Definition 3.36. ([Sti88], p. 140) Letπ :X →S be a morphism of Banach analytic spaces. The space X is called ofrelative finite dimension atx∈X, if there exists an open neighborhood V of x in X, an open subset S0 ⊂ S with π(V) ⊂ S0, an open subset U ⊂ Cn and an embedding i : V ,→ S0 ×U such that the following diagram commutes
V S0 ×U
S0
i
π
,
Proposition 3.37. ([Dou74], p. 596 and [LP75], p. 271) Let π : X → S be a morphism of Banach analytic spaces. Let s ∈ S and x ∈ X(s). If X(s) is of finite dimension at x, then X is of relative finite dimension at x.
3.3 Privileged subspaces of a polycylinder and anaflat
3.3. PRIVILEGED SUBSPACES AND ANAFLAT SHEAVES 45 Let
C(K;F) :={f :K →F|f continuous on K}.
We have B(K;F)⊂C(K;F) andB(K;F) is in fact the uniform closure inC(K;F) of the space of holomorphic maps f : U → F, for an open neighborhood U of K in Cn. LetE, F be Banach spaces. Thenormon the tensor productE⊗F is the norm induced by the norm on the space of linear maps L(F∗, E), where F∗ :=L(F;C), by means of the Kronecker injection E⊗F → L(F∗, E) (see [Dou66], p. 40).
Proposition 3.40. ([Dou66], p. 40) We have
B(K;F)'B(K;C) ˆ⊗F.
From now on we setB(K) := B(K;C). We define now a Banach space of sections B(K;F), where F is a coherent sheaf on a neighborhood of K.
Definition 3.41. ([Dou66], p. 54) LetF be a coherent sheaf on U ⊂Cn and K ⊂U a polycylinder. We define a C-vector space
B(K;F) := B(K)⊗OU(K)F(K).
Definition 3.42. ([Dou66], p. 54) Let K be a polycylinder in U ⊂ Cn. Let F be a coherent sheaf on U. We say that K is F-privileged, if there exists a free resolution
0→ Ln→...→ L1 → L0 → F → 0 (3.1) on an open neighborhood of K, such that the sequence ofC-vector spaces
0→ B(K;Ln)→...→ B(K;L1)→ B(K;L0)→ B(K;F)→0 (3.2) is direct (as C-vector space sequence) and exact.
Remark 3.43.
1. The property of being F-privileged is independent of the choice of the resolu-tion. The exactness of the sequence (3.2) can be expressed by
TorOi K(B(K),F(K)) = 0, for every i >0.
2. By Theorem 3.39 A, any polycylinderK admits a finite free resolution and the resolution can be taken of lengthn. By Theorem 3.39 B, it is enough to require exactness of the complex B(K;L•).
Since (3.1) is a free resolution ofF, we have
B(K;Ln)'B(K;OU⊕rn)'B(K)⊕rn.
Hence, B(K;Ln) is a Banach B(K)-module. Therefore, if K is F-privileged, then B(K;F) = coker (B(K;L1)→ B(K;L0))
is a Banach B(K)-module.
LetU ⊂C2 be an open connected neighborhood of the origin and let h: U →C be holomorphic, with h 6= 0. Let X be the curve given by h, that is (X,OX) = (h−1(0),OU/(h)). We illustrate the notion of OX-privileged polycylinders in C2. We have an exact sequence 0 → OU → Oh· U → OX → 0. Let K = K1 ×K2 ⊂ U be a polycylinder. By definition K is OX-privileged if and only if the induced morphism h :B(K) → B(K) is a direct monomorphism (see Definition 3.12). Let ∂Ki be the boundary of the polycylinder Ki, fori= 1,2. Set ∂K :=∂K1×∂K2.
Proposition 3.44. ([Dou68], p. 68) The following are equivalent 1. h:B(K)→B(K) is a monomorphism.
2. ∃a >0 such that khfk ≥akfk, ∀f ∈B(K).
3. X∩∂K =∅.
Now, let us assume U = U1 ×U2, with Ui ⊂ C, i = 1,2. Let K1 ⊂ U1 be a polycylinder and choose x2 ∈U2.
Proposition 3.45. ([Dou68], p. 68 and [Dou66], p. 61) If X∩(∂K1× {x2}) = ∅, then there exists a polycylinder K2 ⊂U2, with x2 ∈K2, such that K :=K1×K2 is a OX-privileged polycylinder.
Proof. Let us define the function h(x2) : B(K1) → B(K1), sending f to h(x2)·f. By assumption, no zeros of h(x2) is in the boundary of K1. Let l be the number of zeros of h(x2), counted with multiplicity, contained in the interior of K1. Let Pl be the C-vector space of polynomials of degree less than l. By the Weierstrass Division Formula (see, for instance, [KKBB83], p. 72), we get B(K1) ' h(x2)(B(K1))⊕Pl. Now, by [Dou66], p. 60, Proposition 6, there exists V ⊂U2, with x2 ∈V, such that for each polycylinderK2 ⊂V, with x2 ∈K2, we have B(K)'h(B(K))⊕B(K2;Pl), where B(K2;Pl) is the Banach space of continuous functions g :K2 →Pl, which are analytic on the interior of K2 (see Proposition 3.40).
Proposition 3.46. ([Dou66], p. 56) Let 0 → F0 → F → F00 → 0 be an exact sequence of coherent sheaves onU ⊂Cn. If K ⊂U is aF0,F00-privileged polycylinder, then K is F-privileged and
0→B(K;F0)→B(K;F)→B(K;F00)→0 is a direct exact sequence of C-vector spaces.
3.3. PRIVILEGED SUBSPACES AND ANAFLAT SHEAVES 47 LetK ⊂U be a polycylinder and E a coherent sheaf onU. From Proposition 3.46 it follows that ifF is a coherent quotient sheaf ofE such thatK isF-privileged, then B(K;F) is a direct Banach B(K)-submodule of B(K;E).
Theorem 3.47. ([Dou66], p. 62) Let F be a coherent sheaf on an open subset U ⊂ Cn. For any x ∈ U there exists a fundamental system of neighborhoods of x in U, which are F-privileged polycylinders.
Given a polycylinder K in Cn, we can define a sheaf of algebras on K. For any U ⊂K open, we set
BK :U 7→BK(U) := {f :U →C:f is continuous and holomorphic on U∩K◦}.
Therefore, we get a ringed space (K,BK).
Definition 3.48. ([Dou74], p. 577 and [LP75], p. 256) Let F be a sheaf of BK -modules. We say that F is K-privileged if there exists a finite free resolutionL• of F onK such that the complexB(K;L•) := L•(K) is direct and acyclic in degreed >0.
This complex is a resolution of B(K;F) :=F(K). We call aprivileged subspace ofK a ringed space (Y,BY), where Y is a subspace of K and the sheaf BY , extended by 0 over K, is a quotient sheaf of BK by a privileged ideal subsheaf I with support Y. We notice that the sheaf BK restricted to K◦ coincides with OK◦. IfY is a priv-ileged subspace of K, then the space Y ∩K◦ is a complex subspace of K◦. We set Y◦ :=Y ∩K◦.
Definition 3.49. ([Dou74], p. 577 and [LP75], p. 256) Let Y ⊂ K be a privileged subspace. Let L⊂K be another polycylinder. We say that L is Y-privileged if there exists a finite free resolution of BY onK
0→ BrKn → BKrn−1 →...→ BKr1 → BK → BY →0 such that the sequence
0→ BLrn → BrLn−1 →...→ BrL1 → BL, obtained applying − ⊗BK BL, is direct and exact.
In this case (Y ∩L,BY ⊗BK BL) is a privileged subspace of L.
Theorem 3.50. ([Pou75], p. 159) Let K ⊂ Cn be a polycylinder and Y ⊂ K a privileged subspace. Then, everyxinKadmits a fundamental system of neighborhoods which are Y-privileged polycylinders.
Ifx∈K◦, this result is nothing but Theorem 3.47. Ifx /∈K◦, the proof is due to G. Pourcin ([Pou75]).
LetA be a Banach algebra and E a Banach A-module. Let GA(E) be the Grass-mannian of direct A-submodules of E (see Example 3.21). LetGA(Arn, ..., Ar0;E) be the Banach analytic space of direct finite free resolutions of E
0→Arn →dn ... →d1 Ar0 →d0 E see [Dou66], p. 33.
Proposition 3.51. ([Dou66], p. 33) The canonical morphism Im :GA(Arn, ..., Ar0;E)→ GA(E)
(dn, ..., d0)7→Imd0 (3.3) is smooth.
From Proposition 3.51, it follows that the set of all privileged ideals I in B(K) is an open subset of the Grassmannian GB(K)(B(K)) of direct ideals of B(K) (see [Dou66], p. 34).
Definition 3.52. LetK be a polycylinder. The set of all privileged subspaces of K is a Banach analytic space denoted by G(K).
Let (X,Φ) be a Banach analytic space. We naturally get a ringed space (X,OX) setting OX := Φ(C). We want to give an exact definition of what is a coherent sheaf depending analytically on a parameter s belonging to a Banach analytic space.
Definition 3.53. ([Dou66], p. 64) Letf :X →X0 be a morphism of Banach analytic spaces and F a sheaf of OX0-modules on X0. We set:
f∗F :=OX ⊗f−1OX0 f−1F.
If F is a sheaf of OS×X-modules on S×X, for each s∈S, we set:
F(s) := i∗sF,
where is :X ,→S×X is the natural embedding defined by s.
Remark 3.54. IfS is of finite dimension, then F(s)' F/msF 'C⊗OS,s F.
Definition 3.55. ([Dou66], p. 66) Let U ⊂ Cn an open subset and S a Banach analytic space. A sheaf F of S×U- modules on S×U is called S-anaflat if for each (s, x) ∈ S ×U there exists a finite free resolution 0 → L• → F → 0 of F in a neighborhood of (s, x) in S×U such that the complex:
0→ L•(s)→ F(s)→0 is an exact sequence in an neighborhood of xin U.
Theorem 3.56. ([Dou66], p. 66) Let S be a complex space. Let F be a sheaf of OS×U-modules on S×U. Then F is S-anaflat if and only if it is coherent and flat.
3.3. PRIVILEGED SUBSPACES AND ANAFLAT SHEAVES 49 Let S be a Banach analytic space and U an open subset of Cn. Let F be a S-anaflat sheaf on S ×U, s ∈ S and K ⊂ U a F(s)-privileged polycylinder. From Theorem 3.39 A, we get the existence of a finite free resolution on a neighborhood V0×V1 of {s0} ×K in S×U
0→ Ln →...→ L1 → L0 → F →0.
Since F isS-anaflat
0→ Ln(s0)→...→ L1(s0)→ L0(s0)→ F(s0)→0 is exact for each s0 ∈V0. Let us consider the complex
0→ Ln(s0)→...→ L1(s0)→ L0(s0).
It is acyclic in degree d >0 and it defines a complex of Banach spaces 0→B(K;Ln(s0))→...→B(K;L1(s0))→B(K;L0(s0))
for each s0 ∈V0, which, in turn, defines a complex of Banach vector bundles over V0 0→B(K;Ln)→...→B(K;L1)→B(K;L0).
Since K isF(s)-privileged, this complex is direct and acyclic in degree d >0 over a neighborhood Vs of s in V0. Hence, let
B(K;F)Vs :=H0(B(K;L•)) = coker(B(K;L1)→B(K;L0)).
This is a trivial Banach vector bundle over Vs, with fibres
B(K;F(s0))Vs := H0(B(K;L•(s0))) = coker(B(K;L1(s0))→B(K;L0(s0))), for s0 ∈ Vs. The Banach vector bundle B(K;F)Vs does not depend on the choice of the resolution L• of F (see [Dou66], p. 68).
Theorem 3.57. ([Dou68], p. 68)(Flatness and privilege) Let S be a Banach analytic space, U ⊂Cn open, F a S-anaflat sheaf on S×U and K ⊂ U a polycylinder. The set S0 of all points s∈S such that K isF(s)-privileged is open inS and the Banach spaces B(K;F(s)) are the fibres of a Banach vector bundle B(K;F) over S0
This result expresses the idea that if F is S-anaflat, the sheaf F(s) depends continuously on the point s∈S.
Corollary 3.58. ([Dou68], p. 73) Let π : X → S be a morphism of complex spaces and E a coherent OX-module, which is flat as OS-module. Then π|SuppE is an open map.
Proof. Assume that X can be embedded in S ×U, with U ⊂ Cn open, and that E is extended by zero to S×U. Let x0 ∈ Supp E and V a neighborhood of x0 in S×U. Let s0 :=π(x0) and choose an E(s0)-privileged polycylinderK inU, such that {s0} ×K ⊂π−1(W)⊂V, whereW is a neighborhood ofs0 inS. We have the Banach vector bundleB(K;E|π−1(W)), whose fiber oversisB(K;E(s)). Sincex0 ∈Supp E(s0) and K is a neighborhood ofx0, B(K;E(s0))6= 0. Since all the fibres are isomorphic, then for any s ∈ U, B(K;E(s)) 6= 0 and therefore {s} × K ∩ Supp E 6= 0, and s ∈π(Supp E). Thus, π is open.
Proposition 3.59. ([Dou66], p. 69) (Change of base) Let F be a S-anaflat sheaf on S×U. Let S0 be a Banach analytic space and f :S0 →S a morphism. The sheaf
F0 := (f×IdU)∗F is a S0-anaflat sheaf on S0×U.
Definition 3.60. ([Dou66], p. 76) LetSbe a Banach analytic space andX a complex space. LetF be a sheaf ofOS×X-modules on S×X. We say thatF isS-anaflat if for anyx∈X there exists an open subsetU ofCnand a local embeddingi:X ,→S×U, such that i∗F is a S-anaflat sheaf on S×U .
Definition 3.61. Let X, S be Banach anlytic spaces and f : X → S a morphism.
We say that X is anaflat over S if for eachx∈X there exists a neighborhoodV of x inX, an open subsetU ⊂Cnand an embeddingi:V ,→S×U such thatf is induced by the projection of S×U onto S and i∗OV is a S-anaflat sheaf of S×U-modules.
Definition 3.62. ([LP75], p. 258 and [Pou75], p. 183, Theorem 4.13) Let S be a Banach analytic space and K ⊂ Cn a polycylinder. Let G(K) given by Definition 3.52. A morphism S → G(K) is called a S-anaflat subspace of S×K.
Definition 3.63. We denote withY the universalG(K)-anaflat subspace ofG(K)×K defined by the identity morphism Id :G(K)→ G(K).