Internat. J. Math. & Math. Sci.
Vol. 22, No. 4 (1999) 807–815
S 0161-17129922807-4
© Electronic Publishing House
THE RELATIVE DIHEDRAL HOMOLOGY
OF INVOLUTIVE ALGEBRAS
Y. Gh. GOUDA
(Received 23 October 1996 and in revised form 19 May 1997)
Abstract. Let f : A → B be a homomorphism of involutive algebras A, B. The purpose of
this paper is to define a free involutive algebra resolution of algebra B over f and use it
to define and study the relative dihedral homology.
Keywords and phrases. Dihedral homology, free involutive algebra, relative dihedral homology.
1991 Mathematics Subject Classification. Primary 55N91; Secondary 55P91, 55Q91.
1. Introduction. Let A, B be involutive algebras (an involution ∗ is an anti-automorphism of degree zero and order 2) and let f : A → B be a homomorphism. Our first aim
is to find a free involutive algebra resolution R of algebra B over the homomorphism
i
π
R →
B, where i is an inclusion and π is a quasi-isomorphism. The second aim
f : A →
is to define the relative dihedral homology as
f R
,
ᏴᏰ• A → B = Ᏼ• (1.1)
A + [R, R] + Im (1 − r where [R, R] is the commutant of algebra R, r is the involution on R, and study its
main properties.
First, we recall some definitions and facts from [4, 5]. Let A be an associative algebra
over a field k(k = R or C). Define the complex C(A) = (C• (A), Ꮿ• ), where Cn (A) =
A ⊗ · · · ⊗ A is the tensor product of algebra A(n + 1 times) and Ꮿn : Cn (A) → Cn−1 (A)
is the boundary operator
n−1
Ꮿn a0 ⊗ · · · ⊗ an =
(−1)i a0 ⊗ · · · ⊗ ai ai+1 ⊗ · · · ⊗ an + (−1)n an a0 ⊗ · · · ⊗ an+1 .
i=0
(1.2)
It is well known that Ꮿn−1 Ꮿn = 0, that is, the complex C(A) is a chain complex. This
complex is called the Hochschild (simplicial) complex and its homology is called the
Hochschild homology (ᏴᏴ• (A)). If A is a unital involutive algebra, then on the complex C(A), one acts by the operators tn , rn : Cn (A) → Cn (A) by means of
tn a0 ⊗ · · · ⊗ an = (−1)n an ⊗ a0 ⊗ · · · ⊗ an−1 ,
(1.3)
∗
∗
= ±1.
rn a0 ⊗ · · · ⊗ an = (−1)n(n+1)/2 a∗
0 ⊗ an ⊗ · · · ⊗ a 1 ,
Consider the quotient complex C Ᏸn (A) = Cn (A)/ Im (1 − tn ) + Im (1 − rn ) of a complex Cn (A). Following [3] the dihedral homology of algebra, A is the homology of the
complex C Ᏸ• (A).
808
Y. Gh. GOUDA
2. Free involutive algebra resolution. In this part, we discuss the existence of the
∞
free involutive algebra resolution. Let E = n=0 En be a graded involutive vector space
over a field K. Suppose that R is a differential graded (in short DG ) involutive K-algebra
and let RE = R ∗ Tk (E) be the free product of algebras, where Tk (E) = j≥0 E ⊗j is
the tensor algebra over K. We define an involution on algebra RE to be the unique
anti-automorphism on RE which restricts to the given involution on R and E (this is
enough thanks to the universal property of the tensor algebra and the free product).
The product in RE is given by
r1 e1 · · · rn en rn+1 · r1 e1 · · · rk ek rk+1 = r1 e1 · · · rn en (rn+1 r1 )
e1 · · · rk ek rk+1 ,
ri , rj ∈ R,
ei , ej ∈ Tk (E),
(r ∗ )∗ = (e∗ )∗ = e.
(2.1)
Definition 2.1. Let f : R1 → R2 be a homomorphism of involutive differential
graded K-algebras. An algebra R2 is a free algebra over the homomorphism f if there
exists an isomorphism α : R1 E R2 , where E is an involutive differential graded
vector space with the following commutative diagram:
f
/ R2
JJ
α
JJ
JJ
i
$
R1 J
J
(2.2)
R1 E,
where i is the inclusion map.
Lemma 2.2. Let f : A → B be a morphism of involutive K-algebra. Then there exists
an involutive differential graded algebra R = i=0 Ri with the following properties
(i) π is surjection and the following diagram is commutative
R
~?
~
i ~
π
~~
~~
/ B,
A
f
(2.3)
where i is an inclusion map.
There is an amorphism j : R → A such that j ◦ i = 1A .
(ii) π is quasi-isomorphism, i.e., π• : Ᏼ• (R) → Ᏼ• (B) = B, where B is a differential
graded algebra,
(B)i =

B,
i = 0,
0,
i > 0, and the differential ∂ B = 0.
(2.4)
(iii) The involutive DG algebra R is free over the homomorphism i : A → R.
Definition 2.3. The involutive DG algebra which satisfies the conditions (i), (ii),
and (iii) of Lemma 2.2 is called a free involutive algebra resolution of algebra B over f .
809
THE RELATIVE DIHEDRAL HOMOLOGY . . .
Proof of Lemma 2.2.
First step. We construct a commutative diagram of involutive algebra
(0)
R
|>
i0 ||
|
π0
||
||
/ B,
A
f
(2.5)
where R (0) is free over the homomorphism i0 : A → R (0) , π0 an involutive surjection.
Define Ati = AE(ti ), where E(ti ) is an involutive vector space generated by {ti },
or generated by the family {ti , ti∗ }. The automorphism ∗ : E(ti ) → E(ti ) is given as
(0)
follows: ∗(ti ) = (ti∗ ), ∗(ti∗ ) = ti . We choose a system {Ꮿi } of generators in algebra B.
This family is assumed to be closed under an involutive on B.
(0)
(0)
(0)
Now, let R (0) = Ati , where ti is equivalent to the generator {Ꮿi } of algebra B,
(0)
(0)
(0)
and suppose that βi = ti or (ti )∗ . We may define π0 using the universal property
of R (0) . Let π0 be the unique homomorphism of involutive algebras R (0) → B which
(0)
(0)
restricts to f on A and sends ti to Ꮿi .
(0)
Since i0 : A → Ati is an inclusion map, i0 (a) = a, i0 is an involutive algebra
homomorphism and π0 i0 (a) = π0 (a) = f (a). Hence, diagram (2.5) is commutative
and π0 is surjective.
Let j0 : R (0) → A be the unique homomorphism involutive algebra restricting to
(0)
the identity on A and mapping the tn to zero. R (0) is a DG involutive k-algebra:
(0) (0)
(0)
(0)
(0)
(R )0 = R , i = 0, (R )i = 0, i > 0, ∂ R Bij = 0. The algebra R (0) is free over the
homomorphism i0 : A → R (0) since R (0) = At (0) .
Second step. We construct the second commutative diagram
(1)
R
|>
i1 ||
|
π1
||
||
/ B,
A
f
(2.6)
where R (1) is a free algebra over i1 and π1 is an involutive surjection. Choose a system
(1)
(1)
Ꮿj of generators of ker π0 which is closed under the involution. Let tj be indeter(1)
(0)
(1)
(0)
minates which are in bijection with the Ꮿj . Define R (1) = Ati , tj , where ti
(1)
βj
(1)
tj
is
(1)
(tj )∗ .
defined above. Suppose that
denotes
or
The homomorphism π1 is
defined to be the unique homomorphism of involutive algebra R (1) → B restricting
(1)
to π0 on R (0) and sending tj to 0. We can see, from the above discussion, that the
(0)
homomorphism π1 can be defined as π0 and that π1 is surjective since π1 (β1 ) = Ꮿi ,
(1)
(0)
(1)
π1 (βj ) = 0. The homomorphism i1 : A → Ati , tj is inclusion. The diagram (2.6)
is commutative since (π1 i1 )(a) = π1 (a) = f (a). The homomorphism j1 is defined to
be unique homomorphisms: R (1) → A, of involutive algebras restricting to identity on
(1)
(0)
(1)
A and mapping ti to zero. The algebra R (1) = Ati , tj is free over i1 . Finally, we
have a differential graded algebra
R (1) = R (1) 0 ⊕ R (1) 0 ⊕ · · · ,
(1)
deg βi
= 0,
(1)
deg βj
= 1.
(2.7)
810
Y. Gh. GOUDA
Note that the algebra R (1) also has a universal property with respect to derivations
(not only homomorphism). This property should be used to define the differential.
(1)
The differential ∂ R of R (1) is the unique derivation on R (1) satisfying the graded
Leibniz rule and commuting with the involution which restricts to zero on R (1) and
(1)
(1) (0)
(1) (0)
(1)
(1)
(1)
sends tj to Ꮿj . So, ∂ R βi = 0, ∂ R βi = Ꮿj ∈ ker π0 , ∂iR = 0, i > 1.
In the same manner, we can construct the commutative diagram
(2)
R
|>
i2 ||
|
π2
||
||
/ B,
A
f
(0)
(1)
(2.8)
(2)
where R (2) = Ati , tj , tk is a DG algebra, free over i2 , R (2) = (R (2) )0 ⊕ (R (2) )1 ⊕
(R (2) )2 ⊕ · · · ,
(0)
deg βi
(1)
= 0, deg βj
(2)
= 1, deg βk = 2, the differential of algebra R (2) is
also defined by using a universal property and, hence, ∂0R
R (2)
(2)
(2)
(2)
R (2)
(0)
βi
= 0, ∂1R
(2)
(0)
βj
(1)
= Ꮿj ,
= 0, i > 2.
∂1 βj = Ꮿk , ∂i
Consequently, we can construct an involutive algebra R (i) , i ≥ 0 with the following
commutative diagram:
(0)
R
=
P0
P1
/ R (1)
|
i0 ||
|
π0
|
|
||
/B
A
/ ...
Pn−1
/ R (n)
πi
...
B
P0
/ ...
(2.9)
πn
...,
B
where πi is an involutive surjection, i ≥ 0, in = Pn−1 ◦ · · · ◦ P0 ◦ i0 is an inclusion map
from A to R (n) , Pi is also an inclusion map from
(0) (1)
(i)
(0) (1)
(i) (i+1)
Pi : A tm
, tm1 , . . . , tm
, tm1 , . . . , tm
, tmi +1 .
to
A tm
(2.10)
0
0
i
i
(0)
(1)
Define in = qn ◦· · ·◦qi ◦j0 , where qn is the projection of the map from Atm0 , tm1 , . . . ,
(0) (1)
(i) (i+1)
(n)
(i)
tmi on Atm0 , tm1 , . . . , tmi , tmi+1 . The diagram (2.9) is commutative since in+1 (βi ) =
(n)
πn (βi ) = 0, n ≥ 0. Define R = lim Rn , π = lim πn , i = lim in , j = lim jn . Then the DG
algebra R satisfies the items of Lemma 2.2 since
(1) π = lim in is an involutive surjection, the diagram
(2)
R
=
||
|
|
π2
||
||
/B
A
f
i2
(2.11)
is commutative since i(a) = a, π (a) = f (a).
(2) π is a quasi-isomorphism of DG algebras
(R)0 o
π0
B o
∂0R
(R)1 o
∂1R
... o
π1
0o
... o
R
∂n
(R)n o
R
∂n+1
...
(2.12)
πn
0o
...,
THE RELATIVE DIHEDRAL HOMOLOGY . . .
811
where ∂iR = lim ∂iR , (R)0 = ker (π )0 = B, Im ∂ R = ker ∂0R , i.e., Ᏼ0 (R) = B, Ᏼi (R) = 0.
(3) The DG involutive algebra R is free over the homomorphism i : A → R since
R = E, E is an involutive vector space generated by the system
(0) (1)
(n)
ti0 , ti1 , . . . , tin , . . . .
(2.13)
3. The relative dihedral homology. In this part, we define the relative dihedral
homology and study its properties. Let f be a morphism of involutive algebras A and
B over a field K with characteristic zero. Let RfB be a free involutive algebra resolution
of algebra B over f and, for r1 , r2 ∈ RfB , let [r1 , r2 ] = r1 r2 −(−1)|r1 ||r2 | r2 r1 , where |ri | =
deg ri , i = 1, 2. Let Ꮿ = [RfB , RfB ] be the linear space generated by [r1 , r2 ], r1 , r2 ∈ RfB .
It is clear that, Ꮿ = [RfB , RfB ] is a K-submodule of a K-module RfB . We construct the
complex ([RfB , RfB ] + Im (1 − r )), where r (P ) = (−1)|P |(|P |−1)/2 P ∗ , ∗ is an involutive
on RfB , = ±1. It is clear, from the definition of RfB , that Im (1−r ) is a subcomplex of
RfB . We have
∂[r1 r2 ] = r1 r2 − (−1)|r1 ||r2 | r2 r1
= ∂r1 r2 + (−1)|r1 | r1 ∂r2 − (−1)|r1 ||r2 | ∂r2 r1 + (−1)|r2 | r2 ∂r1
= ∂r1 r2 − (−1)|r2 |(|r1 |+1) r2 ∂r1 + (−1)|r1 | r1 ∂r2 − (−1)|r1 |(|r2 |+1) ∂r2 r1
= [∂r1 , r2 ] + (−1)|r1 | [r1 , ∂r2 ], |∂ri | = |ri | − 1,
(3.1)
i = 1, 2.
Then RfB , RfB is a subcomplex in RrB . Therefore, the chain complex of the K-module
B B
Rf , Rf + Im (1 − r ) is a subcomplex of RfB .
Definition 3.1. Let f : A → B be an involutive K-algebra (char K = 0) homomorphism, RfB be a free involutive algebra resolution of algebra B over f . Then the relative
dihedral homology is defined as follows:


RfB
f
.

(3.2)
ᏴᏰi A →
B = Ᏼi RfB , RfB + Im (1 − r )
The main properties of the relative dihedral homology are submitted in Theorems 3.2,
3.6, and 3.7.
Theorem 3.2. Let A be an involutive algebra. Then ᏴᏰi (A → 0) = ᏴᏰi−1 (A),
where ᏴᏰi (A) is the dihedral homology of k-algebra A (char (k) = 0).
Proof. To do this, we need the following definition and lemmas.
Definition 3.3. The K-algebra At, generated by the elements a0 ta1 t · · · tan ,
n ≥ 0, can be considered as an involutive DG algebra by requiring that the morphism
A → At is a morphism of involutive differential graded algebras (A is viewed as a
DG algebra concentrated in degree 0) and that deg t = 1, ∂t = 0, and t ∗ = t.
Lemma 3.4. The algebra At is splitable and is a free involutive algebra resolution
of the algebra B = 0 over the homomorphism A → 0.
812
Y. Gh. GOUDA
Proof. Define the following chain complex
∂
∂
∂
∂
∂
A ← AtA ← AtAtA ← · · · ← At · · · tA ← · · · ,
(3.3)
where At · · · tA (t-n-times) is a K-module and the boundary operator ∂ is given by
n−1
(−1)i a0 ta1 t · · · tai (∂t)ai+1 t · · · tan
∂ a0 ta1 t · · · tan−1 tan =
i=0
=
n−1
(−1) a0 ta1 t · · · t ai ai+1 t · · · tan .
(3.4)
i
i=0
Note that the differential ∂ in at is equivalent to the operator Ꮿn : Cn (A) → Cn−1 (A)
(see [4]), defined by
n−1
Ꮿn a0 ⊗ · · · ⊗ an =
(−1)i a0 ⊗ · · · ⊗ ai ai+1 ⊗ · · · ⊗ an .
(3.5)
i=0
Following [4], the complex (Cn (A), Ꮿn ) is splitable and so the complex At is also
splitable, that is, Ᏼ• (At) = 0. Therefore, the algebra At is a free involutive algebra
resolution of the algebra B = 0 over the homomorphism A → 0.
Lemma 3.5. The complex At/[A, At] is the standard simplicial (Hochschild) complex.
Proof. Consider the factor complex At/[A, At]. The complex At/[A, At]
is generated by the elements a0 ta1 t · · · an−1 t, since a0 ta1 t · · · tan−1 tan = an a0 ×
ta1 t · · · tan−1 t(mod[A, At]). The action of the differential ∂ on the complex
At/[A, At] is given by
n−1
∂ a0 ta1 t · · · tan−1 tan =
(−1)i a0 ta1 t · · · t(ai ai+1 )t · · · an−1 tan t
i=0
+ (−1)n a0 ta1 t · · · an−1 tan
=
n−1
(3.6)
i
(−1) a0 ta1 t · · · t(ai ai+1 )t · · · an−1 tan t
i=0
+ (−1)n an a0 ta1 t · · · an−1 t.
Consider the complex
id
Ꮿ
Ꮿ
Ꮿ
Ꮿ
A ← A ← A⊗2 ← · · · ← A⊗n ← · · · ,
(3.7)
where Ꮿ is the differential in the standard Hochschild complex (see [4]). Since the
space (At/[A, At])n+1 identifies with the space
A⊗n+1 : a0 ta1 · · · tan t → a0 ⊗ a1 ⊗ · · · ⊗ an ,
(3.8)
and the differential in At/[A, At] identifies with the differential in the standard
Hochschild complex, At/A + [A, At] is the Hochschild (simplicial) complex of algebra A.
813
THE RELATIVE DIHEDRAL HOMOLOGY . . .
Now, we prove Theorem 3.2. Consider the factor complex: At/[At, At]+Im (1−
r ), such that
a0 ta1 t · · · tan−1 t = (−1)n an ta0 ta1 t · · · an−1 t,
∗
∗
∗
a0 ta1 t · · · tan−1 t = (−1)n(n+1)/2 ta∗
n tan−1 · · · ta1 ta0
= (−1)
n(n+1)/2
(3.9)
∗
∗
ta∗
0 tan t · · · ta1 t,
where = ±1, deg a0 ta1 t · · · tan−1 t = n, deg(an t) = 1, deg(a∗
n ) = 0, deg a0 t · · · an t =
n+1. The dihedral homology of t is the homology of the factor complex At/[At,
At] + Im (1 − r ). By factoring At, first by the subcomplex A ← 0 ← 0 ← · · · and
then by the subcomplex [At, At]+Im (1−r ), we get a homomorphism C Ᏸ• (A →
0) → C Ᏸ•−1 (A), which induces an isomorphism in the dihedral homology groups
ᏴᏰi (A → 0) → ᏴᏰ−1 (A).
Theorem 3.6. Let f : A → B be a homomorphism of involutive algebras over a field
f
B) does not depend on
K (char K = 0). Then the relative dihedral homology ᏴᏰi (A →
the choice of the resolution.
Proof. The homomorphism f induces a homomorphism of chain complexes
f• : C Ᏸ• (A) → C Ᏸ• (B),
(3.10)
where C Ᏸ• (A) is the dihedral complex. Consider the diagram
RfB
?
i
A
f
π
(3.11)
/ B,
where RfB is defined above, i is the inclusion map. The idea of the proof is to show
that the cone of the map i is quasi-isomorphic to an arbitrary category (see [2]), to the
complex: RfB / RfB , RfB + Im (1 − r ). Since
Ᏼi RfB
=

B,
i = 0,
0,
i > 0,
(3.12)
then the isomorphism π• : C Ᏸ• (RfB ) → C Ᏸ• (B) induces an isomorphism of the homology of these complexes. Since i• : C Ᏸ• (A) → C Ᏸ• (RfB ) is an inclusion, then
M(i• ) ≈ C Ᏸ• (RfB )/ C Ᏸ(A), where M(i• ) is the cone of the map i (see [6]).
Note that the symbol ≈ always denotes a quasi-isomorphism. It is clear, from the
above discussion, that the following diagram is commutative
C Ᏸ• RfB
r8
rrr
r
π•
r
r
rrr
/ C Ᏸ• (B),
CD• (A)
i•
f•
(3.13)
814
Y. Gh. GOUDA
and, hence, M(f• ) ≈ C Ᏸ• (RfB )/ C Ᏸ• (A). Following [1], we have CC• (RfB )/CC• (A) ≈
2
RfB /A + [RfB , RfB ] and by using the spectral sequence Eij
= Ᏼ• (Z/2, ᏴCj (RfB )) =
ᏴDi+j (RfB ), we have
C Ᏸ• RfB
CᏰ
• (A)
≈
RfB
.
A + RfB , RfB + Im (1 − r )
(3.14)
f
B) does not depend on the
So, M(f• ) ≈ RfB /A + [RfB , RfB ] + Im (1 − r ). Then ᏴᏰi (A →
choice of RfB .
Theorem 3.7. Let A, B, and C be involutive algebras. Then the following sequence
f
g
B →
C induces the long exact sequence of the relative dihedral homology
A →
f g◦f g f B → ᏴᏰi A →
C → ᏴᏰi B →
C → ᏴᏰi−1 A →
B → · · · .
· · · → ᏴᏰi A →
(3.15)
Proof. In Theorem 3.6, it has been proved that any homomorphism f : A → B of
involutive algebras in an arbitrary category is equivalent to an inclusion i : A → RfB
RB
∼
f
?
i 

 f
/ B.
A
(3.16)
g
f
B →
C of involutive algebras, we have the folThen, for an arbitrary sequence A →
lowing complex
A?
?
i
/ RfB
??
??
??
B ?
?
i
/ RgC
??
??
??
(3.17)
g
f
B
C.
Consider the following sequence of mapping cones
0 → M(i• ) → M i• → M i• 0i• → 0.
(3.18)
In general, the sequence (3.18) is not exact. In fact, the composition of two morphisms
will be non zero. However, the cone over the morphism M(i• ) → M(i• ) is canonically
homotopy equivalent to M(i• 0i• ). So, we get the following long exact sequence of the
relative dihedral homology
g◦f
f
B →
ᏴᏰi A →
C
· · · →
ᏴᏰi A →
g
f
B →
··· .
C →
ᏴᏰi−1 A →
→
ᏴᏰi B →
(3.19)
THE RELATIVE DIHEDRAL HOMOLOGY . . .
815
Acknowledgement. I would like to express my thanks to L. Debnath and the
referee for their valuable comments and help.
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Gouda: Department of Mathematics, Faculty of Science, South Valley University,
Aswan, Egypt