Calabi-Yau Deformations and Negative Cyclic Homology

0.37 MB·English

by  Michel Van den Bergh

CALABI-YAU DEFORMATIONS AND NEGATIVE CYCLIC
HOMOLOGY
2
1 LOUISDETHANHOFFERDEVO¨LCSEYANDMICHELVANDENBERGH
0
2
Abstract. InthispaperwerelatethedeformationtheoryofGinzburgCalabi-

n Yaualgebrastonegativecyclichomology. WedothisbyexhibitingaDG-Lie
a algebrathatcontrolsthisdeformationtheoryandwhosehomologyisnegative
J cyclichomology. Weshowthatthebracketinducedonnegativecyclichomol-

7 ogy coincides with Menichi’s string topology bracket. We show in addition
2 that the obstructions against deforming Calabi-Yau algebras are annihilated
bythemaptoperiodiccyclichomology.

] In the commutative weshow that our DG-Liealgebrais homotopy equiv-
A alentto(Tpoly[[u]],−udiv).
R
.
h
t
a
m
Contents
[
1. Introduction 1
3
2. Acknowledgement 3
v
3. Notation and conventions 3
0
2 4. Preliminaries on the Hochschild and cyclic complexes 3
5 5. Preliminaries on Calabi-Yau algebras 7
1 6. Deformations of Calabi-Yau algebras 9
.
1 7. The Maurer-Cartanformalism 12
0 8. The DG-Lie algebra D•(A,η) 13
2
9. Relation with Hochschild cohomology 16
1 10. Homology of D•(A,η) 16
:
v 11. The commutative case 20
i
X 12. Obstructions 24
Appendix A. A technical result on BV-algebras 26
r
a References 29
1. Introduction
Throughout k is a field of characteristic zero. In this paper we discuss the
deformation theory of Calabi-Yau k-algebras in the sense of Ginzburg [13]. Recall
that a k-algebra A is d-Calabi-Yau if it is perfect as Ae-module and there is an
1991 Mathematics Subject Classification. Primary14J32,16E40,16S80.
Key words and phrases. Calabi-Yaudeformations,negativecyclichomology,stringtopology.
Thesecondauthor isaseniorresearcherattheFWO.
1
2 LOUISDETHANHOFFERDEVO¨LCSEYANDMICHELVANDENBERGH
isomorphism1 in D(Ae)
(1.1) η :RHom (A,Ae)−→Σ−dA
Ae
In the rest of this introduction we fix a d-CY algebra A. Here and throughout the
paper we take the point of view that η is part of the structure of A.
The definition of a d-Calabi-Yau algebra can be “relativized” without any diffi-
culty. Hencethereisanassociateddeformationtheory. Ourfirstresultinthispaper
is the construction of a DG-Lie algebra which controls this deformation theory.
To be moreprecise: letNilp be the categoryofcommutative,finite dimensional,
local k-algebras (R,m) such that R/m = k. For R ∈ Nilp let Def (R) be the
A,η
category of R-algebras B which are d-Calabi-Yau (with respect to R) and which
are in addition equipped with an isomorphism B⊗Rk ∼=A respecting η. We view
Def as a pseudo-functor from Nilp to the category of groupoids Gd.
A,η
For a nilpotent DG-Lie algebra h let MC(h) be the groupoid of solutions to the
Maurer-Cartanequation in h (see §7). For an arbitrary DG-Lie algebra g we have
an associated “deformation functor”
MC(g):Nilp−→Gd:(R,m)7→MC(g⊗ m)
k
In this paper we introduce a DG-Lie algebra D•(A,η) (see §8) which controls the
deformation theory of (A,η).2
Theorem 1.1. (a combination of Prop. 6.2 and Thm 8.1 below) There is a mor-
phismofpseudo-functorsπ :MC(D•(A,η))−→Def whichwhenevaluatedonan
A,η
arbitrary R∈Nilp is essentially surjective on objects and surjective on morphisms.
We obtain in particular for R ∈ Nilp a bijection between MC(D•(A)⊗k m)/∼=
and DefA,η(R)/∼=. In this sense the deformation theory of A is controlled by the
DG-Lie algebra D•(A,η).
D•(A,η)isconstructedasatwistedsemi-directproductoftheHochschildcochain
complexwiththenegativecyclicchaincomplexofA. Sotheconstructionissimilar
in spirit to [26] which treats finite dimensional A -algebraswith a non-degenerate
∞
inner product. However our algebras are not finite dimensional and they do not
carry an inner product.
The construction of D•(A,η) yields a morphism
φ:D•(A,η)−→C¯•(A)
where C¯•(A) is the (shifted) Hochschild cochain complex of A. As is well-known,
C¯•(A) controls the deformation theory of A as algebra. The morphism φ corre-
sponds to “forgetting η” as is explained in §9.
Thenextresultistheconstructionofanexplicitquasi-isomorphismofcomplexes
(1.2) D•(A,η)−∼=→Σ−d+1CC−(A)
•
between D•(A,η) and the shifted negative cyclic complex CC−(A). As a result
•
we obtain the following information about the deformation theory of Calabi-Yau
algebras.
1This isomorphism is sometimes required to satisfy a certain symmetry condition but this
happenstobeautomaticallysatisfied. See[29,App.C].
2In fact this is slightly imprecise as D•(A,η) is only determined up to a non-unique isomor-
phism. TheactualdefinitionofD•(A,η)dependsontheliftofηtoanexplicitcycleinasuitable
complexbutwewillignorethissubtletyintheintroduction.
CALABI-YAU DEFORMATIONS AND NEGATIVE CYCLIC HOMOLOGY 3
Theorem 1.2. (1) The tangent space to the deformation space of a d-Calabi-
Yau algebra is HC− (A).
d−2
(2) Theobstructionsagainstdeformingad-Calabi-Yaualgebraareinker(HC− (A)→
d−3
HCper (A)).
d−3
The first statement is a formal consequence of (1.2). The second statement is
Theorem 12.1 below. It follows in particular that if HC− (A) → HCper (A) is
d−3 d−3
injective then the deformation theory of A as Calabi-Yau algebra is unobstructed.
This happens for example if d≤3 (see Corollary 12.5 and Lemma 12.6 below).
OurnextresultisthedescriptionoftheLiebracketonHC (A)inducedby(1.2):
•
Theorem 1.3. (Theorem 10.2 below) The Lie bracket on negative cyclic homology
induced by (1.2) is the “string topology” Lie bracket introduced in [20] by Menichi.
Let us now specialize to the case where A is commutative. Let Tpoly,•(A) be
the Lie algebra of poly-vector fields on A. Then η in (1.1) may be interpreted as
a volume form (see §11 below). Let div be the divergence operator on Tpoly,•(A)
associated to η. Using Willwacher’s “formality for cyclic chains” [32] (see also
[7, 23, 28]) we show that there is an isomorphism
(Tpoly,•[[u]],−udiv)−∼=→D•(A,η) (|u|=2)
in the homotopy categoryofDG-Lie algebraswhich fits in a commutative diagram
(1.3) (Tpoly,•(A)[[u]],−udiv) ∼= // D•(A)
u7→0 φ
(cid:15)(cid:15) (cid:15)(cid:15)
Tpoly,•(A) // C¯•(A)
∼=
ThelowerarrowisaglobalizedversionofKontsevich’sformalityquasi-isomorphism[16].
ThisdiagramgivesaconceptualexplanationofDolgushev’sresult[8]thattheKont-
sevich ∗-product associated to a divergence free Poissonbracket is Calabi-Yau.3
2. Acknowledgement
We like to thank Damien Calaque and Boris Tsygan for help with references.
3. Notation and conventions
All rings and ring homomorphisms are unital. We mix homological and coho-
mological indices, using the convention X =X−i.
i
4. Preliminaries on the Hochschild and cyclic complexes
In this section remind the reader about the basic operations on the Hochschild
andcyclic complexes. The reasonfor putting this sectionfirst is that it also allows
us to introduce some notation. Readers vaguely familiar with the material may
safely skip to the next section.
3ThisisaspecialcaseDolgushev’sresult. DolgushevdoesnotrestricttotheCalabi-Yaucase.
4 LOUISDETHANHOFFERDEVO¨LCSEYANDMICHELVANDENBERGH
4.1. Notation. Let R be a commutative ring and assume that B is an R-algebra.
LetC (B)andC•(B)denote the usualrelativeHochschildchainandcochaincom-
•
plexes of B/R. Thus
C•(B)= Hom (ΣB⊗n,B)
R
Mn
C (B)= B⊗(ΣB)⊗n
•
Mn
where here and below, all unadorned tensor products are over R. We also use
C•(B)=ΣC•(B)
= Hom (ΣB⊗n,ΣB)
R
Mn
We will also consider the normalized versions of these objects.
C¯•(B)= Hom (Σ(B/R)⊗n,B)
R
Mn
C¯ (B)= B⊗Σ(B/R)⊗n
•
Mn
andasimilardefinitionforC¯•(B). Itiswell-knownthattheobviousmapsC¯•(B)−→
C•(B)andC (B)−→C¯ (B)arequasi-isomorphisms[31,Thm8.3.8,Lemma8.3.7].
• •
If x ∈ Cn(B) then we write |x| = n−1. Thus |x| refers to the cohomological
degree of x.
4.2. The Hochschild cochain complex. The standard algebraic structures on
the Hochschild cochain complex can all be deduced from its structure as a brace
algebra [11]. Recall that the braces are maps
C•(B)⊗...⊗C•(B)−→C•(B):x⊗x ⊗···⊗x 7→x{x ,...,x }
1 m 1 m
defined by
x{x ,...,x }(b ,...b )=
1 m 1 n
(−1)ǫx(b ,...b ,x (b ,...b ),...,b ,x (b ,...,b ),...b )
1 i1 1 i1+1 i1+|x1|+1 im m im+1 im+|xm|+1 n
0≤i1.X..≤im≤n
where ǫ= m|x |i The corresponding Gerstenhaber Lie bracket on C•(B) is
1 k k
P [x,y]=x{y}−(−1)|x||y|y{x}
Letµ∈C1(B)=Hom(ΣB⊗ΣB,ΣB)denotethe“inverse”multiplication µ(b ,b )=
1 2
−b b . Then [µ,µ]=0 and hence
1 2
(4.1) dx=[µ,x]
defines a differential of degree one on C•(B).
The cupproduct on C•(B) is defined by
x∪y =(−1)|x|µ{x,y}
ThisisanassociativeproductofdegreeoneonC•(B),orequivalentlyanassociative
product of degree zero on C•(B). One has [11]
(1) (C•(B),d,[ , ]) is a DG-Lie algebra
(2) (C•(B),d,∪) is a DG-algebra, commutative up to homotopy
CALABI-YAU DEFORMATIONS AND NEGATIVE CYCLIC HOMOLOGY 5
(3) Moregenerally: (C•(B),d,[, ],∪)isaso-calledDG-“Gerstenhaberalgebra”
up to homotopy
4.3. The Hochschild chain complex. When combining the Hochschild cochain
complex with the Hochschild chain complex things becomes much more intricate
[2, 24]. We will content ourselves by giving formulas for the basic operations and
stating some relations between them. We refer to [24] for more details.
The first basic operation is the contraction.
i (b ⊗...⊗b ):=b x(b ,...b )⊗b ⊗...⊗b
x 0 n 0 1 d d+1 n
for x ∈ C•(B) and b ⊗...⊗b ∈ C (B). This is an action of C•(B) on C (B)
0 n • •
satisfying |i |=|x|+1 and
x
(4.2) i i =(−1)(|x|+1)(|y|+1)i
x y y∪x
The contraction is often written as a capproduct: i (−)=x∩−.
x
The second basic operation is the Lie derivative
n−|x|−1
L (b ⊗...⊗b ):= (−1)|x|ib ⊗...⊗b ⊗x(b ,...,b )⊗...⊗b
x 0 n 0 i i+1 i+|x|+1 n
Xi=0
n
+ (−1)n(i+1)+|x|x(b ,...b ,b ,...,b )⊗...⊗b
i+1 n 0 |x|−n+i i
i=Xn−|x|
The Lie derivative defines a graded Lie action of C•(B) on C (B). Explicitly:
•
|L |=|x| and
x
(4.3) [L ,L ]=L
x y [x,y]
The Hochschild differential on C (B) is defined as b = L . From (4.1) and (4.3)
• µ
one obtains
(4.4) [b,L ]=L
x dx
Hence (C (B),b) is a DG-Lie module overC•(B). One also has compatibility with
•
the contraction:
(4.5) [b,i ]+i =0
x dx
The last basic operation we need is the Connes differential.
B:C (B)−→C (B)
• •
with formula
n
B(b ⊗...⊗b )= (−1)ni1⊗b ⊗...⊗b ⊗b ⊗...⊗b
0 n i n 0 i−1
Xi=0
n
+ (−1)n(i+1)b ⊗1⊗b ⊗...⊗b ⊗b ⊗...⊗b
i−1 i n 0 i−2
X0
It is well-known that |B|=−1, bB+Bb=0, B2 =0.
Some of the following identies hold only for normalized chains/cochains. Note
that if x∈C¯•(B) then i ,L are well-defined operations on C¯ (B).
x x •
6 LOUISDETHANHOFFERDEVO¨LCSEYANDMICHELVANDENBERGH
Lemma 4.1. Assume x∈C¯•(B). Then on C (B) we have
•
(4.6) [B,L ]=0
x
The formula (4.6) does not hold for unnormalized cochains.
4.4. The negative cyclic complex. Let u be a variable of degree two and put
CC−(B)=C¯ (B)[[u]]
• •
Equipped with the cyclic differential b+uB, this is the normalized negative cyclic
complex. In the sequel operations on C¯ (B) will be (tacitly) extended to CC−(B)
• •
by making them u-linear. This applies in particular to i and L . Combining
x x
(4.4)(4.6) we obtain
(4.7) [b+uB,L ]=L
x dx
The compatibility of i with the cyclic differential is more subtle. In [24] (see
x
also[12])TamarkinandTsygandefineforx∈C¯•(B)agradedendomorphismS of
x
C¯ (B)(dependinglinearlyonx)suchthat|S |=|x|−1andsuchthatthefollowing
• x
identity holds
(4.8) [b+uB,i +uS ]+i +uS =uL
x x dx dx x
−
on CC (B). This identity will be important for us in the sequel. Note that it
•
implies (4.7).
The following is a special case of [24, Prop. 3.3.4].
Lemma 4.2. Let x,y ∈ C¯•(B) be such that dx = dy = 0. Then [L ,i +uS ] is
x y y
homotopic to (−1)|x|(i +uS ).
[x,y] [x,y]
4.5. A comment on base change. If A is a k-algebra and R is a commutative
k-algebra then for B = A⊗k R it is clear that CR,•(B) ∼= C•(A) ⊗k R (where
contrary to our usual conventions we have now made the base ring explicit in the
notation). Since the negative cyclic complex involves a product this is not true for
CC− (B). However it is true if R is finite dimensional. Similarly in that case we
R,•
have C•R(B) ∼= C•(A)⊗k R. In the sequel we will not mention these base change
isomorphisms explicitly.
4.6. Somecommentsonsigns. Intheprevioussectionstheoperationsi ,L ,S ,b,B
x x x
of degree |x|+1, |x|, |x|−1, 1, −1 were defined as acting on C¯ (B). We define
•
corresponding operations on shifts ΣrC¯ (B) in the usual way.
•
i (srb)=(−1)r(|x|+1)sri (b)
x x
L (srb)=(−1)r|x|srL (b)
x x
S (srb)=(−1)r(|x|−1)srS (b)
x x
b(srb)=(−1)rsrb(b)
B(srb)=(−1)rsrb(B)
where s is the degree change operator |sb|=|b|−1.
The relations between i ,L ,S ,b,B stated in §4.3,§4.3 carry over to all shifts
x x x
ΣrC¯ (B) without any sign changes, since all terms in the identities (necessarily)
•
have the same degree.
CALABI-YAU DEFORMATIONS AND NEGATIVE CYCLIC HOMOLOGY 7
5. Preliminaries on Calabi-Yau algebras
In this section we extend Ginzburg’s definition of Calabi-Yau algebras to the
relative case.
Let R be a commutative ring. For an R-algebra B we put Be = B⊗ B◦. We
R
use without further comment the standard equivalences between the categories of
left Be-modules, right Be-modules and B-bimodules which are R-central.
A Be-module is called perfect if it has a finite resolution by finitely generated
projective Be-modules. If B is R-flat and B is a perfect Be-module then we say
that B is homologically smooth over R. The implicit flatness hypothesis ensures
thatBe =B⊗ B◦ isthecorrectdefinitionfromaderivedpointofview. We could
R
have avoided this hypothesis by first replacing B by an R-flat DG-resolution but
for simplicity we have chosen not to do this.
Definition 5.1. (Ginzburg[13])AnR-Calabi-Yaualgebraofdimensiondisapair
(B,η) where
(1) B is an R-algebra B which is homologically smooth over R;
(2) η is an isomorphism RHom (B,Be)−→Σ−dB in D(Be).
Be
Remark 5.2. Note that the amount of freedom for η is quite limited. If (B,η),
(B,η′) are Calabi-Yau then there exists z ∈Z(B) such that η′ =zη (see [6]).
Recall that if M is a complex of Be-module then its Hochschild homology and
cohomology are respectively defined as
HH (B,M)=H−i(M ⊗LB)
i
HHi(B,M)=Hi(RHom (B,M))
Be
As usual we write HH (B)=HH (B,B) and similarly HHi(B)=HHi(B,B). One
i i
has
HH (B)=H−i(C (B))
i •
and if B is a projective R-module then
HHi(B)=Hi(C•(B))
The operations[ , ],∪,∩,L,B introducedin §4 descend to homologyand make the
pair(HH•(B),HH (B))intoaso-calledcalculus structure[24]. Uptosuitable,and
•
for us irrelevant, signs ∪ is the Yoneda products on HH•(B) = Ext• (B,B) and
Be
∩ is the action of HH•(B) on HH (B)=H−•(B⊗L B) through its action on the
• Be
second factor (see e.g. [3, Prop 11.1, 12.1]).
Lemma 5.3. Let B be a homologically smooth algebra. Then for M a perfect
Be-module there is a canonical isomorphism
(5.1) HHi(B,M)∼=HomBe(ΣiRHomBe(M,Be),B)
in D(R).
Proof. Since M is perfect we may replace it with a complex of finitely generated
projective Be-modules. In this way we reduce to M = Be which is an easy verifi-
cation. (cid:3)
Definition 5.4. LetB beahomologicallysmoothalgebraRandη ∈HH (B). We
d
say that η is nondegenerate if its image under (5.1) is an isomorphism.
8 LOUISDETHANHOFFERDEVO¨LCSEYANDMICHELVANDENBERGH
This allows us to redefine a d-Calabi-Yau algebra over R as a couple (B,η)
whereB is a homologicallysmoothR-algebraandη is anon-degenerateelement of
HH (B). Below we will massage this new definition further. Recall the following
d
Proposition 5.5. (“Poincare duality”) Assume that (B,η) is a d-Calabi-Yau R-
algebra. Then for each i, the map
(5.2) HHi(B)−→HH (B):µ7→µ∩η
d−i
is an isomorphism
Proof. Theexistenceoftheisomorphismwasfirststatedin[30]withouttheexplicit
formula(5.2). Theformula(5.2)isfolklore. Forcompletenessweincludeapossible
proof.
The proof of Lemma 5.3 shows that there is a canonical isomorphism in D(R)
↓ ↓
(5.3) RHomBe(RHomBe(B,Be),B)∼=B⊗LBe B
which is compatible with the RHom (B,B)-actions on the marked copies of B.
Be
By definition η ∈ H−d(B ⊗L B) corresponds under (5.3) to an isomorphism
Be
η+ :RHom (B,Be)→Σ−dB. This yields an isomorphism
Be
↓ ↓
(5.4) RHom (Σ−dB,B)−→RHom (RHom (B,Be),B):θ 7→θ◦η+
Be Be Be
also compatible with the marked RHom (B,B)-actions. Composing (5.3)(5.4)
Be
gives us an isomorphism
↓ ↓
ξ :RHom (Σ−dB,B)−→B⊗L B
Be Be
which sends Id to η.
B
According to the discussion preceding Lemma 5.3, the compatibility with the
RHom (B,B)-actions implies thatξ transforms∪into ∩ onthe levelofcohomol-
Be
ogy. More precisely
ξ(µ∪σ)=±µ∩ξ(σ)
The lemma now follows by taking σ =Id . (cid:3)
B
Corollary 5.6. Assume that (B,η) is a d-Calabi-Yau R-algebra. Then
HHi(B)=0 for i6∈[0,d]
HH (B)=0 for i6∈[0,d] (cid:3)
i
As beforeletCC−(B)=(C (B)[[u]],b+uB)be the negativecycliccomplex and
• •
denote its corresponding homology by HC−(B).
•
Proposition 5.7. Let (B,η) be a d-Calabi-Yau R-algebra. Then HC−(B)=0 for
i
i>d and furthermore the map
π :CC−(B)−→C (B): b ui 7→b
• • i 0
X
induces an isomorphism HC−d(B)∼=HHd(B).
Proof. Weuseaspectralsequenceargument. WeviewCC−(B)asadoublecomplex
•
with b pointing vertically upwards and uB pointing horizontally to the right. By
CALABI-YAU DEFORMATIONS AND NEGATIVE CYCLIC HOMOLOGY 9
Corollary5.6 we have HH (B)=0 for i>d. Hence if we filter CC−(B) by column
i •
degree then the E1 term of the resulting spectral sequence looks like
0 HH (B) uB // uHH (B) uB // u2HH (B)
d−2 d−1 d
0 HH (B) uB // uHH (B) 0
d−1 d
0 HH (B) 0 0
d
0 0 0 0
From this the result follows. (cid:3)
Definition 5.8. Let B be a homologically smooth R-algebra. We say that an
element η ∈HC−(B) is non-degenerate if π(η) is non-degenerate.
d
The leads to the following redefinition of a Ginzburg d-Calabi-Yau R-algebra
which we use below.
Definition 5.9. (Restatement of Definition 5.1.) A Calabi-Yau algebra of dimen-
sion d over R is a couple (B,η) where B is a homologically smooth R-algebra and
η is a non-degenerate element of HC−(B).
d
WehaveshownthatthisdefinitionisequivalenttoGinzburg’soriginaldefinition.
InthemoregeneralsettingofDG-algebrasthisisnolongerthecase. Itisgenerally
believed that Definition 5.9 is the “correct” definition for a d-Calabi-Yau algebra
in the DG-case. This is the point of view of Kontsevich-Soibelmanin [17] and also
of Keller [15].
6. Deformations of Calabi-Yau algebras
In this section we fix a d-Calabi-Yau k-algebra (A,η ) as in Definition 5.9. We
0
will study the deformations of A as a Calabi-Yau algebra.
Let Nilp be the category of commutative, finite dimensional, local k-algebras
(R,m) such that R/m = k. For (R,m) ∈ Nilp we define a groupoid Def (R)
A,η0
as follows: the objects in Def (R) are triples (B,s,η) such that B is an R-flat
A,η0
R-algebra, s : B →A is an R-algebra map inducing an isomorphism B⊗ k →A
R
and η is an element of HC−(B) such that s(η)=η .
d 0
A morphism (B ,s ,η )→(B ,s ,η ) is a commutative diagram
1 1 1 2 2 2
B φ //B
1❆s❆1❆❆❆❆❆❆ ~~⑥⑥⑥⑥⑥⑥s⑥2⑥ 2
A
such η = φ(η ). One sees that Def becomes a pseudo-functor Nilp → Gd in
2 1 A,η0
the obvious way.
TobeabletorightfullyclaimthatDef describestheCalabi-Yaudeformations
A,η0
of (A,η ) we need the following elementary lemma.
0
10 LOUISDETHANHOFFERDEVO¨LCSEYANDMICHELVANDENBERGH
Lemma 6.1. Assume that (B,s,η)∈Def (R). Then (B,η) is d-Calabi-Yau.
A,η0
Proof. We have to show that B is a perfect Be-module and that η induces an
isomorphism η+ :RHom (B,Be)→Σ−dB.
Be
Since R is finite dimensional every flat R-module is R-projective. This applies
in particular to B and Be. Let
0→P →···→P →A→0
u 0
be a finite resolution of A by finitely generated projective Ae-modules. It is easy
to see that this resolution can be lifted step by step to a resolution
0→Q →···→Q →B →0
u 0
wherethe Qi arefinitely generatedprojectiveBe-modules satisfying Qi⊗Rk ∼=Pi.
In particular B is perfect.
It also follows that H =coneη+ is perfect. It it easy to that η+⊗Lk ∼=η+ and
0
hence (coneη+)⊗Lk ∼=cone(η+⊗Lk)∼=coneη+ =0. If now suffices to note that
0
if H is perfect and H ⊗Lk =0 then H =0. (cid:3)
WenowintroduceavariantofthegroupoidDef (R)whichiseasiertodescribe
A,η0
cohomologically. We remind the reader of the base change conventionexhibited in
§4.5whichwewillusethroughout. Asin§4.2let−µ ∈C1(A)bethemultiplication
0
−
map on A and let ηˆ be a lift of η to CC (A). We define an associated groupoid
0 0 d
Def♭ (R) as follows. The objects are couples (µ,η) where
A,ηˆ0
(1) µ∈C1(A)⊗ R is such that −µ defines a unital associative multiplication
k
on A⊗ R;
k
(2) µ mod m=µ ;
0
−
(3) η ∈CC (A)⊗ R;
d k
(4) (L +uB)(η)=0;
µ
(5) η mod m=ηˆ .
0
For (4) recall that L +uB is the cyclic differential for the algebra (A⊗ R,µ). A
µ k
morphism (µ ,η )→(µ ,η ) in Def♭ (R) is a couple (φ,ξ) where
1 1 2 2 A,µ0,ηˆ0
(1) φ is a unital map of R-algebras φ:(A⊗ R,−µ )−→(A⊗ R,−µ );
k 1 k 2
(2) φ is the identity modulo m;
−
(3) ξ is an element of CC (A)⊗ m;
d+1 k
(4) (L +uB)(ξ)=φ(η )−η .
µ2 1 2
The composition of morphisms
(φ′,ξ′) (φ,ξ)
(µ ,η )−−−−→(µ ,η )−−−→(µ ,η )
1 1 2 2 2 2
is defined by
(6.1) (φ,ξ)◦(φ′,ξ′)=(φ◦φ′,φ(ξ′)+ξ)
Below we will often use the notation η¯for the cohomology class of a cocycle η.
Proposition 6.2. The morphism of groupoids
Ob(Def♭ (R))−→Ob(Def (R)):(µ,η)7→((A⊗ R,−µ),“modm”,η¯)
A,ηˆ0 A,η0 k
Mor(Def♭ (R))−→Mor(Def (R)):(φ,ξ)7→φ
A,ηˆ0 A,η0
is essentially surjective on objects and surjective on morphisms.