Table Of ContentSingular Deformation Theory and the Invariance of
Gerstenhaber Algebra Structure on the Singular
Hochschild Cohomology
Zhengfang WANG ∗
6
1
0
Abstract
2
n Keller proved in1999 thattheGerstenhaberalgebra structureof theHochschild
a
J cohomology of an algebra is an invariant of the derived category. In this paper, we
7 adapt his approach and develop the singular infinitesimal deformation theory. As
a consequence, we show that the Gerstenhaber algebra structure on the singular
]
T Hochschild cohomology of an algebra is preserved under singular equivalences of
R Morita type with level.
.
h
t
a 1 Introduction
m
[
In the recent paper [Wang2], we defined the singular Hochschild cohomology group
1
HHi (A,A) of an associative algebra A over a field k as morphisms from A to A[i] in
v sg
1 D (Aop ⊗ A) for any i ∈ Z. Similar to the case of the Hochschild cohomology ring
sg k
4 HH∗(A,A), we proved that the singular Hochschild cohomology ring HH∗ (A,A) has a
6 sg
1 Gerstenhaber algebra structure (cf. [Ger]). Moreover, the natural morphism
0
1. HH∗(A,A) → HH∗ (A,A)
sg
0
6
is a homomorphism of Gerstenhaber algebras.
1
: Keller proved in [Kel1] that the Gerstenhaber algebra structures on Hochschild co-
v
i homology rings are preserved under derived equivalences of standard type. That is, let
X
A and B be two k-algebras and if X is a complex of A-B-bimodules such that the to-
r
a tal derived tensor product by X is an equivalence D(A) → D(B) between the derived
categories of A and of B, then X yields a natural isomorphism of Gerstenhaber alge-
bras from HH∗(A,A) to HH∗(B,B). In this paper, we will show that the Gerstenhaber
algebra structure on the singular Hochschild cohomology ring is also preserved under
derived equivalences of standard type. In fact, we will prove a stronger result, that is,
the Gerstenhaber algebra structure on the singular Hochschild cohomology ring is pre-
served under singular equivalences of Morita type with level (cf. [Wang1] and Section 6
below). Recall that a derived equivalence of standard type induces a singular equivalence
of Morita type with level (cf. [Wang1]).
∗[email protected],Universit´eParisDiderot-Paris7,InstitutdeMath´ematiquesdeJussieu-
Paris Rive Gauche CNRS UMR 7586, Bˆatiment Sophie Germain, Case 7012, 75205 Paris Cedex 13,
France
1
The reminder of this paper is organized as follows. In Section 2, we recall the bar res-
olution of an associative algebra A and give some natural lifts of elements in HH∗ (A,A)
sg
along the bar resolution. The bullet product • is the key ingredient of the Gerstenhaber
algebra structure on the singular Hochschild cohomology. We will give a derived interpre-
tation of this bullet product • in Section 3. Section 4 is devoted to recall some notions on
R-relative derived categories and R-relative derived tensor products. In Section 5, we will
generalize the infinitesimal deformation theory in [Kel1] to what we call the singular in-
finitesimal deformation theory. As a result, we give an interpretation of the Gerstenhaber
bracket on the singular Hochschild cohomology ring from the point of view of singular
infinitesimal deformation theory. This interpretation relies on the derived interpretation
of the bullet product • in Section 3. In the last section, we will show our main theorem.
Namely,
Theorem 1.1 (=Theorem 6.3). Let k be a field and let A and B be two finite dimensional
k-algebras. Suppose that ( M , N ) defines a singular equivalence of Morita type with
A B B A
level l ∈ Z . Then the functor
≥0
M ⊗ −⊗ N : D (B ⊗Bop) → D (A⊗Aop)
B B sg sg
induces an isomorphism of Gerstenhaber algebras between the singular Hochschild coho-
mology rings HH∗ (A,A) and HH∗ (B,B).
sg sg
Throughout this paper, we fix by k as the commutative base ring with a unit. For
simplicity, thesymbol ⊗alwaysrepresents thetensorproduct⊗ over thebaseringk. For
k
a k-algebra A, we denote the element (a ⊗a ⊗···⊗a ) ∈ A⊗j−i+1(i ≤ j) sometimes by
i i+1 j
a for short. We frequently use some notions on differential graded algebras and relative
i,j
derived categories in this paper, for more details, we refer to [Kel1, Kel2, BeLu]. We also
refer to [KeVo, Ric, Wei, Zim] for some notions on triangulated categories and derived
categories.
Acknowledgement
This paper is a part of author’s PhD thesis, I would like to thank my PhD supervisor
Alexander Zimmermann for introducing this interesting topic and for his many valuable
suggestions for improvement. I also would like to thank Gufang Zhao for many useful
discussions when I just started to prepare this paper.
2 Bar resolution of an associative k-algebra
2.1 Definition
Let A be an associative algebra over a commutative ring k such that A is projective as a
k-module. Then there is a projective resolution of A as an A-A-bimodule,
Bar (A) : ··· //A⊗r+2 dr //A⊗r+1 dr−1 //··· // A⊗3 d1 // A⊗2 //0, (1)
∗
where the differential d : A⊗r+2 → A⊗r+1 is defined by
r
r+1
d (a ⊗···⊗a ) := (−1)i−1a ⊗a a ⊗a ,
r 1 r+2 1,i−1 i i+1 i+2,r+2
i=1
X
2
for any a ⊗···⊗a ∈ A⊗r+2. Here we denote, for i < j,
1 r+2
a := a ⊗a ⊗···⊗a .
i,j i i+1 j
This is the so-called (un-normalized) bar resolution of A. We also have the normalized
bar resolution Bar of A. It is defined by
∗
⊗r
Bar (A) := A⊗A ⊗A,
r
where A = A/(k ·1), and with the induced differential from that of Bar (A). Note that
∗
Bar (A) is also a projective resolution of A as an A-A-bimodule.
∗
For any r ∈ Z , let us denote by Ωr(A), the image of the differential
≥1
d : A⊗r+2 → A⊗r+1
r
r
in the un-normalized bar resolution Bar (A). Similarly, we denote by Ω (A), the image
∗
of the differential
⊗r ⊗r−1
d : A⊗A ⊗A → A⊗A ⊗A
r
in the normalized bar resolution Bar (A). In particular, we will use the notation
∗
Ω0(A) = Ω0(A) := A.
Observe that we have the un-normalized bar resolution Bar (Ωr(A)) of the A-A-bimodule
∗
Ωr(A):
Bar (Ωr(A)) : ··· //A⊗r+s+2 dr+s //A⊗r+s+1dr+s−1// ··· //A⊗r+2 //0
∗
r r
and the normalized bar resolution Bar (Ω (A)) of the A-A-bimodule Ω (A):
∗
Bar (Ωr(A)) : ··· // A⊗A⊗r+s ⊗A dr+s //A⊗A⊗r+s−1⊗Adr+s−1// ··· //A⊗A⊗r ⊗A // 0
∗
In the rest of this paper, let us just consider the un-normalized bar resolution. For
the case of normalized bar resolution, we always have the analogous results.
Remark 2.1. Note that the following chain complex
(Bar (Ωp(A)),d ) := Bar (Ωp(A)) dp ////Ωp(A)
∗ p ∗
is exact for any fixed p ∈ Z . In this remark, we will show that the identity morphism
≥0
on (Bar (Ωp(A)),d ) is homotopy equivalent to zero, where we use the homotopy
∗ p
s : Bar (Ωp(A)) → Bar (Ωp(A))
p r r+1
s (x) := (−1)rpx⊗1.
p
Namely, we have the following diagram.
··· //A⊗p+3 dp+1 //A⊗p+2 dp //Ωp(A) //0
id ✉✉✉s✉p✉✉✉✉✉✉ id ✉✉✉s✉p✉✉✉✉✉ id
(cid:15)(cid:15) zz (cid:15)(cid:15) zz (cid:15)(cid:15)
··· //A⊗p+3 //A⊗p+2 //Ωp(A) //0
dp+1 dp
3
It is straightforward to verify that
s ◦d+d◦s = id.
p p
Soitfollowsthattheidentitymorphismishomotopyequivalenttozeroon(Bar (Ωp(A)),d ).
∗ p
Note that s is a morphism of graded left A-modules rather than a morphism of graded
p
(A⊗Aop)-modules. Therefore the complex (Bar (Ωp(A)),d ) vanishes in the homotopy
∗ p
categoryK−(A-mod)anddoesnot,ingeneralinthehomotopycategoryK−(A⊗Aop-mod).
For any p,q ∈ Z , we will construct a morphism of complexes of A-A-bimodules
≥0
between Bar (Ωp+q(A)) and Bar (Ωp(A))⊗ Bar (Ωq(A)). Recall that for any r ∈ Z ,
∗ ∗ A ∗ ≥0
we have
Bar (Ωp(A)) = A⊗p+r+2,
r
r
(Bar (Ωp(A))⊗ Bar (Ωq(A))) = Bar (Ωp(A))⊗ Bar (Ωq(A)).
∗ A ∗ r i A r−i
i=0
M
Let us define
∆ : Bar (Ωp+q(A)) → Bar (Ωp(A))⊗ Bar (Ωq(A)) (2)
p,q ∗ ∗ A ∗
as follows. For any r ∈ Z and a ∈ Bar (Ωp+q(A)), define
≥0 1,p+q+r+2 r
r
∆ (a ) := (a ⊗1)⊗ (1⊗a ).
p,q;r 1,p+q+r+2 1,p+i+1 A p+i+2,p+q+2
i=0
X
Then it is straightforward to verify that ∆ is a well-defined homomorphism of chain
p,q
complexes of A-A-bimodules.
Lemma 2.2. If pq=0, then ∆ is an isomorphism of K(A⊗Aop).
p,q
Proof. Note that ∆ is a lift of the identity morphism id : Ωq(A) → Ωq(A).
0,q Ωq(A)
Hence ∆ is a quasi-isomorphism. Since Bar (Ωq(A)) is a complex of projective A-
0,q ∗
A-bimodules, it follows that ∆ is also an isomorphism in K(A ⊗ Aop). By the same
0,q
argument as above, we have that ∆ is an isomorphism of K(A⊗Aop).
p,0
Remark 2.3. ∆ plays a quite important rˆole in our discussion below. Recall that ∆
p,q 0,0
is induced from the coproduct ∆ in the tensor coalgebra
TcA := A⊗n
n≥0
M
defined in [Sta] and [Kel1].
2.2 Two types of lift
Let A be an associative k-algebra such that A is projective as a k-module. Let m,p ∈ Z
≥0
and f ∈ HHm(A,Ωp(A)). By definition, f can be represented by an element
f ∈ Hom (A⊗m,Ωp(A))
k
such that
δ(f) = 0,
4
where δ is the differential of the Hochschild cochain complex Hom (A⊗∗,Ωp(A)). Denote
k
by ǫ the graded vector space defined as follows,
m
k if i = m
(ǫ ) :=
m i
(0 otherwise.
ThenwehavethefollowingtwotypesofliftinthecategoryC(A⊗ Aop)ofchaincomplexes
k
of A-A-bimodules.
θ (f) :Bar (A) → Bar (Ωp(A))⊗ǫ [1],
1 ∗ ∗ m−1
θ (f) :Bar (A) → Bar (Ωp(A))⊗ǫ [1],
2 ∗ ∗ m−1
where θ (f) and θ (f) are defined, respectively, as follows,
1 2
θ (f)(a ) := (−1)p+1+(m−p−1)ra f(a )⊗a ,
1 1,m+r+2 1 2,m+1 m+2,m+r+2
θ (f)(a ) := (−1)r+1a ⊗f(a )a .
2 1,m+r+2 1,r+1 r+2,r+m+1 r+m+2
Note that θ (f) and θ (f) are indeed morphisms of chain complexes of A-A-bimodules.
1 2
It is well-known in homological algebra (cf. e.g. [Wei, Comparison Theorem 2.2.6]) that
θ (f) and θ (f) are homotopy equivalent to each other. Moreover, there exists a specific
1 2
chain homotopy
s(f) : Bar (A) → Bar (Ωp(A))⊗ǫ
∗ ∗ m−1
from θ (f) to θ (f), defined as follows: for any r ∈ Z ,
1 2 ≥0
sr(f) : A⊗m+r+2 → A⊗p+r+3 (3)
sends a ∈ A⊗m+r+2 to
1,m+r+2
r
(−1)i−1a ⊗f(a )⊗a .
1,i i+1,i+m+1 i+m+2,m+r+2
i=1
X
Namely, we have the following diagram,
··· //A⊗m+3 dm+1 //A⊗m+2 dm // A⊗m+1 dm−1 // ···
θ1(f) θ2(sfs)sssss0s(sfs)ssssssθs1s(sf) θ2(sfs)ssssssss0ssssssss
(cid:15)(cid:15) (cid:15)(cid:15) yy (cid:15)(cid:15) (cid:15)(cid:15) yy (cid:15)(cid:15)
··· //A⊗p+3 // A⊗p+2 //0
−dp+1
which means that
θ (f)−θ (f) = s(f)◦d+d◦s(f). (4)
1 2
We will denote the r-th lift of θ (f) by Ωr(θ (f)), namely,
1 1
Ωr(θ (f)) : A⊗m+r+2 → Ωp+r(A)
1
a 7→ (−1)p+(m−p)rd(a f(a )⊗a ).
1,m+r+2 1 2,m+1 m+2,m+r+2
Similarly, we denote the r-th lift of θ (f) by Ωr(θ (f)),
2 2
Ωr(θ (f)) : A⊗m+r+2 → Ωp+r(A)
2
a 7→ d(a ⊗f(a )a ).
1,m+r+2 1,r+1 r+2,r+m+1 r+m+2
Remark 2.4. Since s(f) is a morphism of graded (Aop ⊗A)-modules, it follows that
θ (f) = θ (f)
1 2
inthehomotopycategoryK−(Aop ⊗A-mod)andbothofthemrepresentthesameelement
f ∈ Hom (A,Ωp(A)[m]).
Db(Aop⊗A)
5
3 A derived interpretation of the bullet product •
3.1 Definition of R-relative derived category
Letusstartwiththegeneralsetting. LetRbeacommutativedifferentialgradedk-algebra
and E be a differential graded R-algebra. The R-relative (unbounded) derived category
D (E) is a k-linear category with objects differential graded E-modules and morphisms
R
obtained from morphisms of differential graded E-modules by formally inverting all R-
relative quasi-isomorphisms, i.e. all morphisms s : L → M of differential graded E-
modules whose restriction to R is a homotopy equivalence. For example, the k-relative
derived category D (E) is exactly the usual derived category D(E) of the differential
k
graded algebra E. The R-relative derived category D (R) is just the homotopy category
R
K(R) of R. For more details on R-relative derived categories, we refer to [Kel1, Kel2].
For convenience, from now on, the term modules refers to right modules.
Remark 3.1. We also consider the R-relative bounded derived category Db (E), which
R
is by definition, the full subcategory of D (E) consisting of those objects X such that
R
there are only finitely many integers i such that H (X) 6= 0.
i
Let A be an associative algebra over a commutative ring k such that A is projective as
a k-module and R be any commutative differential graded k-algebra. Then Aop⊗ A⊗ R
k k
is a differential graded R-algebra.
We denote by R the commutative differential graded algebra k[ǫ ]/(ǫ2), where ǫ is of
i i i i
degree i and the differential d = 0. We also denote by ǫ the kernel of the augmentation
i
R → k. Let us consider the R -relative derived category D (Aop ⊗ A⊗ R ).
i i Ri k k i
Let α : X → Y be a morphism of complexes. Let us construct a complex C(α)
associated to α. As graded spaces,
C(α) := X ⊕Y[−1]
and the differential is dX α . For simplicity, we use the following diagram to repre-
0 dY[−1]
sent the complex C(α)(cid:16): (cid:17)
α
""
X ⊕ Y[−1],
Let m,p ∈ Z and f ∈ HHm(A,Ωp(A)). Recall that in Section 2.2 we defined two
≥0
lifts θ (f) and θ (f) associated to f. Now let us construct two differential graded right
1 2
(Aop ⊗ A ⊗ R )-modules C(θ (f)) and C(θ (f)) associated to θ (f) and θ (f),
k k m−p−1 1 2 1 2
respectively. As a graded right (Aop ⊗ A⊗ R )-module,
k k m−p−1
p−1
C(θ (f)) := (A⊗i+2 ⊗k[i]) Bar (Ωp(A))⊗R [p] (5)
1 ∗ m−p−1
i=0
M M
where k is considered as a right R -module, and the differential is illustrated as
m−p−1
follows,
θ1(f)
&&
Bar (A) ⊕ Bar (Ωp(A))⊗ǫ ,
∗ ∗ m−1
6
where we used the fact that
C(θ (f)) = Bar (A)⊕Bar (Ωp(A))⊗ǫ
1 ∗ ∗ m−1
as graded k-modules. Similarly, as a graded right (Aop ⊗ A⊗ R )-module,
k k m−p−1
p−1
C(θ (f)) := (A⊗i+2 ⊗k[i]) Bar (Ωp(A))⊗R [p] (6)
2 ∗ m−p−1
i=0
M M
and the differential is illustrated as follows,
θ2(f)
&&
Bar (A) ⊕ Bar (Ωp(A))⊗ǫ .
∗ ∗ m−1
Remark 3.2. Denote
f
%%
C(f) := Bar (A) ⊕ Ωp(A)⊗ǫ .
∗ m−1
Then we claim that C(f) is also a differential graded right (Aop ⊗A⊗R )-module.
m−p−1
Indeed, as a graded right (Aop ⊗A⊗R )-module,
m−p−1
C(f) ∼= (A⊗i+2[i]⊗k) (A⊗p+2[p]⊕Ωp(A)[m−1])
i6=p
M M
where (A⊗p+2[p]⊕Ωp(A)[m−1]) is a graded right (Aop ⊗A⊗R )-module defined
m−p−1
as follows,
x·ǫ := dx ∈ Ωp(A)[m−1]
m−p−1
for any x ∈ A⊗p+2[p].
Lemma 3.3. Let m,p ∈ Z and f ∈ HHm(A,Ωp(A)), then we have the following
≥0
isomorphism in D (Aop ⊗A⊗R ),
Rm−p−1 m−p−1
∼ ∼
C(θ (f)) = C(θ (f)) = C(f).
1 2
Proof. Let us prove the first isomorphism. Consider the following morphism of chain
complexes,
id s(f) : C(θ (f)) → C(θ (f))
0 id 1 2
where s(f) is the chain hom(cid:0)otopy(cid:1)defined in (3). Clearly, it is also a morphism of
differential graded right (Aop ⊗A⊗R )-modules. On the other hand, we note that
m−p−1
the morphism id s(f) is an isomorphism of dg right (Aop ⊗A⊗R )-modules with
0 id m−p−1
inverse
(cid:0) (cid:1)
id −s(f) : C(θ (f)) → C(θ (f)).
0 id 2 1
Hence we have an isomorph(cid:0)ism in D(cid:1)Rm−p−1(Aop ⊗A⊗Rm−p−1),
∼
C(θ (f)) = C(θ (f)).
1 2
7
It remains to verify the second isomorphism. Consider the following morphism of chain
complexes of right Aop ⊗A-modules,
id 0
σp:=(cid:16)0 σp(cid:17)
C(θ2(f)) b // C(f)
where
σ : Bar (Ωp(A))⊗ǫ → Ωp(A)⊗ǫ
p ∗ m−1 m−1
is the canonical surjection induced by the surjection d : Ap+2 → Ωp(A). Note that it is
p
also a morphism of differential graded (Aop⊗A⊗R )-modules. Let us consider the
m−p−1
mapping cone Cone(σ ). We claim that
p
Cone(σ ) = 0
b p
inD (Aop⊗A⊗R ), equivalently, Cone(σ )isisomorphictozerointhecategory
Rm−p−1 m−p−1 b p
K(R ). Indeed, let us write Cone(σ ) as follows,
m−p−1 p
b
b θ2(f)
**
Cone(σ ) ∼= Bar (A)[1] ⊕ Bar (A) ⊕ (Bar (Ωp(A)),d )⊗ǫ [1]
p ∗ ∗ ∗ p m−p−1
77
f
b
Then we have the following homotopy,
sp
Cone(σ ) //Cone(σ )[−1]
p b p
where s is defined as follows
p b b
s0 0 0
sp := 0 s0 f⊗1
b 0 0 sp
(cid:18) (cid:19)
where s is defined in Remark 2.1 and
p b
a f(a )⊗1 if r = m+1,
1 2,m+1
(f ⊗1)(a ) =
1,r
(0 otherwise.
Clearly, s is a well-defined homotopy of differential graded R -modules, so that the
p m−p−1
identity on Cone(ǫˆ ) is homotopy equivalent to 0.
p
b
s ◦d +d ◦s = id .
p Cone(σp) Cone(σp) p Cone(σp)
b b b
So we have
b b
Cone(σ ) = 0
p
in D (Aop ⊗ A ⊗ R ) and thus σ is an R -relative quasi-isomorphism.
Rm−p−1 m−p−1 p m−p−1
Therefore, we have the following isomorphisbms in D (Aop ⊗A⊗R ),
Rm−p−1 m−p−1
b
∼ ∼
C(θ (f)) = C(θ (f)) = C(f).
1 2
8
3.2 A derived interpretation
First, let us recall the bullet product • defined in [Wang2]. Forany m ∈ Z and p ∈ Z ,
>0 ≥0
denote
Cm(A,Ωp(A)) := Hom (A⊗m,Ωp(A)).
k
Then we have a double complex C∗(A,Ω∗(A)) with the horizontal map δ induced from
the differential of the bar resolution, and the vertical map zero.
Give f ∈ Cm(A,Ωp(A)) and g ∈ Cn(A,Ωq(A)), denote
d((f ⊗id⊗q)(id⊗i−1⊗g ⊗id⊗m−i)⊗1) if 1 ≤ i ≤ m,
f • g :=
i (d((id⊗−i⊗f ⊗id⊗q+i)(g ⊗id⊗m−1)⊗1) if −q ≤ i ≤ −1,
where d is the differential of the bar resolution Bar (A), and
∗
m q
f •g := (−1)p+q+(i−1)(q−n−1)f • g + (−1)p+q+i(p−m−1)f • g
i −i
i=1 i=1
X X
Then we define
[f,g] := f •g −(−1)(m−p−1)(n−q−1)g •f. (7)
Note that
[f,g] ∈ Cm+n−1(A,Ωp+q(A)).
Then from Proposition 4.6 in [Wang2], it follows that [·,·] defines a differential graded
Lie algebra structure on the total complex
Cm(A,Ωp(A))
m∈Z>M0,p∈Z≥0
and thus [·,·] defines a graded Lie algebra structure on the cohomology groups
HHm(A,Ωp(A)).
m∈Z>M0,p∈Z≥0
We also recall that the bullet product • has the following property (cf. [Wang2,
Proposition 4.9]): for any f ∈ HHm(A,Ωp(A)) and g ∈ HHn(A,Ωq(A)),
f ∪g −(−1)(m−p)(n−q)g ∪f = (−1)m−pδ(g •f) = −(−1)(m−p−1)(n−q)δ(f •g). (8)
In the following, we will give a derived interpretation of Identity (8). Now let us fix two
elements f ∈ HHm(A,Ωp(A)) and g ∈ HHn(A,Ωq(A)), where m,n ∈ Z and p,q ∈ Z .
>0 ≥0
Let us consider the following chain complex (denoted by C (f,g)) associated to f and g,
1
θ2(g)
θ1(f) $$ ""
Bar (A) ⊕ Bar (Ωp(A))⊗ǫ ⊕ Bar (Ωq(A))⊗ǫ ⊕ Bar (Ωp+q(A))⊗ǫ
∗ ∗ m−1 ∗ n−1 ∗ m+n−2
;;;;
θ1(Ωq(θ1(f)))
θ2(Ωp(θ2(g)))
(9)
9
it is straightforward to verify that this is a well-defined chain complex of A-A-bimodules.
Weneedtoconstruct thefollowing chaincomplex (denotedbyC (f,g))ofA-A-bimodules
2
associated to f and g.
θ2(g)
(10)
θ1(f) $$ ""
Bar (A) ⊕ Bar (Ωp(A))⊗ǫ ⊕ Bar (Ωq(A))⊗ǫ ⊕ Ωp+q(A)⊗ǫ .
∗ ∗ m−1 ∗ n−1 m+n−2
<<<<
Ωq(θ1(f))
Ωp(θ2(g))
We will also construct another chain complex (denoted by C (f,g)) associated to f and
3
g.
θ1(g)
(11)
θ1(f) $$ ""
Bar (A) ⊕ Bar (Ωp(A))⊗ǫ ⊕ Bar (Ωq(A))⊗ǫ ⊕ Ωp+q(A)⊗ǫ
∗ ∗ m−1 ∗ n−1 m+n−2
99::<<
Ωq(θ2(f))
Ωp(θ2(g))
f•g
From Identity (8), it follows that the complex C (f,g) above is indeed a well-defined
3
complex of (Aop ⊗A)-modules.
Remark 3.4. Each of the three complexes C (f,g),C (f,g) and C (f,g) can be consid-
1 2 3
ered as differential graded right (Aop ⊗A⊗R)-modules, where
R := R ⊗R
m−p−1 n−q−1
is the tensor algebra of the differential graded algebras R and R . Moreover
m−p−1 n−q−1
we have the following.
Lemma 3.5. For any f ∈ HHm(A,Ωp(A)) and g ∈ HHn(A,Ωq(A)), we have
∼ ∼
C (f,g) = C (f,g) = C (f,g)
1 2 3
in the R-relative bounded derived category Db (Aop ⊗A⊗R), where
R
R := R ⊗R .
m−p−1 n−q−1
Proof. Let usprove thefirst isomorphism. Notethat we have thefollowing morphism
of dg (Aop ⊗A⊗R)-modules,
id 0 0 0
0 id 0 0
σp+q:= 0 0 id 0
C (f,ge) 0 0 0 σp+q//C (f,g).
1 2
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