arXiv:math/0609575v3 [math.QA] 8 Dec 2006
DEFORMATIONS OF AZUMAYA ALGEBRAS
PAUL BRESSLER, ALEXANDER GOROKHOVSKY, RYSZARD NEST,
AND BORIS TSYGAN
1. Introduction.
In this paper we compute the deformation theory of a special class of
algebras, namely of Azumaya algebras on a manifold (C ∞ or complex
analytic).
Deformation theory of associative algebras was initiated by Gerstenhaber in [G]. A deformation of an associative algebra A over an Artinian ring a is an a-linear associative algebra structure on A ⊗ a such
that, for the maximal ideal m of a, A ⊗ m is an ideal, and the quotient algebra on A is the original one. Gerstenhaber showed that the
Hochschild cochain complex of an associative algebra A has a structure
of a differential graded Lie algebra (DGLA), and that deformations of
A over an Artinian ring a are classified by Maurer-Cartan elements of
the DGLA C • (A, A)[1] ⊗ m. A Maurer-Catan element of a DGLA L•
with the differential δ is by definition an element λ of L1 satisfying
1
(1.0.1)
δλ + [λ, λ] = 0
2
Isomorphic deformations correspond to equivalent Maurer-Cartan elements, and vice versa.
In subsequent works [GM], [SS], [Dr] it was shown that deformation
theories of many other objects are governed by appropriate DGLAs in
the same sense as above. Moreover, if two DGLAs are quasi-isomorphic,
then there is a bijection between the corresponding sets of equivalence
classes of Maurer-Cartan elements. Therefore, to prove that deformation theories of two associative algebras are isomorphic, it is enough
to construct a chain of quasi-isomorphisms of DGLAs whose endpoints
are the Hochschild complexes of respective algebras.
This is exactly what is done in this paper. We construct a canonical isomorphism of deformation theories of two algebras: one is an
Azumaya algebra on a C ∞ manifold X, the other the algebra of C ∞
functions on X. The systematic study of the deformation theory of the
A. Gorokhovsky was partially supported by NSF grant DMS-0400342. B. Tsygan
was partially supported by NSF grant DMS-0605030 .
1
2
P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN
latter algebra was initiated in [BFFLS]. In addition to the definition
above, it is required that the multiplication on A ⊗ a be given by bidifferential expressions. The corresponding Hochschild cochain complex
consists of multi-differential maps C ∞ (X)⊗n → C ∞ (X). The complete
classification of deformations of C ∞ (X) is known from the formality
theorem of Kontsevich [K1]. This theorem asserts that there is a chain
of quasi-isomorphisms connecting the two DGLAs C • (C ∞ (X), C ∞ (X))[1]
and Γ(X, ∧• (T X))[1], the latter being the DGLA of multivector fields
with the Schouten-Nijenhuis bracket. For the proofs in the case of a
general manifold, cf. also [Do], [Ha], [DTT].
An Azumaya algebra on a manifold X is a sheaf of algebras locally
isomorphic to the algebra of n×n matrices over the algebra of functions.
Such an algebra determines a second cohomology class with values in
×
OX
(here, as everywhere in this paper, in the C ∞ case OX denotes
the sheaf of smooth functions). This cohomology class c is necessarily
×
n-torsion, i.e. nc = 0 in H 2 (X, OX
). The main result of this paper
is that there is a chain of quasi-isomorphisms between the Hochschild
complexes of multidifferential cochains of the algebra of functions and
of an arbitrary Azumaya algebra. This chain of quasi-isomorphisms
does depend on some choices but is essentially canonical. More precisely, what we construct is a canonical isomorphism in the derived
category [H] of the closed model category [Q] of DGLAs.
Note that in [BGNT] we considered a related problem of deformation
×
theory. Namely, any cohomology class in H 2 (X, OX
), whether torsion
or not, determines an isomorphism class of a gerbe on X, cf. [Br]. A
gerbe is a partial case of an algebroid stack; it is a sheaf of categories
on X satisfying certain properties. In [BGNT] we showed that deformation theory of algebroid stacks is governed by a certain DGLA. If an
algebroid stack is a gerbe, we constructed a chain of quasi-isomorphisms
between this DGLA and another one, much more closely related to the
Hochschild complex. When the gerbe is an Azumaya algebra, this latter DGLA is just the Hochschild complex itself. In ther words, we
solved a different deformation problem for an Azumaya algebra, and
got the same answer.
Now let us turn to the case of a complex analytic manifold X. In
this case, let OX stand for the sheaf of algebras of holomorphic functions. It is natural to talk about deformations of this sheaf of algebras.
Multidifferential multiholomorphic Hochschild cochains form a sheaf,
and it can be shown that the corresponding deformation theory is governed by the DGLA of Dolbeault forms Ω0,• (X, C •>0 (OX , OX )[1]). The
full DGLA Ω0,• (X, C • (OX , OX )[1]) governs deformations of OX as an
DEFORMATIONS OF AZUMAYA ALGEBRAS
3
algebroid stack. We show that our construction of a chain of quasiisomorphisms can be carried out for this full DGLA in the holomorphic
case, or in a more general case of a real manifold with a complex integrable distribution. This construction does not seem to work for the
DGLA governing deformations of OX as a sheaf of algebras. As shown
in [NT] and [BGNT], such deformation theory can be in general more
complicated.
Our motivations for studying deformations of gerbes and Azumaya
algebras are the following:
1) A fractional index theorem from [MMS]. The algebra of pseudodifferential operators which is used there is closely related to formal
deformations of Azumaya algebras.
2) Index theory of Fourier integral operators (FIOs). Guillemin and
Sternberg [GS] have studied FIOs associated to a coisotropic submanifold of a cotangent bundle. It appears that higher index theorems for
such operators are related to algebraic index theorems [BNT] for deformations of the trivial gerbe on a symplectic manifold with an étale
groupoid. A similar algebraic index theorem in the holomorphic case
should help establish a Riemann-Roch theorem in the setting of [KS],
[PS].
3) Dualities between gerbes and noncommutative spaces ([MR1],
[MR2], [MR3], [Bl], [BBP], [Ka]).
Hochschild and cyclic homology of Azumaya algebras were computed
in [CW, S] and in the more general case of continuous trace algebras
(in the cohomological setting) in [MS]. Here we require, however, much
more precise statement which involves the whole Hochschild complex
as a differential graded Lie algebra, rather then just its cohomology
groups. It is conjectured in [S] that algebras which are similar in the
sense of [S] have the same deformation theories. Some of our results
can be considered as a verification of this conjecture in the particular
case of Azumaya algebras.
A very recent preprint [Do1] contains results which have a significant
overlap with ours. Namely, a chain of quasi-isomorphisms similar to
ours is established in a partial case when the Azumaya algebra is an
algebra of endomorphisms of a vector bundle. On the other hand, a
broader statement is proven, namely that the above chain of quasiisomorphisms extends to Hochschild chain complexes viewed as DGL
modules over DGLAs of Hochschild cochains. This work, like ours, is
motivated by problems of index theory.
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P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN
2. Azumaya algebras.
Let X be a smooth manifold. In what follows we will denote by OX
the sheaf of complex-valued C ∞ functions on X.
Definition 1. An Azumaya algebra on X is a sheaf of central OX algebras locally isomorphic to Matn (OX ).
Thus, by definition, the unit map OX ֒→ A takes values in the center
of A.
Let A0 := [A, A] denote the OX -submodule generated by the image
of the commutator map.
We will now consider A, A0 and OX as Lie algebras under the commutator bracket. Note that the bracket on A takes values in A0 and
the latter is a Lie ideal in A.
Lemma 2. The composition OX ֒→ A → A/A0 is an isomorphism.
Proof. The issue is local so we may assume that A = Matn (OX ) in
which case it is well known to be true.
Corollary 3. The map OX ⊕ A0 → A induced by the unit map and
the inclusion is an isomorphism of Lie algebras.
Lemma 4. The sequence
ad
0 → OX → A → DerOX (A) → 0
ad
is exact. Moreover, the composition A0 ֒→ A → DerOX (A) is an
isomorphism of Lie algebras.
Let C(A) denote the sheaf of (locally defined) connections on A with
respect to which the multiplication on A is horizontal; equivalently,
such a connection ∇ satisfies the Leibniz rule ∇(ab) = ∇(a)b + a∇(b)
in Ω1X ⊗OX A for all (locally defined) a, b ∈ A. The sheaf C(A) is a
torsor under Ω1X ⊗OX DerOX (A).
Any connection ∇ ∈ C(A) satisfies ∇(A0 ) ⊂ Ω1X ⊗ A0 .
For ∇ ∈ C(A) there exists a unique θ = θ(∇) ∈ Ω2X ⊗ A0 such that
ad(θ) = ∇2 ∈ Ω2X ⊗ DerOX (A).
3. Jets
Let JX be the sheaf of infinite jets of smooth functions on X. Let
p : JX → OX denote the canonical projection. Suppose now that A
is an Azumaya algebra. Let J (A) denote the sheaf of infinite jets of
A. Let pA : J (A) → A denote the canonical projection. The sheaves
J (A) and A ⊗OX JX have canonical structures of sheaves of central
JX -algebras locally isomorphic to Matn (JX ).
DEFORMATIONS OF AZUMAYA ALGEBRAS
5
Let Isom0 (A ⊗ JX , J (A)) denote the sheaf of (locally defined) JX algebra isomorphisms A⊗JX → J (A) such that the following diagram
is commutative:
A ⊗ JX
/
J (A)
pA
Id⊗p
A
Id
/
A
Similarly denote by Aut0 (A ⊗ JX ) the sheaf of (locally defined) JX algebra automorphisms of A ⊗ JX such that the following diagram is
commutative:
A ⊗ JX
/
A ⊗ JX
Id⊗p
Id⊗p
A
Id
/
A
Lemma 5. The sheaf Isom0 (A ⊗JX , J (A)) is a torsor under the sheaf
of groups Aut0 (A ⊗ JX ).
Proof. Since both J (A) and A⊗OX JX are locally isomorphic to Matn (JX ),
the sheaf Isom0 (A⊗JX , J (A)) is locally non-empty, hence a torsor.
Lemma 6. The sheaf of groups Aut0 (A ⊗ JX ) is soft.
Proof. Let DerJX ,0 (A ⊗ JX ) denote the sheaf of JX -linear derivations
of the algebra A ⊗ JX which reduce to the zero map modulo J0 . The
exponential map
(3.0.2)
exp : DerJX ,0 (A ⊗ JX ) → Aut0 (A ⊗ JX ) ,
δ 7→ exp(δ), is an isomorphism of sheaves (the inverse map is given by
P
(a−1)n
a 7→ log a = ∞
). Therefore, it suffices to show that the sheaf
n=1
n
DerJX ,0 (A ⊗ JX ) is soft, but this is clear since it is a module over the
sheaf OX of C ∞ -functions.
Corollary 7. The torsor Isom0 (A⊗JX , J (A)) is trivial, i.e. Isom0 (A⊗
JX ), J (A)) := Γ(X; Isom0 (A ⊗ JX , J (A))) 6= ∅.
Proof. Since the sheaf of groups Aut0 (A⊗JX ) is soft we have H 1 (X, Aut0 (A⊗
JX ) = 1 ([DD], Lemme 22, cf. also [Br], Proposition 4.1.7). Therefore
every Aut0 (A ⊗ JX ) torsor is trivial.
In what follows we will use ∇can
to denote the canonical flat connecE
tion on J (E). A choice of σ ∈ Isom0 (A ⊗ JX , J (A)) induces the flat
connection σ −1 ◦ ∇can
A ◦ σ on A ⊗ JX .
A choice of ∇ ∈ C(A)(X) give rise to the connection ∇ ⊗ Id + Id ⊗
∇can
O on A ⊗ JX .
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P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN
Lemma 8.
(1) For any σ ∈ Isom0 (A ⊗ JX , J (A)), ∇ ∈ C(A)(X),
the difference
(3.0.3)
can
σ −1 ◦ ∇can
A ◦ σ − (∇ ⊗ Id + Id ⊗ ∇O )
is JX -linear.
(2) There exists a unique F ∈ Γ(X; Ω1X ⊗ A0 ⊗ JX ) (depending on
σ and ∇) such that (3.0.3) is equal to ad(F ).
(3) Moreover, F satisfies
(3.0.4)
1
(∇ ⊗ Id + Id ⊗ ∇can
O )F + [F, F ] + θ = 0
2
Proof. We leave the verification of the first claim to the reader. If
follows that (3.0.3) is a global section of Ω1X ⊗ DerJX (A ⊗ JX ). Since
ad ⊗Id
the map A0 ⊗ JX −→ DerJX (A ⊗ JX ) is an isomorphism the second
claim follows.
We have
(3.0.5)
σ −1 ◦ ∇can ◦ σ = ∇ ⊗ Id + Id ⊗ ∇can
O + ad F
where F ∈ Γ(X; Ω1X ⊗ A ⊗ JX ).
Since (σ −1 ◦ ∇can ◦ σ)2 = 0 the element (∇A ⊗ Id + Id ⊗ ∇can )F +
1
[F, F ] + θ must be central; it also must lie in A0 ⊗ J . Therefore, it
2
vanishes, i.e. the formula (3.0.4) holds.
4. Deformations of Azumaya algebras.
4.1. Review of formal deformation theory. Consider a DGLA L•
with the differential δ. A Maurer-Catan element of L• is by definition
an element λ of L1 satisfying
(4.1.1)
1
δλ + [λ, λ] = 0
2
Now assume that L0 is nilpotent. Then exp(L0 ) is an algebraic group
over the ring of scalars k. This group acts on the set of Maurer-Cartan
elements via
∞
X
1
(4.1.2)
eX (λ) = AdeX (λ) −
adnX (δX)
(n
+
1)!
n=0
Informally,
δ + eX (λ) = AdeX (δ + λ).
We call two Maurer-Cartan elements equivalent if they are in the same
orbit of the above action.
DEFORMATIONS OF AZUMAYA ALGEBRAS
7
Theorem 9. ([GM]) Let a be an Artinian algebra with the maximal
ideal m. A quasi-isomorphism of DGLAs L•1 → L•2 induces a bijection
between the sets of equivalence classes of Maurer-Cartan elements of
L•1 ⊗ m and of L•2 ⊗ m.
Given a DGLA L• and an Artinian algebra a with the maximal ideal
m, denote by MC(L• ) the set of equivalence classes of Maurer-Cartan
elements of L• ⊗ m.
4.2. Let A be an Azymaya algebra on X. Let Def(A) denote the
formal deformation theory of A as a sheaf of associative C-algebras,
i.e. the groupoid-valued functor of (commutative) Artin C-algebras
which associates to an Artin algebra a the groupoid whose objects
e φ) consisting of a flat OX ⊗C a-algebra Ae and a map
are pairs (A,
e a C → A. The morphism
φ : Ae → A which induces an isomorphism A⊗
in Def(A)(a) are morphisms of OX ⊗C a-algebras which commute with
the respective structure maps to A.
Theorem 10 is the main result of this note.
Theorem 10. Suppose that A is an Azumaya algebra on X. There
∼
exists a canonical equivalence Def(A) = Def(OX ).
The proof of Theorem 10 will be given in 4.8. The main technical
ingredient in the proof is Theorem 20.
4.3. Hochschild cochains. Recall that the sheaf C n (A) of Hochschild
cochains of degree n is defined by
C n (A) := HomC (A⊗C n , A) .
In the case of Azumaya algebras, we will always consider the subcomplex of local sections of C n (A), i.e. of multidifferential operators. The
link from the deformation theory to the Hochschild theory is provided
by the following
Proposition 11. [G] For every algebra A, there exists a canonical
∼
equivalence Def(A) = MC(C • (A)[1])).
In particular, for an Azumaya algebra A there exists a canonical
∼
equivalence Def(A) = MC(Γ(X; C •(A)[1])).
Proof. By definition, an element of degree one in the DGLA C • (A)[1]⊗
m is a map λ : A ⊗ A → A ⊗ m. Put a ∗ b = ab + λ(a, b). Extend
∗ a binary a-linear operation on A ⊗ a. Modulo m, this operation is
the multiplication in A. Its associativity is equivalent to the MaurerCartan equation (1.0.1). Two Maurer-Cartan elements are equivalent
8
P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN
if and only if there is a map X : A → A ⊗ m such that, if one extends it
to an a-linear map X : A ⊗ m → A ⊗ a, its exponential expad(X) is an
isomorphism of the corresponding two associative algebra structures on
A ⊗ a. On the other hand, any such isomorphism of algebra structures
(which is identical modulo m is of the form expad(X) for some X.
Let
⊗OX n
, J (A))
C n (J (A)) := Homcont
OX (J (A)
There exists a canonical map j ∞ : C n (A) → C n (J (A)) which to a
multidifferential operator associates its linearization. The canonical
flat connection on J (A) induces a flat connection, denoted ∇can , on
C n (J (A)). The de Rham complex DR(C n (J (A))) satisfies H i (DR(C n (J (A)))) =
∼
0 for i 6= 0 while the map j ∞ induces an isomorphism C n (A) =
H 0 (DR(C n (J (A)))).
The Hochschild differential, denoted δ and the Gerstenhaber bracket
endow C • (A)[1] (respectively, C • (J (A))[1]) with a structure of a DGLA.
The connection ∇can acts by derivations of the Gerstenhaber bracket
on C • (J (A))[1]. Since it acts by derivations on J (A) the induced
connection on C • (J (A))[1] commutes with the Hochschild differential.
Hence, the graded Lie algebra Ω•X ⊗ C • (J (A))[1] equipped with the
differential ∇can + δ is DGLA.
Proposition 12. The map
(4.3.1)
j ∞ : C • (A)[1] → Ω• ⊗ C • (J (A))[1]
is a quasi-isomorphism of DGLA.
Proof. It is clear that the map (4.3.1) is a morphism of DGLA.
Let Fi C • (?) = C ≥−i (?). Then, F• C • (?) is a filtered complex and the
differential induced on Gr•F C • (?) is trivial. Consider Ω•X as equipped
with the trivial filtration. Then, the map (4.3.1) is a morphism of
filtered complexes with respect to the induced filtrations on the source
and the target. The induced map of the associated graded objects a
quasi-isomorphism, hence, so is (4.3.1).
Corollary 13. The map
MC(Γ(X; C • (A)[1])) → MC(Γ(X; Ω•X ⊗ C • (J (A))[1]))
induced by (4.3.1) is an equivalence.
n
4.4. The cotrace map. Let C (JX ) denote the sheaf of normalized
Hochschild cochains. It is a subsheaf of C n (JX ) whose stalks are given
DEFORMATIONS OF AZUMAYA ALGEBRAS
9
by
n
C (JX )x = HomOX,x ((JX,x /OX,x · 1)⊗n , JX,x ) ⊂
⊗n
Hom OX,x (JX,x
, JX,x ) = C n (JX )x
•
•
The sheaf C (A)[1] (respectively, C (JX )[1] is actually a sub-DGLA of
C • (A)[1] (respectively, C • (JX )[1] ) and the inclusion map is a quasiisomorphism.
•
The flat connection ∇can preserves C (JX )[1], and the (restriction
•
to C (OX )[1] of the) map j ∞ is a quasi-isomorphism of DGLA.
Consider now the map
•
cotr : C (JX )[1] → C • (A ⊗ JX )[1]
(4.4.1)
defined as follows:
(4.4.2)
cotr(D)(a1 ⊗ j1 , . . . , an ⊗ jn ) = a0 . . . an D(j1 , . . . , jn ).
Proposition 14. The map cotr is a quasiisomorphism of DGLAs.
Proof. It is easy to see that cotr is a morphism of DGLAs. Since the
fact that this is a quasiisomorphism is local it is enough to verify it
when A = Matn (OX ). In this case it is a well-known fact (cf. [Lo],
section 1.5.6).
4.5. Comparison of deformation complexes. Let σ ∈ Isom0 (A ⊗
JX , J (A)), ∇ ∈ C(A). The isomorphisms of algebras σ induces the
isomorphism of DGLA
σ∗ : C • (A ⊗ JX )[1] → C • (J (A))[1]
which is horizontal with respect to the flat connection (induced by)
∇can on C • (J (A))[1] and the induced flat connection given by (3.0.5).
Therefore, it induces the isomorphism of DGLA (the respective de
Rham complexes)
(4.5.1)
σ∗ : Ω•X ⊗ C • (A ⊗ JX )[1] → Ω•X ⊗ C • (J (A))[1].
Here, the differential in Ω•X ⊗ C • (A ⊗ JX )[1] is ∇ ⊗ Id + Id ⊗ ∇can +
ad F + δ and the differential in Ω•X ⊗ C • (J (A))[1] is ∇can + δ.
Let ιG denote the adjoint action of G ∈ Γ(X; ΩkX ⊗ C 0 (A ⊗ JX ))
(recall that C 0 (A ⊗ JX ) = A ⊗ JX ). Thus, ιG is a map ΩpX ⊗ C q (A ⊗
q−1
JX ) → Ωp+k
(A ⊗ JX ).
X ⊗C
10
P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN
Lemma 15. For any H, G ∈ Γ(X; Ω•X ⊗ C 0 (A ⊗ JX )) we have:
(4.5.2)
[δ, ιH ] = ad H
(4.5.3)
[ad H, ιG ] = ι[H,G]
(4.5.4)
[ιH , ιG ] = 0
[∇ ⊗ Id + Id ⊗ ∇can
O , ιH ] = ι(∇⊗Id+Id⊗∇can
O )H
(4.5.5)
Proof. Direct calculation.
P∞ 1 k
2
Let exp(tιF ) = k=0 n! t (ιF )◦k = Id + tιF + t2 ιF ◦ ιF + · · · . Note
that this is a polynomial in t since ΩpX = 0 for p > dim X. Since ιF is
a derivation, the operation exp(ιF ) is an automorphism of the graded
Lie algebra Ω•X ⊗ C • (A ⊗ JX )[1]. The automorphism exp(ιF ) does not
commute with the differential.
Lemma 16.
exp(ιF ) ◦ (∇ ⊗ Id + Id ⊗ ∇can
O + δ + ad F ) ◦ exp(−ιF ) =
∇ ⊗ Id + Id ⊗ ∇can
O + δ + ιθ .
Proof. Consider the following polynomial in t: p(t) = exp(tιF ) ◦ δ ◦
exp(−tιF ). Then using the identities from the Lemma 15 we obtain
p′ (t) = exp(tιF ) ◦ [ιF , δ] ◦ exp(−tιF ) = − exp(tιF ) ◦ ad F ◦ exp(−tιF ),
p′′ (t) = − exp(tιF )◦[ιF , ad F ]◦exp(−tιF ) = exp(tιF )◦ι[F,F ] ◦exp(−tιF ),
2
and p(n) = 0 for n ≥ 3. Therefore p(t) = δ − t ad F + t2 ι[F,F ]. Setting
t = 1 we obtain
1
exp(ιF ) ◦ δ ◦ exp(−ιF ) = δ − ad F + ι[F,F ] .
2
Similarly we obtain
exp(ιF ) ◦ ad F ◦ exp(−ιF ) = ad F − ι[F,F ]
and
exp(ιF ) ◦ (∇ ⊗ Id + Id ⊗ ∇can
O ) ◦ exp(−ιF ) =
(∇ ⊗ Id + Id ⊗ ∇can
O ) − ι(∇⊗Id+Id⊗∇can
O )F
Adding these formulas up and using the identity (3.0.4) we obtain the
desired result.
Lemma 17. The map
(4.5.6)
•
Id ⊗ cotr : Ω•X ⊗ C (JX )[1] → Ω•X ⊗ C • (A ⊗ JX )[1] .
is a quasiisomorphism of DGLA, where the source (respectively, the
target) is equipped with the differential ∇can
O + δ (respectively, ∇ ⊗ Id +
Id ⊗ ∇can + δ + ιθ ).
DEFORMATIONS OF AZUMAYA ALGEBRAS
11
Proof. It is easy to see that Id ⊗ cotr is a morphism of graded Lie
algebras, which satisfies (∇ ⊗ Id + Id ⊗ ∇can
O ) ◦ (Id ⊗ cotr) = (Id ⊗
can
cotr) ◦ ∇O and δ ◦ (Id ⊗ cotr) = (Id ⊗ cotr) ◦ δ. Since the domain of
(Id ⊗ cotr) is the normalized complex, we also have ιθ ◦ (Id ⊗ cotr) = 0.
This implies that (Id ⊗ cotr) is a morphism of DGLA.
Introduce filtration on Ω•X by Fi Ω•X = Ω≥−i
(the “stupid” filtration)
X
•
and consider the complexes C (JX )[1] and C • (A ⊗ JX )[1] equipped
with the trivial filtration. The map (4.5.6) is a morphism of filtered
complexes with respect to the induced filtrations on the source and the
target. The differentials induced on the associated graded complexes
are δ (or, more precisely, Id ⊗δ) and the induced map of the associated
graded objects is Id ⊗ cotr which is a quasi-isomorphism in virtue of
Proposition 14. Therefore, the map (4.5.6) is a quasiisomorphism as
claimed.
Proposition 18. For a any choice of σ ∈ Isom0 (A ⊗ JX , J (A)), ∇ ∈
C(A), the composition Φσ,∇ := σ∗ ◦ exp(ιFσ,∇ ) ◦ (Id ⊗ cotr) (where F is
as in Lemma 8),
(4.5.7)
•
Φσ,∇ : Ω•X ⊗ C (JX )[1] → Ω•X ⊗ C • (J (A))[1]
is a quasi-isomorphism of DGLA.
Proof. This is a direct consequence of the Lemmata 16, 4.5.6.
4.6. Independence of choices. According to Proposition 18, for any
choice of σ ∈ Isom0 (A ⊗ JX , J (A)), ∇ ∈ C(A) we have a quasiisomorphism of DGLA Φσ,∇ .
Proposition 19. The image of Φσ,∇ in the derived category is independent of the choices made.
Proof. For i = 0, 1 suppose given σi ∈ Isom0 (A ⊗ JX , J (A)), ∇i ∈
C(A). Let Φi = Φσi ,∇i . The goal is to show that Φ0 = Φ1 in the
derived category.
There is a unique θi ∈ Γ(X; Ω2X ⊗ A0 ⊗ JX )) such that ∇2i = ad(θi ).
By Lemma 8 there exist unique Fi ∈ Γ(X; Ω1X ⊗ A0 ⊗ JX ) such that
can
ad(Fi ) = σi−1 ◦ ∇can
A ◦ σi − (∇i ⊗ Id + Id ⊗ ∇O )
Let I := [0, 1]. Let ǫi : X → I × X denote the map x 7→ (i, x). Let
pr : I × X → X denote the projection on the second factor.
It follows from Lemma 5, the isomorphism (3.0.2) and the isomorphism A0 ⊗ JX → DerJX,0 (A ⊗ JX ) that there exists a unique f ∈
Γ(X; A0 ⊗ JX ) such that σ1 = σ0 ◦ exp(ad(f )). For t ∈ I let σt = σ0 ◦
exp(ad(tf )). Let σ
e denote the isomorphism pr∗ (A ⊗ JX ) → pr∗ J (A)
which restricts to σ(t) on {t} × X. In particular, σi = ǫ∗i (e
σ ).
12
P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN
e denote the connection on
For t ∈ I let ∇t = t∇0 + (1 − t)∇1 . Let ∇
∗
e
pr A which restricts to ∇t on {t} × X. In particular, ∇i = ǫ∗i (∇).
Suppose that
(1) σ
e : pr∗ (A ⊗ JX ) → pr∗ J (A) is an isomorphism of pr∗ JX algebras which reduces to the identity map on pr∗ A modulo J0
σ );
and satisfies σi = ǫ∗i (e
∗
e
e
(2) ∇ ∈ C(pr A) satisfies ∇i = ǫ∗i (∇).
(Examples of such are constructed above.)
Then, there exists a unique Fe ∈ Γ(I × X; Ω1I×X ⊗ pr∗ (A0 ⊗ JX ))
such that
e ⊗ Id + Id ⊗ pr∗ (∇can
e
σ
e−1 ◦ pr∗ (∇can
e=∇
A )◦σ
O ) + ad(F )
It follows from the uniqueness that ǫ∗ (Fe) = Fi .
There exists a unique θe ∈ Γ(I × X; Ω2I×X ⊗ pr∗ (A0 ⊗ JX )) such that
e It follows from the uniqueness that ǫ∗ (θ)
e = θi .
e 2 = ad(θ).
∇
i
e := σ
The composition Φ
e∗ ◦ exp(ιFe ) ◦ (Id ⊗ cotr),
(4.6.1)
e : Ω• ⊗ pr∗ C • (JX ))[1] → Ω• ⊗ pr∗ C • (J (A))[1]
Φ
I×X
I×X
is a quasi-isomorphism of DGLA, where the differential on source (re∗ e can
spectively, target) is pr∗ (∇can
O ) + δ (respecively, pr (∇A ) + δ + ιθe).
The map (4.6.1) induces the map of direct images under the projection pr
(4.6.2)
e : pr Ω• ⊗ C • (JX )[1] → pr Ω• ⊗ C • (J (A))[1]
Φ
∗ I×X
∗ I×X
which is a quasi-isomorphism (since all higher direct images vanish).
The pull-back of differential forms pr∗ : Ω• → pr∗ Ω•I×X is a quasiisomorphism of commutative DGA inducing the quasi-isomorphism of
•
•
DGLA pr∗ ⊗ Id : Ω•X ⊗ C (JX )[1] → pr∗ Ω•I×X ⊗ C (JX )[1] (with
∗
can
differentials ∇can
O + δ and pr (∇O ) + δ respectively).
The diagram of quasi-isomorphisms of DGLA
(4.6.3)
Ω•X ⊗ C • (J (A))[1]
fff3
fffff
f
f
f
f
f
fffff
fffff Φ◦(pr
∗ ⊗Id)
e
/ pr Ω•
⊗ C • (JX )[1]
∗ I×X
XXXXX
XXXXX
XXXXX
XXXXX
Φ1
XXX+
O
Φ0
Ω•X
ǫ∗0 ⊗Id
⊗ C • (J (A))[1]
ǫ∗1 ⊗Id
Ω•X ⊗ C • (J (A))[1]
DEFORMATIONS OF AZUMAYA ALGEBRAS
13
is commutative. Since ǫ∗i ◦ pr∗ = Id for i = 0, 1 and pr∗ ⊗ Id is a
quasiisomorphism, ǫ∗0 ⊗ Id and ǫ∗1 ⊗ Id represent the same morphism in
the derived category. Hence so do Φ0 and Φ1 .
4.7. The main technical ingredient. To each pair (σ, ∇) with σ ∈
Isom0 (A⊗JX , J (A)) and ∇ ∈ C(A) we associated the quasi-isomorphism
of DGLA (4.5.7) (Proposition 18). According to Proposition 19 all of
these give rise to the same isomorphism in the derived category. We
summarize these findings in the following theorem.
Theorem 20. Suppose that A is an Azumaya algebra on X. There
exists a canonical isomorphism in the derived category of DGLA Ω•X ⊗
•
∼
C (JX )[1] = Ω•X ⊗ C • (J (A))[1].
4.8. The proof of Theorem 10. The requisite equivalence is the
composition of the equivalences
∼
•
Def(OX ) = MC(Γ(X; C (OX )[1]))
∼
•
= MC(Γ(X; Ω•X ⊗ C (JX )[1]))
∼
= MC(Γ(X; Ω•X ⊗ C • (J (A))[1]))
∼
= MC(Γ(X; C •(A)[1]))
∼
= Def(A)
The first and the last equivalences are those of Proposition 11, the
second and the fourth are those of Corollary 13, and the third one
is induced by the canonical isomorphism in the derived category of
Theorem 10.
5. Holomorphic case
5.1. Complex distributions. Let TX denote the sheaf of real valued
vector fields on X and let TXC := TX ⊗R C.
Definition 21. A (complex) distribution on X is a sub-bundle of TXC 1
For a distribution P on X we denote by P ⊥ ⊆ Ω1X the annihilator
of P (with respect to the canonical duality pairing).
Definition 22. A distribution P of rank r on X is called integrable if,
locally on X, there exist functions f1 , . . . , fr ∈ OX such that df1 , . . . , dfr
form a local frame for P ⊥ .
Lemma 23. An integrable distribution is involutive, i.e. it is a Lie
subalgebra of TXC (with respect to the Lie bracket of vector fields).
1A
sub-bundle is an OX -submodule which is a direct summand locally on X
14
P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN
5.2. Differential calculus on X/P. In this subsection we briefly review relevant definitions and results of the differential calculus in the
presence of integrable complex distribution. We refer the reader to
[Ko], [R] and [FW] for details and proofs. Suppose that P is an integrable distribution. Let F• Ω•X denote the filtration by the powers of the
V
j−i
differential ideal generated by P ⊥ , i.e. F−i ΩjX = i P ⊥ ∧ ΩX
⊆ ΩjX .
Let ∂ denote the differential in Gr F Ω•X . The wedge product of differential forms induces a structure of a commutative DGA on (Gr F Ω•X , ∂).
F
F
Lemma 24. The complex Gr−i
Ω•X satisfies H j (Gr−i
Ω•X , ∂) = 0 for
j 6= i.
F
Let ΩiX/P := H i (Gr−i
Ω•X , ∂), OX/P := Ω0X/P . We have OX/P :=
V
Ω0X/P ⊂ OX , Ω1X/P ⊂ P ⊥ ⊂ Ω1X and, more generally, ΩiX/P ⊂ i P ⊥ ⊂
ΩiX . The wedge product of differential forms induces a structure of a
graded-commutative algebra on Ω•X/P := ⊕i ΩiX/P [−i] = H • (Gr F Ω•X , ∂).
P
If f1 , . . . , fr are as in 22, then Ω1X/P = ri=1 OX/P · dfi , in particular, Ω1X/P is a locally free OX/P -module. Moreover, the multiplication
V
induces an isomorphism iOX/P Ω1X/P → ΩiX/P .
The de Rham differential d restricts to the map d : ΩiX/P → Ωi+1
X/P
and the complex Ω•X/P := (ΩiX/P , d) is a commutative DGA. Moreover,
the inclusion Ω•X/P → Ω•X is a quasi-isomorphism.
Example 25. Suppose that P = P. Then, P = D ⊗R C, where D
a subbundle of TX . Then, D is an integrable real distribution which
defines a foliation on X and Ω•X/P is the complex of basic forms.
Example 26. Suppose that P̄ ∩ P = 0 and P̄ ⊕ P = TXC . In this case
again P is integrability is equivalent to involutivity, by NewlanderNirenberg theorem. Ω•X/P in this case is a holomorphic de Rham complex.
5.3. ∂-operators. Suppose that E is a vector bundle on X, i.e. a
locally free OX -module of finite rank. A connection along P on E
is, by definition, a map ∇P : E → Ω1X /P ⊥ ⊗OX E which satisfies the
Leibniz rule ∇P (f e) = f ∇P (e)+∂f ·e. A connection along P gives rise
to the OX -linear map ∇P(•) : P → EndC (E) defined by P ∋ ξ 7→ ∇Pξ ,
with ∇Pξ (e) = ιξ ∇P (e).
Conversely, an OX -linear map ∇P(•) : P → EndC (E) which satisfies
the Leibniz rule ∇Pξ (f e) = f ∇Pξ (e) + ∂f · e determines a connection
along P. In what follows we will not distinguish between the two
DEFORMATIONS OF AZUMAYA ALGEBRAS
15
avatars of a connection along P described above. Note that, as a
consequence of the ∂-Leibniz rule a connection along P is OX/P -linear.
A connection along P on E is called flat if the corresponding map
∇P(•) : P → EndC (E) is a morphism of Lie algebras. We will refer to a
flat connection along P on E as a ∂-operator on E.
Example 27. The differential ∂ in Gr F Ω•X gives rise to canonical ∂V
operators on i P ⊥ , i = 0, 1, . . ..
Example 28. The adjoint action of P on TXC preserves P, hence descends
to an action of the Lie algebra P on TXC /P. The latter action is easily
seen to be a connection along P, i.e. a canonical ∂-operator on TXC /P
which is easily seen to coincide with the one induced on TXC /P via the
duality pairing between the latter and P ⊥ . In the situation of Example
25 this connection is known as the Bott connection.
Example 29. Suppose that F is a locally free OX/P -module of finite
rank. Then, OX ⊗OX/P F is a locally free OX -module of rank rkOX/P F
and is endowed in a canonical way with the ∂-operator, namely, Id ⊗ ∂.
A connection on E along P extends uniquely to a derivation of the
2
graded Gr0F Ω•X -module Gr0F Ω•X ⊗OX E. A ∂-operator ∂ E satisfies ∂ E =
0. The complex (Gr0F Ω•X ⊗OX E, ∂ E ) is referred to as the (corresponding)
∂-complex. Since ∂ E is OX/P -linear, the sheaves H i (E, ∂ E ) are OX/P modules.
Lemma 30. Suppose that E is a vector bundle and ∂ E is a ∂-operator
on E. Then, H i (Gr0F ⊗OX E, ∂ E ) = 0 for i 6= 0, i.e. the ∂-complex
is a resolution of ker(∂ E ). Moreover, ker(∂ E ) is locally free over OX/P
of rank rkOX E and the map OX ⊗OX/P ker(∂ E ) → E (the OX -linear
extension of the inclusion ker(∂ E ) → E) is an isomorphism.
Remark 31. In the notations of Example 29 and Lemma 30, the assignments F 7→ (OX ⊗OX/P F , Id ⊗ ∂) and (E, ∂ E ) 7→ ker(∂ E ) are mutually
inverse equivalences of suitably defined categories.
By the very definition, the kernel of the canonical ∂-operator on
P
coincides with TXC /P (the subsheaf of P-invariant sections,
see Example 28). We denote this subsheaf by TX/P .
The duality pairing restricts to a non-degenerate OX/P -bilinear pairing between Ω1X/P and TX/P giving rise to a faithful action of TX/P on
OX/P by derivations by the usual formula ξ(f ) = ιξ df , for ξ ∈ TX/P
and f ∈ OX/P .
TXC /P
16
P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN
Let pri : X × X → X denote the projection on the ith factor and let
∆X : X → X × X denote the diagonal embeding. The latter satisfies
∆∗ (P × P) = P. Therefore, the induced map ∆∗X : OX×X → OX
satisfies
Im( ∆∗X |OX×X/P×P ) ⊂ OX/P .
Let ∆∗X/P := ∆∗X |OX×X/P×P , IX/P := ker(∆∗X/P ).
For a locally-free OX/P -module of finite rank E let
k
k+1
−1
J (E) := (pr1 )∗ OX×X/P×P /IX/P ⊗pr−1
pr2 E ,
2 OX/P
k
k
let JX/P
:= J k (OX/P ). It is clear from the above definition that JX/P
k
is, in a natural way, a commutative algebra and J k (E) is a JX/P
module.
We regard J k (E) as OX/P -modules via the pull-back map pr∗1 :
OX/P → (pr1 )∗ OX×X/P×P (the restriction to OX/P of the map pr∗1 :
k
OX → (pr1 )∗ OX×X ) with the quotient map (pr1 )∗ OX×X/P×P → JX/P
.
l+1
k+1
For 0 ≤ k ≤ l the inclusion IX/P → IX/P induces the surjective map
πl,k : J l (E) → J k (E). The sheaves J k (E), k = 0, 1, . . . together with
the maps πl,k , k ≤ l form an inverse system. Let J (E) = J ∞ (E) :=
limJ k (E). Thus, J (E) carries a natural topology.
←−
Let j k : E → J k (E) denote the map e 7→ 1 ⊗ e, j ∞ := limj k .
←−
Let
d1 : OX×X/P×P ⊗pr−1
pr−1
2 E →
2 OX/P
1
→ pr−1
OX×X/P×P ⊗pr−1
pr−1
1 ΩX/P ⊗pr−1
2 E
1 OX/P
2 OX/P
denote the exterior derivative along the first factor. It satisfies
k+1
d1 (IX/P
⊗pr−1
pr−1
2 E) ⊂
2 OX/P
1
k
pr−1
IX/P
⊗pr−1
pr−1
1 ΩX ⊗pr−1
2 E
1 OX/P
2 OX/P
for each k and, therefore, induces the map
(k)
d1 : J k (E) → Ω1X/P ⊗OX/P J k−1 (E)
(k)
The maps d1 for different values of k are compatible with the maps
πl,k giving rise to the canonical flat connection
∇can
: J (E) → Ω1X/P ⊗OX/P J (E)
E
which extends to the flat connection
∇can
: OX ⊗OX/P J (E) → Ω1X ⊗OX (OX ⊗OX/P J (E))
E
DEFORMATIONS OF AZUMAYA ALGEBRAS
17
Here and below by abuse of notation we write (•) ⊗OX/P JX/P (E) for
lim(•) ⊗OX/P J k (E).
←−
5.4. Hochschild cochains. Suppose that A is an OX/P -Azumaya algebra , i.e. a sheaf of algebras on X locally isomorphic to Matn (OX/P ).
For n > 0 let C n (A) denote the sheaf of multidifferential operators
×n
A
→ A; Let C 0 (A) = A. The subsheaf of normalized cochains
n
C (A) consists of those multidifferential operators which yield zero
whenever one of the arguments is in C · 1 ⊂ A.
With the Gerstenhaber bracket [ , ] and the Hochschild differential,
•
denoted δ, defined in the standard fashion, C • (A)[1] and C (A)[1] are
•
DGLA and the inclusion C (A)[1] → C • (A)[1] is a quasi-isomorphism
of such.
For n > 0 let C n (OX ⊗OX/P J (A)) denote the sheaf of continuous
OX -multilinear maps (OX ⊗OX/P J (A))×n → OX ⊗OX/P J (A). The
canonical flat connection ∇can
A on OX ⊗OX/P J (A) induces the flat concan
nection, still denoted ∇A , on C n (OX ⊗OX/P J (A)). Equipped with the
Gerstenhaber bracket and the Hochschild differential δ, C • (OX ⊗OX/P
J (A))[1] is a DGLA. Just as in the case P = 0 (see 4.3) we have the
DGLA Ω•X ⊗OX C • (OX ⊗OX/P J (A))[1] with the differential ∇can + δ.
Lemma 32. The de Rham complex DR(C n (OX ⊗OX/P J (A))) := (Ω•X ⊗OX
C n (OX ⊗OX/P J (A)), ∇can
A ) satisfies
(1) H i DR(C n (OX ⊗OX/P J (A))) = 0 for i 6= 0
(2) The map j ∞ : C n (A) → C n (OX ⊗OX/P J (A)) is an isomorphism onto H 0 DR(C n (OX ⊗OX/P J (A))).
Corollary 33. The map j ∞ : C • (A) → Ω•X ⊗OX C • (OX ⊗OX/P J (A))[1]
is a quasi-isomorphism of DGLA.
5.5. Azumaya. Suppose that A is an OX/P -Azumaya algebra.
The sheaves J (A) and A ⊗OX/P JX/P have canonical structures of
sheaves of (central) JX/P -algebras locally isomorphic to Matn (JX/P )
and come equipped with projections to A.
Let Isom0 (OX ⊗OX/P A⊗OX/P JX/P , OX ⊗OX/P J (A)) denote the sheaf
of (locally defined) OX ⊗OX/P JX/P -algebra isomorphisms OX ⊗OX/P
A ⊗OX/P JX/P → OX ⊗OX/P J (A) which induce the identity map on
OX ⊗OX/P A. Let Aut0 (OX ⊗OX/P A ⊗OX/P JX/P ) denote the sheaf of
(locally defined) OX ⊗OX/P JX/P -algebra automorphisms of OX ⊗OX/P
A ⊗OX/P JX/P which induce the identity map on OX ⊗OX/P A.
18
P.BRESSLER, A.GOROKHOVSKY, R.NEST, AND B.TSYGAN
Just as in Corollary 7 we may conclude that
Isom0 (OX ⊗OX/P A ⊗OX/P JX/P , OX ⊗OX/P J (A)) :=
Γ(X; Isom0 (OX ⊗OX/P A ⊗OX/P JX/P , OX ⊗OX/P J (A)))
is non-empty.
A choice of σ ∈ Isom0 (OX ⊗OX/P A ⊗OX/P JX/P , OX ⊗OX/P J (A))
and ∇ ∈ C(OX ⊗OX/P A) give rise to a unique F ∈ Γ(X; Ω1X ⊗OX/P
A0 ⊗OX/P JX/P ) and θ ∈ Γ(X; Ω2X ⊗OX/P A0 ) such that ∇2 = ad θ and
the equation (3.0.5) holds. Such a σ provides us with the isomorphism
of DGLA
(5.5.1) σ∗ : Ω•X ⊗OX C • (OX ⊗OX/P A ⊗OX/P JX/P )[1] →
→ Ω•X ⊗OX C • (OX ⊗OX/P J (A))[1]
where the latter is equipped with the differential ∇can
A + δ and the
former is equipped with the differential ∇ ⊗ Id + Id ⊗ ∇can
O + ad(F ) + δ.
The operator exp(ιF ) is an automorphism of the graded Lie algebra
Ω•X ⊗OX C • (OX ⊗OX/P A ⊗OX/P JX/P )[1]. It does not commute with
the differential ∇ ⊗ Id + Id ⊗ ∇can
O + ad(F ) + δ. Instead, the formula
of Lemma 16 holds. Hence, the composition σ∗ ◦ exp(ιF ) is a quasiisomorphism of DGLA as in (5.5.1) but with the source equipped with
the differential ∇ ⊗ Id + Id ⊗ ∇can
O + δ + ιθ .
The cotrace map
•
cotr : C (OX ⊗OX/P JX/P )[1] →
→ C • (OX ⊗OX/P A ⊗OX/P JX/P )[1]
defined as in (4.4.2) gives rise to the quasi-isomorphism of DGLA
•
Id ⊗ cotr : Ω•X ⊗OX C (OX ⊗OX/P JX/P )[1] →
→ Ω•X ⊗OX C • (OX ⊗OX/P A ⊗OX/P JX/P )[1]
where the source (respectively, the target) is equipped with the differcan
ential ∇can
O + δ (respectively, ∇ ⊗ Id + Id ⊗ ∇O + δ + ιθ ).
The proof of 19 shows that the image of the composition σ∗ ◦exp(ιF )◦
(Id⊗cotr) in the derived category does not depend on the choices made.
We summarize the above in the following theorem.
Theorem 34. Suppose that P is an integrable (complex) distribution
on X and A is an OX/P -Azumaya algebra. There is a canonical iso•
morphism in the derived category of DGLA Ω•X ⊗OX C (OX ⊗OX/P
JX/P )[1] ∼
= Ω•X ⊗OX C • (OX ⊗OX/P J (A))[1].
DEFORMATIONS OF AZUMAYA ALGEBRAS
19
Corollary 35. Under the assumptions of Theorem 34, there is a canon•
ical isomorphism in the derived category of DGLA C (OX/P ) ∼
= C • (A).
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DEFORMATIONS OF AZUMAYA ALGEBRAS
Department of Mathematics, University of Arizona
E-mail address: bressler@math.arizona.edu
Department of Mathematics, University of Colorado
E-mail address: Alexander.Gorokhovsky@colorado.edu
Department of Mathematics, University of Copehagen
E-mail address: rnest@math.ku.dk
Department of Mathematics, Northwestern University
E-mail address: tsygan@math.northwestern.edu
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