Homological dimensions of modules of holomorphic functions on submanifolds of Stein manifolds

0.26 MB·English

by  A. Yu. Pirkovskii

HOMOLOGICAL DIMENSIONS
OF MODULES OF HOLOMORPHIC FUNCTIONS
ON SUBMANIFOLDS OF STEIN MANIFOLDS
A. YU. PIRKOVSKII
2
1
0 Abstract. LetX beaSteinmanifold,andletY ⊂X beaclosedcomplexsubmanifold.
2 DenotebyO(X)thealgebraofholomorphicfunctionsonX. Weshowthattheweak(i.e.,
n flat) homologicaldimension of O(Y) as a Fr´echet O(X)-module equals the codimension
a of Y in X. In the case where X and Y are of Liouville type, the same formula is proved
J for the projective homological dimension of O(Y) over O(X). On the other hand, we
3 showthatifX isofLiouvilletypeandY ishyperconvex,thentheprojectivehomological
1
dimension of O(Y) over O(X) equals the dimension of X.

]
A
F
. 1. Introduction
h
t
a This paper is motivated by the following fact from commutative algebra. Let X be a
m
nonsingular affine algebraic variety over C, and let O(X) denote the algebra of regular
[
functions on X. It is well known and easy to show that, for each nonsingular closed

1 algebraic subvariety Y ⊂ X, the projective homological dimension of O(Y) considered as
v a module over O(X) is equal to the codimension of Y in X:
8
82 dhO(X)O(Y) = codimX Y. (1)
2 Indeed, let I ⊂ O denote the ideal sheaf of Y in X. Since Y is a local complete
. Y X
1 intersection in X [17, 8.22.1], it follows that for each y ∈ Y the ideal I of the local
0 Y,y
ring O is generated by a regular sequence of length m = codim Y. Hence we have
2 X,y X
v:1 dThhOeXo,ryeOmY,1y1=]. mNo[t2e9,al3s.o8,thTahte,osrienmce2O2](.XT)hies gNloobetahlefroiramn,uwlae(a1l)sonohwavfeollows from [28, 9.2,
i
X w.dhO(X)O(Y) = codimX Y, (2)
r
a where w.dh stands for the weak (i.e., flat) homological dimension.
In [32], we proved (1) in the situation where X is a smooth real manifold, O(X) =
C∞(X) is the algebra of smooth functions on X, and Y ⊂ X is a closed smooth subman-
ifold. Now the formula (1) should be understood in the context of “Topological Homol-
ogy”, i.e., a relative homological algebra in categories of Fr´echet modules over Fr´echet
algebras [18]. The above-mentioned localization technique is not applicable in this case,
so we had to develop an essentially different proof which heavily relied on the softness of
the structure sheaf C∞. The formula (2) also easily follows from the results of [32] (see
X
Remark 5.14 for details).
Our goal here is to study complex analytic analogues of (1) and (2) in the context of
Topological Homology. Suppose that X is a complex Stein manifold, O(X) is the Fr´echet
2010 Mathematics Subject Classification. Primary 46M18; secondary 46H25, 46E25, 16E10, 16E30,
16E40.
This work was partially supported by the Ministry of Education and Science of Russia (programme
“Development of the scientific potential of the Higher School”, grant no. 2.1.1/2775).
1
2 A.YU. PIRKOVSKII
algebra of holomorphic functions on X, and Y ⊂ X is a closed analytic submanifold.
In Section 3, we show that (2) holds. As a byproduct, we obtain a complex analytic
version of the Hochschild-Kostant-Rosenberg Theorem [22]. Our proof essentially uses
the nuclearity of O(X) and some results of O. Forster [11].
Surprisingly, the validity of (1) turns out to depend on some special properties of X
and Y. In Section 4, we show that (1) holds provided that both X and Y are of Liouville
type (i.e., each bounded above plurisubharmonic function on X and Y is constant). In
particular, this is true whenever both X and Y are affine algebraic. On the other hand,
we show in Section 6 that (1) fails in the general case. Specifically, assume that X
is of Liouville type and that Y is hyperconvex (this means that there exists a negative
plurisubharmonic exhaustion function on Y). We show that dhO(X)O(Y) = dimX in this
case. Forexample, if Y is theopendisc embedded into C2 (cf. [1]), then dhO(C2)O(Y) = 2.
Our proofs heavily rely on the linear topological invariants (DN), (Ω), and (Ω), in-
troduced by D. Vogt in the 1970ies (see, e.g., [25, Chapter 8]). We essentially use the
splitting theorem due to D. Vogt and M. J. Wagner [41], the results of D. Vogt [44] on
bounded linear maps between Fr´echet spaces, and the results of V. P. Zakharyuta [47],
D. Vogt [45], and A. Aytuna [2] on linear topological properties of spaces of holomorphic
functions. Another essential ingredient is the Van den Bergh isomorphism [39] between
the Hochschild homology and cohomology of O(X), which is proved in Section 5.
2. Preliminaries
This section gives a brief account of some basic facts from Topological Homology. Our
main reference is [18]; some details can also be found in [9,19,21,33,38].
Throughout, allvectorspacesandalgebrasareassumedtobeoverthefieldCofcomplex
numbers. All algebras are assumed to be associative and unital. By a Fr´echet algebra we
mean an algebra A endowed with a complete, metrizable locally convex topology (i.e., A
is an algebra and a Fr´echet space simultaneously) such that the product map A×A → A
is continuous. If, in addition, A is locally m-convex (i.e., the topology on A can be
determined by a family {k·k : λ ∈ Λ} of seminorms satisfying kabk ≤ kak kbk for all
λ λ λ λ
a,b ∈ A), then A is said to be a Fr´echet-Arens-Michael algebra.
Let Abea Fr´echet algebra. Aleft Fr´echet A-moduleisaleft A-moduleM endowed with
a complete, metrizable locally convex topology in such a way that the action A×M → M
is continuous. We always assume that 1 ·x = x for all x ∈ M, where 1 is the identity
A A
of A. Left Fr´echet A-modules and their continuous morphisms form a category denoted
by A-mod. Given M,N ∈ A-mod, the space of morphisms from M to N will be denoted
by h (M,N). We always endow h (M,N) with the topology of uniform convergence
A A
on bounded subsets of M; note that this topology, in general, is not metrizable. The
categories mod-A and A-mod-A of right Fr´echet A-modules and of Fr´echet A-bimodules
are defined similarly. Note that A-mod-A ∼= Ae-mod ∼= mod-Ae, where Ae = A⊗Aop, and
where Aop stands for the algebra opposite to A.
If M is a right Fr´echet A-module and N is a left Fr´echet A-module, then theirbA-module
tensor product M ⊗ N is defined to be the quotient (M ⊗N)/L, where L ⊂ M ⊗N is
A
the closed linear span of all elements of the form x · a ⊗ y − x ⊗ a · y (x ∈ M, y ∈ N,
a ∈ A). As in purbe algebra, the A-module tensor producbt can be characterized bby the
universal property that, for each Fr´echet space E, there is a natural bijection between
HOMOLOGICAL DIMENSIONS OF MODULES OF HOLOMORPHIC FUNCTIONS 3
the set of all continuous A-balanced bilinear maps from M ×N to E and the set of all
continuous linear maps from M ⊗ N to E.
A
A chain complex C = (Cn,dn)n∈Z in A-mod is admissible if it splits in the category of
topological vector spaces, i.e., ifbit has a contracting homotopy consisting of continuous
linear maps. Geometrically, this means that C is exact, and Kerd is a complemented
n
subspace of C for each n.
n
Let Vect denote the category of vector spaces and linear maps. A left Fr´echet A-module
P is projective (respectively, strictly projective) if the functor h (P,−): A-mod → Vect
A
takes admissible (respectively, exact) sequences of Fr´echet A-modules to exact sequences
of vector spaces. Similarly, a left Fr´echet A-module F is flat (respectively, strictly flat)
if the tensor product functor (−)⊗ F: mod-A → Vect takes admissible (respectively,
A
exact) sequences of Fr´echet A-modules to exact sequences of vector spaces. Clearly, each
strictly projective (respectively, strbictly flat) Fr´echet module is projective (respectively,
flat). It is also known that every projective Fr´echet module is flat.
A resolution of M ∈ A-mod is a pair (P,ε) consisting of a nonnegative chain complex
P in A-mod and a morphism ε: P → M making the sequence P −→ε M → 0 into an
0
admissible complex. The length of P is the minimum integer n such that P = 0 for all
i
i > n, or ∞ if there is no such n. If all the P ’s are projective (respectively, flat), then
i
(P,ε)is called a projective resolution (respectively, a flat resolution) of M. 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.
If M,N ∈ A-mod, then the space Extn(M,N) is defined to be the nth cohomology
A
of the complex h (P,N), where P is a projective resolution of M. Similarly, if M ∈
A
mod-A and N ∈ A-mod, then the space TorA(M,N) is defined to be the nth homology
n
of the complex M ⊗ F, where F is a flat resolution of N. The spaces Extn(M,N)
A A
and TorA(M,N) do not depend on the particular choice of P and F and have the usual
n
functorial propertiebs (see [18] for details). If M ∈ A-mod-A, then the nth Hochschild
cohomology(respectively, homology)ofAwithcoefficientsinM isdefinedbyH n(A,M) =
Extn (A,M) (respectively, H (A,M) = TorAe(M,A)).
Ae n n
For each M ∈ mod-A and each N ∈ A-mod the tensor product N ⊗M is a Fr´echet
A-bimodule in a natural way, and there exist topological isomorphisms
TorA(M,N) ∼= H (A,N ⊗M). b (3)
n n
If M,N ∈ A-mod and M is a Banach module, then L(M,N) is a Fr´echet A-bimodule in
a natural way, and we have vector space isomorphismsb
Extn(M,N) ∼= H n(A,L(M,N)). (4)
A
In some cases, the spaces TorA(M,N) can be computed via nonadmissible resolutions.
n
Let N ∈ A-mod, and let (F,ε) be a pair consisting of a nonnegative chain complex F
in A-mod and a morphism ε: F → N making the sequence F −→ε N → 0 into an exact
0
complex. Suppose also that all the F ’s are flat Fr´echet modules (so that F is “almost” a
i
flat resolution of N). Take M ∈ mod-A, and assume that either all the F ’s are nuclear,
i
or both A and M are nuclear. Then we have TorA(M,N) ∼= H (M ⊗ F) (see [38]
n n A
or [9, 3.1.13]).
The projective homological dimension of M ∈ A-mod is the minimubm integer n =
dh M ∈ Z ∪ {∞} with the property that M has a projective resolution of length
A +
n. Similarly, the weak homological dimension of M ∈ A-mod is the minimum integer
4 A.YU. PIRKOVSKII
n = w.dh M ∈ Z ∪ {∞} with the property that M has a flat resolution of length n.
A +
Equivalently,
dh M = min{n ∈ Z |Extn+1(M,N) = 0 ∀N ∈ A-mod}
A + A
= min{n ∈ Z |Extp+1(M,N) = 0 ∀N ∈ A-mod, ∀p ≥ n};
+ A
w.dh M = min n ∈ Z TorAn+1(N,M) = 0, and TorAn(N,M)
A + is Hausdorff ∀N ∈ mod-A
(cid:26) (cid:12) (cid:27)
(cid:12)
= min n ∈ Z (cid:12) TorAp+1(N,M) = 0, and TorAn(N,M) .
+(cid:12) is Hausdorff ∀N ∈ mod-A, ∀p ≥ n
(cid:26) (cid:12) (cid:27)
(cid:12)
Note that dh M = 0 if and only if(cid:12)M is projective, and w.dh M = 0 if and only if M
A (cid:12) A
is flat. Since each projective module is flat, we clearly have w.dh M ≤ dh M.
A A
The global dimension and the weak global dimension of A are defined by
dgA = sup{dh M |M ∈ A-mod},
A
w.dgA = sup{w.dh M |M ∈ A-mod}.
A
The bidimension and the weak bidimension of A are defined by dbA = dh A and
Ae
w.dbA = w.dh A, respectively. We clearly have w.dgA ≤ dgA and w.dbA ≤ dbA. It
Ae
is also true (but less obvious) that dgA ≤ dbA and w.dgA ≤ w.dbA.
Throughout the paper, all complex manifolds are assumed to be connected (although
mostofourresultscaneasilybeextendedtomanifoldshavingfinitelymanycomponentsof
the same dimension). The structure sheaf of acomplex manifoldX will always be denoted
by O . The phrase “an O -module” will mean “a sheaf of O -modules”. Recall that
X X X
O(X) is a nuclear Fr´echet algebra with respect to the compact-open topology (i.e., the
topologyof uniformconvergence oncompact subsets ofX). Recall also (see, e.g., [13, 5.6])
that, for each coherent O -module F, the space of global sections F(X) = Γ(X,F) has
X
a canonical topology making F(X) into a nuclear Fr´echet O(X)-module.
If X is a Stein manifold, then a Fr´echet O(X)-module F is a Stein module if it is
topologically isomorphic to F(X) for some coherent O -module F. By a result of
X
O. Forster [11, 2.1], the functor Γ(X, ·) of global sections is an equivalence between
the category of coherent O -modules and the full subcategory of O(X)-mod consisting of
X
Stein O(X)-modules.
3. Forster resolution and weak dimension
The following lemma is an easy consequence of O. Forster’s results [11]. We formulate
it here for the reader’s convenience.
Lemma 3.1. Let X be a Stein manifold, and let Y ⊂ X be a closed submanifold of
codimension m. There exists an exact sequence
0 ← O ← P ← P ← ··· ← P ← 0 (5)
Y 0 1 m
where P ,...,P are locally free O -modules.
0 m X
Proof. By [11, 6.4], there exists an exact sequence
0 ← O ← F ←d−0 F ← ··· ← F ←d−m−−−1 F ←dm− ···
Y 0 1 m−1 m
HOMOLOGICAL DIMENSIONS OF MODULES OF HOLOMORPHIC FUNCTIONS 5
where F ,F ,... are free O -modules. Letting P = Imd , we obtain an exact
0 1 X m m−1
sequence
0 ← O ← F ← F ← ··· ← F ← P ← 0. (6)
Y 0 1 m−1 m
To complete the proof, it remains to show that P is locally free. Fix x ∈ X, and
m
consider the exact sequence
0 ← O ← F ← F ← ··· ← F ← P ← 0 (7)
Y,x 0,x 1,x m−1,x m,x
of O -modules. If x ∈/ Y, then O = 0, whence (7) splits, and so P is projective.
X,x Y,x m,x
Now suppose that x ∈ Y, and let I ⊂ O denote the ideal sheaf of Y. We have
X
O ∼= O /I . Choose a local coordinate system z1,...,zn in a neighborhood U of x
Y,x X,x x
such that Y ∩U = {z ∈ U : z1 = ··· = zm = 0}. The O -module I is generated by
X,x x
the regular sequence (z1,...,zm), which implies that dhOX,xOY,x = m (see, e.g., [29, 3.8,
Theorem 22])1. By looking at (7), we conclude that P is projective.
m,x
Thus for each x ∈ X the O -module P is projective. Since O is local, this
X,x m,x X,x
means that P is free. Therefore P is a locally free O -module, as required. (cid:3)
m,x m X
Corollary 3.2. Let X be a Stein manifold, and let Y ⊂ X be a closed submanifold of
codimension m. There exists an exact sequence
0 ← O(Y) ← P ← P ← ··· ← P ← 0 (8)
0 1 m
of Fr´echet O(X)-modules, where P ,...,P are finitely generated and strictly projective.
0 m
Proof. Applying the global section functor Γ(X, ·) to (5), we obtain (8). Using [11, 6.2
and 6.3], we conclude that P ,...,P are finitely generated and strictly projective. (cid:3)
0 m
Definition 3.3. Any sequence of the form (5) (respectively, (8)) will be called a Forster
resolution of O over O (respectively, of O(Y) over O(X)).
Y X
Remark 3.4. Since the global section functor is an equivalence between the category of
coherent O -modules and the category of Stein O(X)-modules, it yields a 1-1 correspon-
X
dence between Forster resolutions of O over O and Forster resolutions of O(Y) over
Y X
O(X).
Remark 3.5. In fact, it easily follows from Forster’s construction [11] that resolution (8)
can be chosen in such a way that P = O(X) and that the arrow P → O(Y) is the
0 0
restriction map. We will not use this in the sequel.
Remark 3.6. Note that (8) is not necessarily a resolution in the sense of Topological
Homology, i.e., it need not be admissible. The first counterexample was given in [27,
Proposition 5.3]; a more general situation will be discussed in Remark 6.4 below.
Proposition 3.7. Let X be a Stein manifold, and let Y be a closed submanifold of X.
Then for each coherent O -module F and each p ∈ Z we have a topological isomorphism
X +
TorO(X)(O(Y),F(X)) ∼= Γ(X,TorOX(O ,F)).
p p Y
1Here,ofcourse,theprojectivehomologicaldimensiondhOX,xOY,x shouldbeunderstoodinthepurely
algebraic context.
6 A.YU. PIRKOVSKII
Proof. Let P → O → 0 be a Forster resolution of O over O . Applying the global
Y Y X
section functor, we get a Forster resolution P(X) → O(Y) → 0 of O(Y) over O(X) (see
Remark 3.4). Since all the modules P (X) are nuclear, it follows that
i
TorO(X)(O(Y),F(X)) ∼= H (P(X) ⊗ F(X)).
p p
O(X)
We also have P(X)⊗O(X)F(X) ∼= Γ(X,P⊗OX F) (sebe, e.g., [9, 4.2.4] or [31, 2.2]).
Hence
TorO(X)(O(Yb),F(X)) ∼= H (Γ(X,P ⊗ F))
p p
O
X
∼= Γ(X,H (P ⊗ F)) = Γ(X,TorOX(O ,F)). (cid:3)
p p Y
O
X
Corollary 3.8. Let X be a Stein manifold, and let Y be a closed submanifold of X.
Denote by I ⊂ O the ideal sheaf of Y. Then for each p ∈ Z we have
X +
TorO(X)(O(Y),O(Y)) ∼= Γ(X, p(I/I2)).
p
Proof. By [24, 3.5], we have TorOX(O ,O ) ∼= p(IV/I2) (the proof of this fact, given
p Y Y
in[24]forregularschemes, appliestocomplexmanifoldswithoutchanges). Nowitremains
V (cid:3)
to apply Proposition 3.7.
As a byproduct, we obtain the following analytic version of the Hochschild–Kostant–
Rosenberg Theorem [22].
Corollary 3.9. Let X be a Stein manifold. Then for each p ∈ Z we have
+
H (O(X),O(X)) ∼= Ωp(X),
p
where Ωp(X) is the space of holomorphic p-forms on X.
Proof. Recall from [14, II.3.3] that there exists a topological isomorphism
O(X)e = O(X)⊗O(X) ∼= O(X ×X), f ⊗g 7→ ((x,y) 7→ f(x)g(y)).
Under this identification, the O(X)e-module O(X) becomes the O(X×X)-module O(∆),
where ∆ = {(x,x) : x ∈ Xb} is the diagonal of X ×X. Let I ⊂ O be the ideal sheaf
X×X
of ∆. Identifying X with ∆ via the map x 7→ (x,x) induces a sheaf isomorphism between
(I/I2)| and the cotangent sheaf Ω1 (see, e.g., [15]). Now Corollary 3.8 implies that
∆ X
H (O(X),O(X)) = TorO(X)e(O(X),O(X)) ∼= TorO(X×X)(O(∆),O(∆))
p p p
∼= Γ(X ×X, p(I/I2)) ∼= Ωp(X). (cid:3)
Theorem 3.10. Let X be a Stein manifold, and let Y be aVclosed submanifold of X.
Then w.dh O(Y) = codim Y.
O(X) X
Proof. Let M be a Fr´echet O(X)-module, and let P → O(Y) → 0 be a Forster res-
olution of O(Y) over O(X). Using the nuclearity argument (see the proof of Proposi-
tion 3.7), we see that TorpO(X)(O(Y),M) is topologically isomorphic to Hp(P ⊗O(X)M),
which vanishes for p > m and is Hausdorff for p = m, where m = codim Y. Therefore
X
w.dhO(X)O(Y) ≤ m. Toobtaintheoppositeestimate, recallfrom[24,3.4]that(bI/I2)|Y
is locally free of rank m. In particular, m(I/I2) 6= 0, and it follows from Corollary 3.8
that TorO(X)(O(Y),O(Y)) 6= 0. The rest is clear. (cid:3)
m V
HOMOLOGICAL DIMENSIONS OF MODULES OF HOLOMORPHIC FUNCTIONS 7
Corollary 3.11 ( [31,36]). Let X be a Stein manifold. Then w.dgO(X) = w.dbO(X) =
dimX.
Proof. As in the proof of Corollary 3.9, identify X with the diagonal ∆ ⊂ X × X.
Theorem 3.10 implies that
w.dbO(X) = w.dhO(X)e O(X) = w.dhO(X×X)O(∆) = codimX×X ∆ = dimX.
Hence w.dgO(X) ≤ dimX. Applying Theorem 3.10 to the singleton Y = {x } yields the
0
opposite estimate w.dgO(X) ≥ codim {x } = dimX. (cid:3)
X 0
4. Liouville-type property and projective dimension
Let X be a complex manifold. Following [30], we say that X is of Liouville type if
each bounded above plurisubharmonic function on X is constant. For example, each
nonsingular affine algebraic variety is of Liouville type. More examples can be found
in [3,4] (note that Stein manifolds of Liouville type are called parabolic in [4]).
Starting from this section, we will make use of the linear topological invariants (DN)
and (Ω) introduced by D. Vogt [40,41] (see also [25]). Let E be a Fr´echet space. By
definition, E has property (DN) if the topology on E can be determined by an increasing
sequence {k · k : n ∈ N} of seminorms satisfying the following condition: there exists
n
p ∈ N such that for each k ∈ N and each 0 < r < 1 there exist n ∈ N and C > 0 such
that
kxk ≤ Ckxkrkxk1−r (x ∈ E).
k p n
Given a continuous seminorm k·k on a Fr´echet space E, define the dual “seminorm”
k·k∗: E∗ → [0,+∞] by kyk∗ = sup{|y(x)| : kxk ≤ 1}. Note that k·k∗ can take the value
+∞ as well. By definition, E has property (Ω) if the topology on E can be determined
by an increasing sequence {k·k : n ∈ N} of seminorms satisfying the following condition:
n
for each p ∈ N there exists q ∈ N such that for each k ∈ N there exist C > 0 and r ∈ (0,1)
satisfying
kyk∗ ≤ C(kyk∗)r(kyk∗)1−r (y ∈ E∗).
q p k
Property (DN) is inherited by subspaces, while (Ω) is inherited by quotients. A basic
fact about (DN) and (Ω) is the following Splitting Theorem due to Vogt and Wagner [41]
(see also [25]): an exact sequence 0 → E → F → G → 0 of nuclear Fr´echet spaces splits
provided that E has (Ω) and G has (DN).
The following lemma is an easy consequence of the Splitting Theorem.
Lemma 4.1. Let E = (E ,d ) be a chain complex of nuclear Fr´echet spaces. Suppose that
i i
all the spaces E have properties (DN) and (Ω). Then E splits.
i
Proof. For each i ∈ Z we have an exact sequence
0 ← K ←d−i E ← K ← 0, (9)
i i+1 i+1
where K = Im(d : E → E ). Note that each K , being a subspace of E and a quotient
i i i+1 i i i
of E , has properties (DN) and (Ω). Now the Splitting Theorem implies that (9) splits
i+1
for all i ∈ Z, which means exactly that E splits. (cid:3)
Let now X be a Stein manifold. As was observed in [42], O(X) always has property
(Ω). Indeed, for X = Cn this can be seen directly, and in the general case X can be
embedded into Cn for sufficiently large n, so O(X) becomes a quotient of O(Cn). On the
other hand, it was shown independently by Zakharyuta [47], Vogt [45], and Aytuna [2]
8 A.YU. PIRKOVSKII
that O(X) has (DN) if and only if X is of Liouville type. In particular, this is true
provided that X is affine algebraic [46]. For more results in this direction, see [3,4,30].
Corollary 4.2. Let X be a Stein manifold, and let Y be a closed submanifold of X.
Suppose that both X and Y are of Liouville type. Then each Forster resolution of O(Y)
over O(X) is admissible.
Proof. SinceX andY areofLiouvilletype, itfollowsthatO(X)andO(Y)haveproperties
(DN) and (Ω). Each finitely generated, strictly projective Fr´echet O(X)-module also has
properties (DN) and (Ω), because it is isomorphic to a direct summand of O(X)p for
some p ∈ N. Now we see that resolution (8) satisfies the conditions of Lemma 4.1. (cid:3)
Theorem 4.3. Let X be a Stein manifold, and let Y be a closed submanifold of X.
Suppose that both X and Y are of Liouville type. Then dh O(Y) = codim Y.
O(X) X
Proof. Corollary 4.2 implies that dhO(X)O(Y) ≤ codimX Y, and the opposite estimate is
(cid:3)
immediate from Theorem 3.10.
Arguing in the same way as in the proof of Corollary 3.11, we obtain the following.
Corollary 4.4. Let X be a Stein manifold of Liouville type. Then dgO(X) = dbO(X) =
dimX.
5. Van den Bergh isomorphisms for O(X)
The Van den Bergh isomorphisms are certain relations between Hochschild homology
and cohomology of associative algebras. Special cases of such isomorphisms can be traced
back to the origins of homological algebra, but systematically they were studied only in
1998 by M. Van den Bergh [39]. For more recent results involving the Van den Bergh
isomorphisms, see [5,7,8,10,12,23,26], to cite a few. Below we will use a Fr´echet algebra
version of the Van den Bergh isomorphisms, which was introduced and applied in [32] to
some problems of Topological Homology (see also [35]).
Let A be a Fr´echet algebra. A Fr´echet A-bimodule M is invertible if there exists a
Fr´echet A-bimodule M−1 such that M ⊗ M−1 ∼= M−1⊗ M ∼= A in A-mod-A. For
A A
example, if X is a Stein manifold, then for each line sheaf L of O -modules the O(X)-
X
bimodule L(X) is invertible, and L(X)b−1 = L−1(X), wbhere L−1 = HomOX(L,OX).
More examples can be found in [35]. By [32, Prop. 5.2.6], each invertible Fr´echet A-
bimodule is projective in A-mod and in mod-A.
Following [10] (see also [35] for the Fr´echet algebra case), we say that A satisfies the
Van den Bergh condition VdB(n) if there exists an invertible Fr´echet A-bimodule L (a
dualizing bimodule) such that for each Fr´echet A-bimodule M and each i ∈ Z there is a
vector space isomorphism
H i(A,M) ∼= H (A,L⊗M).
n−i
A
For example, if X is a smooth manifold, then C∞(X) satisfies VdB(n) with n = dimX
b
and L = Tn(X), the module of smooth n-polyvector fields on X [32]. We refer to [35] for
more examples.
The goal of this section is to establish the Van den Bergh isomorphisms for A = O(X),
where X is a Stein manifold of Liouville type.
HOMOLOGICAL DIMENSIONS OF MODULES OF HOLOMORPHIC FUNCTIONS 9
Lemma 5.1. Let A be a Fr´echet algebra, and let C = (Ci,di) be a cochain complex in
mod-A such that the cohomology spaces Hi(C) are Hausdorff. Let F ∈ A-mod be a flat
Fr´echet module, and assume that either all the Ci’s are nuclear or A and F are nuclear.
Then the cohomology spaces Hi(C⊗ F) are also Hausdorff, and there exist canonical
A
topological isomorphisms
Hi(Cb)⊗F ∼= Hi(C⊗F). (10)
A A
Specifically, if ξ ∈ Hi(C) is represented by an i-cocycle c ∈ Ci, then (10) takes ξ ⊗ x to
b b A
the cohomology class of c⊗ x.
A
Proof. Let Zi = Zi(C) denote the space of i-cocycles of C. Write also Hi = Hi(C) for
short. For each i we have an exact sequence
0 → Zi → Ci −→di Zi+1 → Hi+1 → 0
of right Fr´echet A-modules. Since F is flat, it follows from the nuclearity assumptions
(see, e.g., [9, 3.1.12]) that the tensored sequence
0 → Zi⊗F → Ci⊗F −d−i⊗−A−1→F Zi+1⊗F → Hi+1⊗F → 0 (11)
A A A A
is exact. Using the exactness of (11) first for i, and then for i+1, we obtain topological
b b b b
isomorphisms
Zi⊗F ∼= Ker Ci⊗F −d−i⊗−A−1→F Zi+1⊗F
A A A
(12)
b = Ker(cid:0)Ci⊗b F −d−i⊗−A−1→F Ci+1⊗b F(cid:1) = Zi(C⊗F).
A A A
Now it follows from the exactn(cid:0)ess of (11) for i−1 and f(cid:1)rom (12) that
b b b
Hi⊗F ∼= Coker Ci−1⊗F −d−i−−1−⊗b−A−1→F Zi⊗F
A A A
b ∼= Coker(cid:0)Ci−1⊗b F −d−i−−1−⊗b−A−1→F Zi(bC⊗(cid:1)F) = Hi(C⊗F).
A A A
Itfollowsfromtheconstructio(cid:0)nthat, foreachc ∈ Zi andx ∈ F(cid:1),theresultingisomorphism
b b b
(10) indeed takes (c+Imdi−1)⊗ x to c⊗ x+Im(di−1 ⊗ 1 ), as required. (cid:3)
A A A F
In what follows, we will need some topological modules which are not Fr´echet modules.
Let A be a Fr´echet algebra. A left A-⊗-module is a left A-module M endowed with a
complete locally convex topologyinsuch a way that the actionA×M → M is continuous.
Right A-⊗-modules and A-⊗-bimodulesbare defined similarly. If M is a right A-⊗-module
and N is a left A-⊗-module, then their A-module tensor product M ⊗ N is defined
A
similarlybto the case of Frb´echet modules (see Section 2). The only exceptiobn is that
the quotient (M ⊗Nb)/L need not be complete in the nonmetrizable casbe, so M ⊗AN is
defined to be the completion of (M ⊗N)/L.
Recallnowacobnstructionfrom[32]. LetAbeaFr´echet algebra, andletM,N ∈bA-mod.
Fix a projective resolution P → Mb→ 0 of M in A-mod. Each hA(Pi,A) has a natural
structure of a right A-⊗-module given by (ϕ·a)(p) = ϕ(p)a for a ∈ A, p ∈ P. We have a
map of complexes
h (P,bA)⊗N → h (P,N), ϕ⊗ n 7→ (p 7→ ϕ(p)·n). (13)
A A A
A
b
10 A.YU. PIRKOVSKII
If, for some n, the spaces Extn(M,A) and Extn(M,N) are Hausdorff and complete, then
A A
Extn(M,A) becomes a right A-⊗-module in a natural way, and we have continuous linear
A
maps
Extn(M,A)⊗Nb → Hn(h (P,A)⊗N) → Extn(M,N). (14)
A A A
A A
Here, for each ξ ∈ Extn(M,A) represented by an n-cocycle ϕ ∈ h (P ,A), the first arrow
A b b A n
in (14) takes ξ ⊗ n to the cohomology class of ϕ ⊗ n. The second arrow in (14) is
A A
induced by (13). The composite map
Extn(M,A)⊗N → Extn(M,N) (15)
A A
A
does not depend on the choice of the projective resolution P.
b
Lemma 5.2. Let A be a Fr´echet algebra, and let P be a finitely generated, strictly pro-
jective left Fr´echet A-module. Then for each N ∈ A-mod h (P,N) is a Fr´echet space.
A
Moreover, if N is nuclear, then so is h (P,N).
A
Proof. If P = A, then h (A,N) is topologically isomorphic to N via the map ϕ 7→ ϕ(1);
A
the continuity of the inverse map n 7→ (a 7→ a · n) is easily checked. The general case
(cid:3)
follows by additivity and functoriality.
Following [35], we say that a left Fr´echet A-module M is of strictly finite type if M
has a resolution P → M → 0 consisting of finitely generated, strictly projective Fr´echet
A-modules. The Fr´echet algebra A is said to be of finite type if A is of strictly finite type
in Ae-mod. For example, if X is a Stein manifold, Y ⊂ X is a closed submanifold, and
both X and Y are of Liouville type, then O(Y) is of strictly finite type over O(X) (see
Corollary 4.2). Identifying O(X)e with O(X×X), we see, in particular, that the algebra
O(X) is of finite type.
Corollary 5.3. Let A be a Fr´echet algebra, and let M ∈ A-mod be of strictly finite type.
If Extn(M,N) is Hausdorff for some n, then Extn(M,N) is a Fr´echet space.
A A
Lemma 5.4. Let A be a nuclear Fr´echet algebra, and let M ∈ A-mod be of strictly
finite type. Suppose that Extn(M,A) is Hausdorff for all n ∈ Z . Then for each flat
A +
N ∈ A-mod and each n ∈ Z Extn(M,N) is Hausdorff as well, and the canonical map
+ A
(15) is a topological isomorphism.
Proof. Fix a resolution P → M → 0 consisting of finitely generated, strictly projective
Fr´echet A-modules. By Lemma 5.2, the cochain complex C = h (P,A) satisfies the
A
conditions of Lemma 5.1. Therefore the first arrow in (14) is a topological isomorphism.
Since each P is finitely generated and strictly projective, it follows that (13) is also a
i
topological isomorphism (cf. the proof of Lemma 5.2). Hence the second arrow in (14) is
(cid:3)
a topological isomorphism as well. This completes the proof.
For the reader’s convenience, we recall two results from [32].
Proposition 5.5 ( [32, Prop. 5.2.1]). Let M be a left Fr´echet module of finite projective
homological dimension over a Fr´echet algebra A. Suppose that there exists n ∈ N such
that for each projective module P ∈ A-mod the following conditions hold:
(i) Exti (M,P) = 0 unless i = n;
A
(ii) Extn(M,P) is Hausdorff and complete;
A