-pseudoflat epimorphisms.
Proposition 3.12.
Let I be a closed two-sided ideal in a nuclear Fr´echet algebra A .Then the quotient map π : A → A/I is a -pseudoflat epimorphism if and only if themultiplication map µ I : I b ⊗ I → I , a ⊗ b ab , is surjective. O. YU. ARISTOV AND A. YU. PIRKOVSKII
Proof.
By Corollary 3.10, π is a 0-pseudoflat epimorphism. Thus π is a 1-pseudoflatepimorphism if and only if Tor A ( A/I, A/I ) = 0. Since A is nuclear, the exact sequence0 ← A/I ← A ← I ← Ai ( A/I, − ) (see [16, Theorem 3.1.12]), whose low-dimensional segment looks as follows:0 ← Tor A ( A/I, A/I ) q ←− Tor A ( A/I, A ) ← Tor A ( A/I, I ) ← Tor A ( A/I, A/I ) ← . (3.7)Clearly, Tor A ( A/I, A ) ∼ = ( A/I ) b ⊗ A A ∼ = A/I . Applying Corollary 3.10 and Lemma 3.7,we see that Tor A ( A/I, A/I ) ∼ = A/I . Under the above identifications, the map q in (3.7)becomes the identity map of A/I . Hence Tor A ( A/I, A/I ) is isomorphic to Tor A ( A/I, I ).Applying Proposition 2.1 to (3.6), we see that Tor A ( A/I, I ) is the cokernel of I b ⊗ A I → A b ⊗ A I ∼ = I , which is isomorphic to the cokernel of µ I . The rest is clear. (cid:3) Example . Let A = O ( C ) be the algebra of holomorphic functions on C , and let I = { f ∈ A : f (0) = 0 } . The multiplication map µ I : I b ⊗ I → I is not surjective, becausethe image of µ I is contained in the ideal J = { f ∈ A : f (0) = f ′ (0) = 0 } , which is strictlysmaller than I . Hence the quotient map A → A/I is not 1-pseudoflat by Proposition 3.12,although it is a 0-pseudoflat epimorphism by Corollary 3.10.
Definition 3.14.
Let ϕ : A → B be a Fr´echet algebra homomorphism. We say that ϕ isa weak homological epimorphism if ϕ is an n -pseudoflat epimorphism for all n ∈ Z + .Thus ϕ : A → B is a weak homological epimorphism if and only if any (hence all) ofthe infinite sequences0 ← B ← P • b ⊗ A B, ← B ← B b ⊗ A Q • , ← B ← B b ⊗ A L • b ⊗ A B (3.8)(where P • , Q • , L • are as in Proposition 3.8) are exact. Definition 3.15.
We say that ϕ is a strong homological epimorphism if any (hence all)of the infinite sequences (3.8) are admissible.The fact that the admissibility of any of the sequences (3.8) implies the admissibilityof the other two follows from [40, Prop. 3.2]. Remark . The notion of a homological epimorphism has a remarkable history. Stronghomological epimorphisms were introduced by J. L. Taylor [52] under the name of “ab-solute localizations”. For nuclear Fr´echet algebras, our notion of a weak homological epi-morphism is equivalent to Taylor’s notion of a “localization” [loc. cit.]; see Section 1. Inthe purely algebraic setting, homological epimorphisms were rediscovered by W. Dicks [12]under the name of “liftings”, by W. Geigle and H. Lenzing [20] (where the current ter-minology was introduced), by A. Neeman and A. Ranicki [35] under the name of “stablyflat homomorphisms”. In [33], R. Meyer introduced strong homological epimorphismsin the setting of nonunital bornological algebras under the name of “isocohomologicalmorphisms”. Finally, O. Ben-Bassat and K. Kremnizer [4] introduced weak homologi-cal epimorphisms (under the name of “homotopy epimorphisms”) in the abstract settingof commutative algebras in symmetric monoidal quasi-abelian categories (cf. also [54]).Amazingly, each of the above-mentioned authors seems to have introduced essentially thesame class of morphisms independently of the earlier literature.The following proposition is an analog of [5, Prop. 5.1].
PEN EMBEDDINGS AND PSEUDOFLAT EPIMORPHISMS 9
Proposition 3.17.
Let ϕ : A → B be a Fr´echet algebra epimorphism, and let n ∈ Z + .Suppose that A and B are nuclear. Then the following conditions are equivalent: (i) ϕ is n -pseudoflat; (ii) M ⊥ nA B for each right Fr´echet B -module M ; (iii) B ⊥ nA N for each left Fr´echet B -module N .Proof. (i) = ⇒ (iii). Let 0 ← B ← P • be a projective resolution in mod - A such that allthe modules P i are nuclear. By Proposition 3.8, (3.3) is an exact sequence. Observe thatall the modules in (3.3) (including B ) are nuclear and projective in mod - B . Therefore,applying ( − ) b ⊗ B N and using Proposition 2.1, we get an exact sequence0 ← N ← P b ⊗ A N ← · · · ← P n +1 b ⊗ A N. (3.9)Since ϕ is an epimorphism, we see that B b ⊗ A N ∼ = N canonically. Hence (3.9) is isomor-phic to (3.1) with M = B . Thus B ⊥ nA N .The implication (i) = ⇒ (ii) is proved similarly; (ii) = ⇒ (i) and (iii) = ⇒ (i) are clearfrom Definition 3.4. (cid:3) Corollary 3.18.
Let ϕ : A → B be a Fr´echet algebra epimorphism. Suppose that A and B are nuclear. Then the following conditions are equivalent: (i) ϕ is a weak homological epimorphism; (ii) M ⊥ A B for each right Fr´echet B -module M ; (iii) B ⊥ A N for each left Fr´echet B -module N .Remark . A similar result [40, Prop. 3.2] on strong homological epimorphisms doesnot involve nuclearity assumptions.Our next goal is to characterize epimorphisms, 0-pseudoflat epimorphisms, and 1-pseudoflat epimorphisms in terms of noncommutative differential forms. Towards thisgoal, let us introduce some notation. Let ϕ : A → B be a Fr´echet algebra homomor-phism. Applying B b ⊗ A ( − ) b ⊗ A B to the canonical sequence (2.5) and composing with B b ⊗ A B → B , we get 0 → B b ⊗ A Ω A b ⊗ A B → B b ⊗ B µ B −→ B → . Identifying Ω B with Ker µ B , we see that there exists a unique B -bimodule morphismˇ ϕ : B b ⊗ A Ω A b ⊗ A B → Ω B making the diagram B b ⊗ A Ω A b ⊗ A B / / ˇ ϕ (cid:15) (cid:15) B b ⊗ B µ B / / B / / / / Ω B j B / / B b ⊗ B µ B / / B / / B -bimodule X we have a linear map e ϕ X : Der( B, X ) → Der(
A, X ) , D Dϕ.
Theorem 3.20.
For a Fr´echet algebra homomorphism ϕ : A → B the following conditionsare equivalent: (i) ϕ is an epimorphism; (ii) e ϕ X : Der( B, X ) → Der(
A, X ) is injective for each X ∈ B - mod - B ; (iii) the image of ˇ ϕ : B b ⊗ A Ω A b ⊗ A B → Ω B is dense in Ω B .Proof. (i) = ⇒ (ii). Given X ∈ B - mod - B , we make the Fr´echet space B ⊕ X into a Fr´echetalgebra by letting ( b, x )( c, y ) = ( bc, by + xc ) ( b, c ∈ B, x, y ∈ X ) . Suppose that ϕ is an epimorphism. For every D ∈ Der(
B, X ) we have a Fr´echet algebrahomomorphism ψ : B → B ⊕ X, ψ ( b ) = ( b, D ( b )) ( b ∈ B ) . If Dϕ = 0, then ψϕ = ψ ′ ϕ , where ψ ′ : B → B ⊕ X, ψ ′ ( b ) = ( b,
0) ( b ∈ B ) . Since ϕ is an epimorphism, we have ψ = ψ ′ , i.e., D = 0.(ii) = ⇒ (i). Suppose that e ϕ X is injective for each X ∈ B - mod - B . Consider thederivation D : B → B b ⊗ A B, b b ⊗ A − ⊗ A b. Since Dϕ = 0, we have D = 0, i.e., b ⊗ A ⊗ A b for each b ∈ B . Then thecontinuous linear map B → B b ⊗ A B , b b ⊗ A
1, is the inverse of the multiplication µ B,A : B b ⊗ A B → B . Thus µ B,A is a topological isomorphism, i.e., ϕ is an epimorphism(see Proposition 3.5).(ii) ⇐⇒ (iii). By the universal properties of Ω and b ⊗ A , for each Fr´echet B -bimodule X there exists a commutative diagramDer( B, X ) e ϕ X / / Der(
A, X ) B h B (Ω B, X ) ˇ ϕ X / / B h B ( B b ⊗ A Ω A b ⊗ A B, X ) (3.11)where ˇ ϕ X induced by ˇ ϕ . Thus (ii) holds if and only if ˇ ϕ is an epimorphism in B - mod - B ,which is equivalent to (iii) by Proposition 3.6. (cid:3) Theorem 3.21.
A Fr´echet algebra homomorphism ϕ : A → B is a -pseudoflat epimor-phism if and only if ˇ ϕ : B b ⊗ A Ω A b ⊗ A B → Ω B is onto.Proof. Since j A is a kernel of µ A in (2.5), we see that the map d : A b ⊗ → A b ⊗ defined by(2.2) factorizes as follows: A b ⊗ A b ⊗ A p A (cid:15) (cid:15) d & & ▲▲▲▲▲▲▲▲▲▲ / / Ω A j A / / A b ⊗ A µ A / / A / / . Applying B b ⊗ A ( − ) b ⊗ A B and combining with (3.10), we get the commutative diagram B b ⊗ A b ⊗ B ˜ p A (cid:15) (cid:15) d ϕ ' ' ❖❖❖❖❖❖❖❖❖❖❖❖ B b ⊗ A Ω A b ⊗ A B / / ˇ ϕ (cid:15) (cid:15) B b ⊗ B µ B / / B / / / / Ω B j B / / B b ⊗ B µ B / / B / / PEN EMBEDDINGS AND PSEUDOFLAT EPIMORPHISMS 11 where d ϕ is defined in Corollary 3.9. Since j B is a kernel of µ B , we see that (3.5) is exactif and only if ˇ ϕ ◦ ˜ p A is onto. Since p A is onto, and since the projective tensor productpreserves surjections of Fr´echet modules, it follows that ˜ p A is onto. Hence ˇ ϕ ◦ ˜ p A is ontoif and only if ˇ ϕ is onto. This completes the proof. (cid:3) To characterize 1-pseudoflat epimorphisms in terms of Ω , we need the following lemma. Lemma 3.22.
Let ϕ : A → B be a -pseudoflat Fr´echet algebra epimorphism, let M ∈ mod - B , and let N ∈ B - mod . Assume that either M or N is flat as a Fr´echet B -module.Then Tor A ( M, N ) is Hausdorff.Proof. Since ϕ is a 0-pseudoflat epimorphism, we see that (3.5) is exact. Since j B is akernel of µ B , there exists a surjective morphism ¯ ϕ : B b ⊗ A b ⊗ B → Ω B such that thediagram 0 B o o B b ⊗ B µ B o o B b ⊗ A b ⊗ B o o ¯ ϕ y y rrrrrrrrrr Ω B j B d d ❍❍❍❍❍❍❍❍❍ (3.12)commutes. (Note that ¯ ϕ = ˇ ϕ ◦ ˜ p A , see the proof of Proposition 3.21.)Assume now that N ∈ B - mod is flat. Since the canonical sequence0 ← B µ B ←− B b ⊗ B j B ←− Ω B ← B - mod , for each M ∈ mod - B the sequence0 ← M ← M b ⊗ B ← M b ⊗ B Ω B ← M b ⊗ B ( − ) is admissible. Applying ( − ) b ⊗ B N , we get an exactsequence 0 ← M b ⊗ B N ← M b ⊗ N ← M b ⊗ B Ω B b ⊗ B N ← . (3.14)Let us now apply M b ⊗ B ( − ) b ⊗ B N to (3.12). We obtain the following commutative dia-gram: 0 M b ⊗ B N o o M b ⊗ N o o M b ⊗ A b ⊗ N o o ¯ ϕ M,N v v ♠♠♠♠♠♠♠♠♠♠♠♠♠ M b ⊗ B Ω B b ⊗ B N g g ❖❖❖❖❖❖❖❖❖❖❖❖ (3.15)Since ¯ ϕ is onto, it follows that ¯ ϕ M,N = 1 M ⊗ B ¯ ϕ ⊗ B N is onto. Together with the ex-actness of (3.14), this implies that the upper row of (3.15) is exact. Identifying M b ⊗ B N with M b ⊗ A N (see Proposition 3.5), we see that the sequence0 ← M b ⊗ A N ← M b ⊗ N ← M b ⊗ A b ⊗ N obtained from the low-dimensional segment of (2.1) via M b ⊗ A ( − ) b ⊗ A N is exact. Equiv-alently, this means that the canonical map Tor A ( M, N ) → M b ⊗ A N is bijective, whichhappens if and only if Tor A ( M, N ) is Hausdorff (see (2.4)).In the case where N is arbitrary and M is flat, the proof is similar. (cid:3) Example . Using Lemma 3.22, it is easy to construct Banach algebra epimorphismsthat are not 0-pseudoflat. Consider, for example, the nonunital Banach sequence algebras ℓ and c (under pointwise multiplication), let A = ℓ and B = ( c ) + denote theirunitizations, and let ϕ : A → B be the tautological embedding. Since ϕ ( A ) is densein B , we see that ϕ is an epimorphism. Assume, towards a contradiction, that ϕ is 0-pseudoflat. By [28], the 1-dimensional B -module C = B/c is flat (in fact, all Banach B -modules are flat [29, VII.2.29], because B is amenable [31]). Hence Lemma 3.22 impliesthat Tor A ( C , c ) is Hausdorff. Consider now the admissible sequence0 → ℓ → A → C → A -modules (where ℓ → A is the tautological embedding). The low-dimensionalsegment of the respective long exact sequence for Tor Ai ( − , c ) looks as follows:0 ← Tor A ( C , c ) ← c j ←− Tor A ( ℓ , c ) ← Tor A ( C , c ) ← . (3.16)By [29, IV.5.9], ℓ is a biprojective Banach algebra (i.e., ℓ is a projective Banach ℓ -bimodule), which implies, in particular, that ℓ is projective in mod - A [29, IV.1.3]. Hencewe may identify Tor A ( ℓ , c ) with ℓ b ⊗ A c = ℓ b ⊗ ℓ c , which is isomorphic to ℓ via themap a ⊗ x ( a n x n ) (cf. [29, II.3.9] or [39, Lemma 4.1]). Under this identification, themap j in (3.16) is nothing but the embedding of ℓ into c . This implies that Tor A ( C , c )is topologically isomorphic to c /ℓ and is therefore non-Hausdorff. The resulting contra-diction shows that ϕ is not 0-pseudoflat.In the purely algebraic context, the following result was discovered by Bergman andDicks [5, Remark 5.4]. Theorem 3.24.
For a Fr´echet algebra homomorphism ϕ : A → B the following conditionsare equivalent: (i) ϕ is a -pseudoflat epimorphism; (ii) e ϕ X : Der( B, X ) → Der(
A, X ) is bijective for each X ∈ B - mod - B ; (iii) ˇ ϕ : B b ⊗ A Ω A b ⊗ A B → Ω B is an isomorphism in B - mod - B .Proof. Since the canonical sequence (2.5) splits in A - mod , the sequence0 ← B ← B b ⊗ A ← B b ⊗ A Ω A ← B b ⊗ A ( − ) is admissible in mod - A . Since B b ⊗ A is projective in mod - A , the low-dimensional segment of the respective long exact sequence for Tor Ai ( − , B )looks as follows:0 ← Tor A ( B, B ) ← B b ⊗ B ← Tor A ( B b ⊗ A Ω A, B ) ← Tor A ( B, B ) ← . We have the following commutative diagram:0 Tor A ( B, B ) o o ¯ µ B,A (cid:15) (cid:15) B b ⊗ B o o Tor A ( B b ⊗ A Ω A, B ) o o α (cid:15) (cid:15) Tor A ( B, B ) o o o o B b ⊗ A Ω A b ⊗ A B ˇ ϕ (cid:15) (cid:15) B o o B b ⊗ B µ B o o Ω B j B o o o o (3.18) PEN EMBEDDINGS AND PSEUDOFLAT EPIMORPHISMS 13 (i) = ⇒ (iii). If ϕ is a 1-pseudoflat epimorphism, then Tor A ( B, B ) = 0, and ¯ µ B,A isbijective by Lemma 3.7. Since both rows in (3.18) are exact, we see that ˇ ϕ ◦ α is bijective.On the other hand, α is bijective by Lemma 3.22 and by (2.4). Hence ˇ ϕ is bijective.(iii) = ⇒ (i). If ˇ ϕ is bijective, then ϕ is a 0-pseudoflat epimorphism by Theorem 3.21.Hence ¯ µ B,A is bijective by Lemma 3.7, and α is bijective by Lemma 3.22 and by (2.4).Since both lines in (3.18) are exact, and since the vertical arrows in (3.18) are bijective,we conclude that Tor A ( B, B ) = 0. Thus ϕ is 1-pseudoflat.(ii) ⇐⇒ (iii). Observe that ˇ ϕ is an isomorphism if and only if for each X ∈ B - mod - B the map ˇ ϕ X in (3.11) is bijective, i.e., if and only if (ii) holds. (cid:3) Remark . Weak and strong homological epimorphisms can be nicely interpreted inthe language of derived categories (cf. [4, 20, 33, 41]). Although we do not need this below,we find it relevant to give at least one of such interpretations (for the convenience of thosereaders who are used to think in terms of derived categories). If A is a Fr´echet algebra,then there are two ways of making A - mod into an exact category (in Quillen’s sense [46]).The first (traditional) exact structure is as follows. Suppose that M i −→ N p −→ P is anexact pair of morphisms in A - mod (i.e., i is a kernel of p and p is a cokernel of i ). We saythat such a pair is admissible if it splits in the category of Fr´echet spaces. It is easy toshow that the collection of all admissible exact pairs makes A - mod into an exact category.We use the same notation A - mod to denote the resulting exact category (this will not leadto a confusion). Alternatively, we can make A - mod into an exact category by declaringthat all exact pairs are admissible. The fact that the collection of all exact pairs in A - mod indeed satisfies the axioms of an exact category follows from the observation that A - mod is quasi-abelian, cf. [43]. The resulting exact category will be denoted by A - mod . We alsolet Fr = C - mod and Fr = C - mod denote the respective categories of Fr´echet spaces.Homological algebra in the exact category A - mod is precisely the “topological homol-ogy” introduced by A. Ya. Helemskii [27] (see also [16, 29, 30]). The main advantage of A - mod over A - mod is that A - mod has enough projectives, which is not the case for A - mod .In fact, by a result of V. A. Geiler [21], even the category Fr of Fr´echet spaces does not haveenough projectives. This is one of the main reasons why homological algebra in A - mod is developed much better than homological algebra in A - mod . Nevertheless, A - mod turnsout to be useful in J. L. Taylor’s homological approach to multivariable spectral theory(cf. [16, 52]).Since A - mod has enough projectives, the functor M b ⊗ A ( − ) : A - mod → Fr is left deriv-able. The left derived functor of M b ⊗ A ( − ) is denoted by M b ⊗ L A ( − ) : D − ( A - mod ) → D − ( Fr ). Exactly as in the algebraic case, b ⊗ L A extends to a bifunctor from D − ( mod - A ) × D − ( A - mod ) to D − ( Fr ). Now it is easy to see that a Fr´echet algebra homomorphism ϕ : A → B is a weak (respectively, strong) homological epimorphism if and only if thecanonical map B b ⊗ L A B → B is an isomorphism in D − ( Fr ) (respectively, in D − ( Fr )). Thismay be compared with condition (ii) of Proposition 3.5, which characterizes epimorphismsof Fr´echet algebras. 4. Stein algebras
For the reader’s convenience, let us recall some standard notation (see, e.g., [25, Chap. 0, §
4, no. 3]). Given a morphism f : ( X, O X ) → ( Y, O Y ) of C -ringed spaces and an O Y -module G , we let f − G denote the sheaf on X associated to the presheaf U G ( f ( U )) =lim −→{ G ( V ) : f ( U ) ⊂ V } . We also let f ∗ G = O X ⊗ f − O Y f − G denote the inverse image of G . Recall that, for each x ∈ X , we have an O X,x -module isomorphism ( f ∗ G ) x ∼ = O X,x ⊗ O Y,f ( x ) G f ( x ) .Throughout, all Stein spaces are assumed to be finite-dimensional. Let ( X, O X ) be aStein space. By Cartan’s Theorem B, the functor Γ of global sections acting from thecategory of coherent O X -modules to the category of all O ( X )-modules is exact. On theother hand, Cartan’s Theorem A easily implies that Γ is faithful. Together with [19, 3.2],this yields the following well-known result. Lemma 4.1.
A sequence F → G → H of coherent O X -modules is exact if and only ifthe sequence of global sections F ( X ) → G ( X ) → H ( X ) is exact. Recall also (see, e.g., [23, V.6]) that, for each coherent O X -module F , the space F ( X )of global sections has a canonical topology making it into a Fr´echet space. Moreover, O ( X ) is a Fr´echet algebra, and F ( X ) is a Fr´echet O ( X )-module. Thus Γ can be viewedas a functor from the category of coherent O X -modules to the category of Fr´echet O ( X )-modules.Given p ∈ X , we denote by C p the one-dimensional O ( X )-module corresponding to theevaluation map O ( X ) → C , a a ( p ). Theorem 4.2.
Let ( X, O X ) and ( Y, O Y ) be Stein spaces, let f : Y → X be a holomorphicmap, and let f • : O ( X ) → O ( Y ) denote the homomorphism induced by f . Then thefollowing conditions are equivalent: (i) f • is a weak homological epimorphism; (ii) f • is a -pseudoflat epimorphism; (iii) f • is an epimorphism, and for each q ∈ Y we have O ( Y ) ⊥ O ( X ) C q ; (iv) f is an open embedding.Proof. (i) = ⇒ (ii): this is immediate from Definition 3.14.(ii) = ⇒ (iii). Since O ( X ) and O ( Y ) are nuclear, we can apply Proposition 3.17.(iii) = ⇒ (iv). We first observe that f is injective. Indeed, since f • is an epimorphism,we see that the map Hom( O ( Y ) , C ) → Hom( O ( X ) , C ) induced by f • is injective. By[18, Satz 1] (see also [23, V.7.3]), for each Stein space Z we have a natural bijection Z ∼ = Hom( O ( Z ) , C ) taking each z ∈ Z to the evaluation map at z . Therefore f isinjective.Given q ∈ Y , let p = f ( q ), and define the ideal sheaf I ⊂ O X by I x = ( m X,x if x = p, O X,x if x = p, where m X,x is the maximal ideal of O X,x . By [18, Satz 6.4], there exists a resolution0 ← O X / I ← O X ← P ← P ← · · · , (4.1)where all the P i ’s are free O X -modules of finite rank, and where O X → O X / I is thequotient map. Taking the sections over X and applying Cartan’s Theorem B, we obtainan exact complex 0 ← C p ← O ( X ) ← P ← P ← · · · (4.2)of Fr´echet O ( X )-modules. Note that C p = C q in O ( X )- mod . PEN EMBEDDINGS AND PSEUDOFLAT EPIMORPHISMS 15
By Proposition 2.1, we can use (4.2) to calculate Tor O ( X ) i ( O ( Y ) , C q ). Condition (iii)implies that Tor O ( X )1 ( O ( Y ) , C q ) = 0 , andTor O ( X )0 ( O ( Y ) , C q ) ∼ = O ( Y ) b ⊗ O ( X ) C q ∼ = O ( Y ) b ⊗ O ( Y ) C q ∼ = C q canonically (see Proposition 3.5). Hence we have an exact sequence0 ← C q ← O ( Y ) b ⊗ O ( X ) O ( X ) ← O ( Y ) b ⊗ O ( X ) P ← O ( Y ) b ⊗ O ( X ) P (4.3)of Fr´echet O ( Y )-modules.Now observe that the functors F O ( Y ) b ⊗ O ( X ) F ( X ) and F ( f ∗ F )( Y ) obviouslyagree on the category of free O X -modules of finite rank. Hence (4.3) is isomorphic to thesequence obtained by applying Γ( Y, − ) to0 ← O Y / I ′ ← O Y ∼ = f ∗ O X ← f ∗ P ← f ∗ P , (4.4)where I ′ ⊂ O Y is the ideal sheaf given by I ′ y = ( m Y,y if y = q, O Y,y if y = q. Applying Lemma 4.1, we conclude that (4.4) is exact.Consider now the stalks of (4.1) over p and the stalks of (4.4) over q . For notationalconvenience, let A = O X,p , B = O Y,q , m A = m X,p , m B = m Y,q , and F i = ( P i ) x . Let also ϕ : A → B denote the homomorphism induced by f . We have two exact sequences0 ← A/ m A ← A ← F ← F ← · · · , (4.5)0 ← B/ m B ← B ⊗ A A ← B ⊗ A F ← B ⊗ A F . (4.6)Comparing (4.5) with (4.6), we see thatTor A ( B, A/ m A ) = 0 , (4.7)where Tor A stands for the purely algebraic Tor-functor. Also, the exactness of (4.5) and(4.6) implies that B ⊗ A ( A/ m A ) ∼ = B/ m B via the map b ⊗ ( a + m A ) bϕ ( a ) + m B . Itis readily verified that the latter condition is equivalent to the equality Bϕ ( m A ) = m B .By [22, 2.2.3], this means that ϕ is onto. In particular, B is a finitely generated A -module. Since A is Noetherian [22, 2.0.1], B is a finitely presented A -module. Combiningthis with (4.7) and applying [7, Chap. II, §
3, no. 2], we see that B is free over A . Sincedim( B/B m A ) = dim( B/ m B ) = 1, it follows from [22, Appendix, 2.7 (i)] that ϕ is anisomorphism. By [17, 0.23], this means exactly that f is locally biholomorphic. Since f is also injective (see above), we conclude that f is an open embedding.(iv) = ⇒ (i). Without loss of generality, we may assume that Y is a Stein opensubset of X and that O Y = O X | Y . Thus f • : O ( X ) → O ( Y ) is the restriction map.Recall from [16, Corollary 4.2.5] that, for each morphism g : ( Z, O Z ) → ( X, O X ) ofStein spaces, and for each coherent O Z -module F , we have O ( Y ) ⊥ O ( X ) F ( Z ), and O ( Y ) b ⊗ O ( X ) F ( Z ) ∼ = F ( g − ( Y )). Letting Z = Y , F = O Y , and g = (inclusion Y ֒ → X ),we obtain O ( Y ) ⊥ O ( X ) O ( Y ) and O ( Y ) b ⊗ O ( X ) O ( Y ) ∼ = O ( Y ). This means exactly thatthe restriction map O ( X ) → O ( Y ) is a weak homological epimorphism. (cid:3) Remark . We have already pointed out in Section 1 that, if f : Y → X is an openembedding of Stein spaces, then f • : O ( X ) → O ( Y ) is not necessarily flat. Thus theclass of 1-pseudoflat Fr´echet algebra epimorphisms is essentially larger than the classof flat epimorphisms, even in the commutative case. It is interesting to compare thiswith recent purely algebraic results from [1, 3]. Namely, a 1-pseudoflat epimorphism A → B of commutative rings is necessarily flat provided that either (a) A is Noetherian [1,Prop. 4.5], or (b) the projective dimension of B over A is ≤ D ⊂ C is the open unit disc, then the restriction map O ( C ) → O ( D ) is a 1-pseudoflat epimorphism by Theorem 4.2 (actually, by [52, Prop. 3.1]), satisfies(b) by [29, Theorem V.1.8], but is not flat by [38]. This shows that the above-mentionedresult of [3] has no analog in the Fr´echet algebra setting.5. Algebras of C ∞ -functions In this section, we prove a C ∞ -analog of Theorem 4.2. Towards this goal, we need twolemmas. Let A , B , C be Fr´echet algebras, N be a Banach B - C -bimodule, and P be aFr´echet A - C -bimodule. Then h C ( N, P ) is a Fr´echet space under the topology of uniformconvergence on the unit ball of N . Moreover, h C ( N, P ) is a Fr´echet A - B -bimodule withrespect to the actions( a · ϕ )( x ) = a · ϕ ( x ) , ( ϕ · b )( x ) = ϕ ( b · x ) ( a ∈ A, b ∈ B, x ∈ N, ϕ ∈ h C ( N, P )) . Lemma 5.1.
Let A , B , C be Fr´echet algebras, M be a Fr´echet A - B -bimodule, N be aBanach B - C -bimodule, and P be a Fr´echet A - C -bimodule. Then there exists a vectorspace isomorphism A h C ( M b ⊗ B N, P ) ∼−→ A h B ( M, h C ( N, P )) , ϕ ( x ( y ϕ ( x ⊗ y ))) . We omit the standard proof (cf. [29, II.5.22], [42, Prop. 3.2]).
Lemma 5.2.
Let ϕ : A → B be a Fr´echet algebra epimorphism, and let ǫ : B → C be acontinuous homomorphism. Let C ǫ denote the one-dimensional B -bimodule correspondingto ǫ . Suppose that B ⊥ A C ǫ . Then e ϕ C ǫ : Der( B, C ǫ ) → Der( A, C ǫ ) is a bijection.Proof. As in the proof of Theorem 3.24, the admissible sequence (3.17) yields a longexact sequence of Tor Ai ( − , C ǫ ), whose low-dimensional segment fits into the followingcommutative diagram:0 Tor A ( B, C ǫ ) o o α (cid:15) (cid:15) B b ⊗ C ǫ o o Tor A ( B b ⊗ A Ω A, C ǫ ) o o α ′ (cid:15) (cid:15) Tor A ( B, C ǫ ) o o o o B b ⊗ A C ǫβ (cid:15) (cid:15) B b ⊗ A Ω A b ⊗ A C ǫγ (cid:15) (cid:15) C ǫ o o B b ⊗ C ǫ o o Ω B b ⊗ B C ǫ o o o o (5.1)Here α and α ′ are the canonical maps from Tor A ( − , − ) to ( − ) b ⊗ A ( − ), β is the canonicalmap from B b ⊗ A C ǫ to B b ⊗ B C ǫ ∼ = C ǫ , and γ corresponds to ˇ ϕ ⊗ B C ǫ under the identifi-cation ( B b ⊗ A Ω A b ⊗ A B ) b ⊗ B C ε ∼ = B b ⊗ A Ω A b ⊗ A C ǫ . Finally, the bottom row of (5.1) isobtained from (3.13) via ( − ) b ⊗ B C ǫ , so it is exact because (3.13) splits in mod - B .Since ϕ is an epimorphism, Proposition 3.5 implies that β is bijective. Since B ⊥ A C ǫ ,we see that α is bijective and that Tor A ( B, C ǫ ) = 0. Together with the fact that both PEN EMBEDDINGS AND PSEUDOFLAT EPIMORPHISMS 17 lines in (5.1) are exact, this implies that γ ◦ α ′ is bijective. Hence α ′ is injective, or,equivalently, bijective (see (2.4)), which in turn implies that γ is bijective. Thus γ is anisomorphism in B - mod .Applying B h ( − , C ǫ ) to γ , we obtain a vector space isomorphism B h (Ω B b ⊗ B C ǫ , C ǫ ) ∼−→ B h ( B b ⊗ A Ω A b ⊗ A C ǫ , C ǫ ) . (5.2)Observe that there is a B -bimodule isomorphism h C ( C ǫ , C ǫ ) ∼ = C ǫ given by f f (1).Together with Lemma 5.1, this implies that B h (Ω B b ⊗ B C ǫ , C ǫ ) ∼ = B h B (Ω B, h C ( C ǫ , C ǫ )) ∼ = B h B (Ω B, C ǫ ) , (5.3)and B h ( B b ⊗ A Ω A b ⊗ A C ǫ , C ǫ ) ∼ = B h (( B b ⊗ A Ω A b ⊗ A B ) b ⊗ B C ε , C ǫ ) ∼ = B h B ( B b ⊗ A Ω A b ⊗ A B, h C ( C ǫ , C ǫ )) ∼ = B h B ( B b ⊗ A Ω A b ⊗ A B, C ǫ ) . (5.4)Under the identifications (5.3) and (5.4), the isomorphism (5.2) becomes B h B (Ω B, C ǫ ) ∼−→ B h B ( B b ⊗ A Ω A b ⊗ A B, C ǫ ) . (5.5)A routine calculation shows that (5.5) is nothing but the map ˇ ϕ C ǫ from diagram (3.11)(in which we let X = C ǫ ). This readily implies that e ϕ C ǫ : Der( B, C ǫ ) → Der( A, C ǫ ) is avector space isomorphism. (cid:3) Let X be a C ∞ -manifold. We denote by C ∞ ( X ) the Fr´echet algebra of infinitelydifferentiable C -valued functions on X . Similarly to the holomorphic case (see Section 4),given p ∈ X , we denote by C p the one-dimensional C ∞ ( X )-module corresponding to theevaluation map C ∞ ( X ) → C , a a ( p ). Theorem 5.3.
Let X and Y be C ∞ -manifolds, let f : Y → X be a smooth map, andlet f • : C ∞ ( X ) → C ∞ ( Y ) denote the homomorphism induced by f . Then the followingconditions are equivalent: (i) f • is a projective epimorphism, i.e., f • is an epimorphism and C ∞ ( Y ) is projectivein C ∞ ( X ) - mod ; (ii) f • is a flat epimorphism, i.e., f • is an epimorphism and C ∞ ( Y ) is flat in C ∞ ( X ) - mod ; (iii) f • is a strong homological epimorphism; (iv) f • is a weak homological epimorphism; (v) f • is a -pseudoflat epimorphism; (vi) f • is an epimorphism, and for each q ∈ Y we have C ∞ ( Y ) ⊥ C ∞ ( X ) C q ; (vii) f is an open embedding.Proof. (i) = ⇒ (ii) = ⇒ (iv), (i) = ⇒ (iii) = ⇒ (iv), (iv) = ⇒ (v): this is trivial.(v) = ⇒ (vi). Since C ∞ ( X ) and C ∞ ( Y ) are nuclear, we can apply Proposition 3.17.(vi) = ⇒ (vii). As in the proof of Theorem 4.2, we first observe that f is injec-tive. Indeed, since f • is an epimorphism, we see that the map Hom( C ∞ ( Y ) , C ) → Hom( C ∞ ( X ) , C ) induced by f • is injective. By [36, Theorem 7.2], for each smooth man-ifold Z we have a natural bijection Z ∼ = Hom( C ∞ ( Z ) , C ) taking each z ∈ Z to theevaluation map at z . Therefore f is injective. To complete the argument we need to show that f is a local diffeomorphism. Bythe Inverse Function Theorem, it suffices to check that for each q ∈ Y the tangentmap df q : T q ( Y ) → T f ( q ) ( X ) is a vector space isomorphism. Identifying T q ( Y ) withDer( C ∞ ( Y ) , C q ) (see, e.g., [24, III.3.1]), we see that df q is nothing but( e f • ) C q : Der( C ∞ ( Y ) , C q ) → Der( C ∞ ( X ) , C q )(as in Theorem 4.2, we identify C q with C f ( q ) in C ∞ ( X )- mod .) By Lemma 5.2, ( e f • ) C q isa bijection. Hence f is an open embedding.(vii) = ⇒ (i). Without loss of generality, we may assume that Y is an open subset of X .Thus f • : C ∞ ( X ) → C ∞ ( Y ) is the restriction map. A standard argument involving bumpfunctions shows that the image of f • is dense in C ∞ ( Y ). Hence f • is an epimorphism.By [37, Theorem 2], C ∞ ( Y ) is projective over C ∞ ( X ). This completes the proof. (cid:3) Concluding remarks and questions
An obvious difference between our main results, i.e., Theorems 4.2 and 5.3, is thatthe strong conditions (i)–(iii) of Theorem 5.3 are missing in Theorem 4.2. We havealready mentioned in Section 1 that open embeddings of Stein spaces usually do not satisfycondition (ii) (and, a fortiori , do not satisfy condition (i)) of Theorem 5.3. However, thesituation with condition (iii) is not that clear. In fact, we do not know the answer to thefollowing question.
Question 6.1.
Let ( X, O X ) be a Stein space, and let ( Y, O Y ) be a Stein open subspace of ( X, O X ) . Is the restriction map O ( X ) → O ( Y ) a strong homological epimorphism? In the special case where X = C n and Y is a polydomain (i.e., a product of one-dimensional open subsets of C ), the answer to Question 6.1 is positive by [52, Prop. 4.3].On the other hand, the answer seems to be unknown already in the case where X = C and Y is the open unit ball.Another difference between Theorems 4.2 and 5.3 is in the “degree of singularity” ofthe objects considered therein. Indeed, Stein spaces are not necessarily reduced (i.e.,their structure sheaves are allowed to have nilpotents), and even reduced Stein spacesare not necessarily smooth (i.e., are not necessarily locally isomorphic to an open subsetof C n ). On the other hand, C ∞ -manifolds are reduced and smooth (in the appropriatesense) by definition. The theory of C ∞ -differentiable spaces [34] studies geometric objectswhich are more general than C ∞ -manifolds, and which can be viewed as “correct” C ∞ -analogs of Stein spaces. In particular, C ∞ -differentiable spaces may have singular pointsand may be non-reduced. (Note that [34] deals with R -valued functions only, but anextension to C -valued functions is straightforward.) It would be interesting to characterizeopen embeddings of C ∞ -differentiable spaces (at least in the affine case) in the spirit ofTheorem 5.3.In its full form, Theorem 5.3 does not extend to C ∞ -differentiable spaces. For example,consider the map π : C ∞ ( R n ) → C [[ x , . . . , x n ]] which takes each smooth function on R n to its Taylor series at 0. Using the Koszul resolution, one can prove that π is a stronghomological epimorphism (cf. [52, Prop. 4.4]). At the same time, the correspondingmap π ∗ of affine C ∞ -differentiable spaces is not an open embedding. On the other hand, In [37], the projectivity of C ∞ ( Y ) over C ∞ ( X ) is proved under the assumption that Y is containedin a coordinate neighborhood. However, the proof readily carries over to the general case. PEN EMBEDDINGS AND PSEUDOFLAT EPIMORPHISMS 19 applying ( − ) b ⊗ A C [[ x , . . . , x n ]] to the inclusion I ֒ → A , where A = C ∞ ( R n ) and I = { f ∈ A : f (0) = 0 } , one can easily show that π is not flat.The C ∞ -differentiable space with one-point spectrum that corresponds to C [[ x , . . . , x n ]]is a special case of W Y/X , the so-called Whitney subspace of Y , where Y is a closedsubset of a C ∞ -differentiable space X [34, Corollary 5.10]. In the case where X is anopen subset of R n , the map W Y/X → X corresponds to the quotient homomorphism C ∞ ( X ) → C ∞ ( X ) /W Y/X , where W Y/X is the ideal of functions whose derivatives of allorders vanish on Y . Normally, W Y/X → X is not an open embedding (in fact, it is alwaysa closed embedding). Nevertheless, we have the following result. Proposition 6.2.
Let X be an open subset of R n , and let Y be a closed subset of X .Then the quotient map C ∞ ( X ) → C ∞ ( X ) /W Y/X is a -pseudoflat epimorphism.Proof. It follows from [55, Lemme 2.4] that any real-valued function f ∈ W Y/X has theform f = f f , where f and f are again in W Y/X . Since C ∞ ( X ) is nuclear, we can applyProposition 3.12. (cid:3) The above remarks lead naturally to the following two questions on C ∞ -differentiablespaces. Question 6.3.
Can open embeddings of affine C ∞ -differentiable spaces be characterizedin terms of projectivity or flatness, as in Theorem ? Question 6.4.
Let ( X, O X ) be an affine C ∞ -differentiable space, and let Y be a closedsubset of X . Is the quotient map O ( X ) → O ( W Y/X ) a -pseudoflat epimorphism? Is ita weak homological epimorphism? Is it a strong homological epimorphism?Acknowledgments. The authors thank the referee for useful comments that significantlyimproved the presentation.
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Oleg Yu. Aristov
E-mail address : [email protected] Alexei Yu. Pirkovskii, Faculty of Mathematics, HSE University, 6 Usacheva, 119048Moscow, Russia
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