IJMMS 27:7 (2001) 429–438
PII. S0161171201000849
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
REFLEXIVE AND DIHEDRAL (CO)HOMOLOGY
OF A PRE-ADDITIVE CATEGORY
YASIEN GH. GOUDA
(Received 18 June 1998)
Abstract. The group dihedral homology of an algebra over a field with characteristic zero
was introduced by Tsygan (1983). The dihedral homology and cohomology of an algebra
with involution over commutative ring with identity, associated with the small category,
were studied by Krasauskas et al. (1988), Loday (1987), and Lodder (1993). The aim of this
work is concerned with dihedral and reflexive (co)homology of small pre-additive category.
We also define the free product of involutive algebras associated with this category and
study its dihedral homology group. Finally, following Perelygin (1990), we show that a
small pre-additive category is Morita equivalence.
2000 Mathematics Subject Classification. 55N91, 55P91, 55Q91.
1. Preliminaries. Suppose that ∆Ᏸ is a small category, with objects, the set {[0], [1],
. . . , [n], . . .}, and the following family of morphisms δin : [n] → [n − 1], 0 ≤ i ≤ n,
j
σn : [n] → [n + 1], 0 ≤ i ≤ n, τn : [n] → [n], ρn : [n] → [n] such that
j
δn+1 δin = δin+1 δi−1
n ,
j
i < j,
j−1
i
= σni σn+1 ,
σn σn+1
i ≤ j,

i−1 j


σn−2 δn−1 , i ≤ j,


j
σn+1 δ1n = Id,
i = j, j + 1,




σ i δi−1 , i > j,
n−2 n−1
τn δin = δi−1
n τn−1 ,
j
j−1
τn σn = σn
0 ≤ i ≤ n,
τn+1 ,
τn
n+1
= 1n ,
(1.1)
2
ρn
= 1,
τn ρn = ρn τn−1 .
0 ≤ j ≤ n,
Definition 1.1. The category ∆Ᏸ is called a dihedral category. Note that the catej
gory generated by only the morphisms δin and σn is called a simplicial category and
j
is denoted by ∆, the category generated by δin , σn , and τn is a cyclic category and is
denoted by ∆C (see [6]), and the category generated by the family of morphisms δin ,
j
σn , and ρn is called a reflexive category and is denoted by ∆R.
Definition 1.2 (see [3]). Let k be a commutative ring with identity and involution.
An algebra over k associated with the category ∆Ᏸ(∆R) is an algebra with identity
j
j
generated by the morphisms δin , σn , τn , and ρn (δin , σn , ρn ).
430
YASIEN GH. GOUDA
Definition 1.3. For an arbitrary category C, following [3], for case of presentation,
a simplicial object in the category C is a functor Ᏺ : ∆op → C (the category ∆op is the
inverse of ∆).
Definition 1.4. Following [2] (see also [6]), for an arbitrary category C, the dihedral (reflexive) object in C is a functor Ᏺ : ∆D op → C, (Ᏺ : ∆᏾op → C). If we drop the
j
morphism ρn from the group family of morphisms (δin , σn , τn , ρn ), we get a cyclic
object of an arbitrary category C (see [5]). Suppose that Ᏺ([n]) = Xn , Ᏺ(δin ) = din ,
j
j
Ᏺ(σn ) = sn , Ᏺ(τn ) = tn , Ᏺ(ρn ) = rn . We write the dihedral (reflexive) object by the famj
j
j
ily (Xn , din , sn , tn , rn ) (Xn , din , sn , rn ). We can easily check that the morphisms din , sn , tn ,
and rn satisfy relations (1.1).
Definition 1.5. Let k be a commutative ring with identity and involution and let C
be a category of k-modules. The dihedral k-module in C is defined to be the dihedral
j
objects (Xn , din , sn , tn , rn ) in the category C.
2. The reflexive and dihedral homology of pre-additive category. In this section,
we define the dihedral k-module associated with a pre-additive category and study its
(co)homology.
Definition 2.1. Let k be a commutative ring with identity and involution. Following
[2], the k-category A with an involution is defined to be a small pre-additive category
with objects the k-modules of set morphisms A(i, j), where, i, j are in A, and the
bilinear maps A(i, j)xA(j, k) → A(i, k), as morphisms. Suppose that, for all objects
i, j ∈ A, there exists a k-linear map ∗ : A(i, j) → A(j, i), such that ∗2 : A(i, j) → A(i, j).
Define the family M = {Mn }n≥0 of k-modules and k-morphisms as follows:
M0 =
⊕ A i0 , i0 , . . . , Mn =
i0 ∈|A|
⊕
i0 ,i1 ,...,in ∈|A|
A i0 , i0 ⊗ A i1 , i2 ⊗ · · · ⊗ A in , i0 .
k
k
k
(2.1)
j
On the family M = (Mn ), define the morphisms din sn tn rn as follows:
j
din : Mn → Mn−1 ,
sn : Mn → Mn−1 ,
rn , tn : Mn → Mn ,
(2.2)
such that
din a0 ⊗ · · · ⊗ an = din a0 ⊗ · · · ⊗ an + (−1)n an a0 ⊗ · · · ⊗ an−1 ,
n−1
(−1)k a0 ⊗ · · · ⊗ ak ak+1 ⊗ · · · ⊗ an ,
din a0 ⊗ · · · ⊗ an =
j
sn a0 ⊗ · · · ⊗ an
k=0
= a0 ⊗ a1 ⊗ · · · ⊗ ai ⊗ 1 ⊗ ai+1 ⊗ · · · ⊗ an ,
(2.3)
tn a0 ⊗ · · · ⊗ an = (−1)n an ⊗ a0 ⊗ · · · ⊗ an−1 ,
∗
∗
rn a0 ⊗ · · · ⊗ an = α(−1)n(n+1)/2 a∗
0 ⊗ an ⊗ · · · ⊗ a 1 ,
α = ±1,
where a∗
i ’s are the image of the elements ai (0 ≤ i ≤ n) under the involution ∗.
Clearly, the module M = {Mn } under the last morphisms is a dihedral k-module. Now,
REFLEXIVE AND DIHEDRAL (CO)HOMOLOGY . . .
431
we define the dihedral homology group. Suppose that M = (Mn ) = C p,q,r , p, q, r > 0,
and consider the complex (C p,q,r , δi ), i = 1, 2 (see [3]), where
δ1 : C p,q,r → C p−1,q,r ,
δ2 : C p,q,r → C p,q−1,r ,
(2.4)
are defined by
δ1 =


1 − Tn ,
n = 1 (mod 2),

Nn = 1 + Tn + · · · + T n ,
n
δ2 =
n = 0 (mod 2),
(2.5)


bn+1 = (−1)i din+1 ,
n = 0 (mod 2),

−b = (−1)i di ,
n = 1 (mod 2).
Clearly, by definition, δi · δi = 0, i = 1, 2. The complex (C p,q,r , δi ) can be illustrated
by Tsygan’s bicomplex C(M) (see [5]):
..
.
..
.
−b
b
C 0,2,2
1−T
1−T
N
C 1,1,1
1−T
C 1,0,0
−b
C 2,2,2
1−T
N
C 2,1,1
1−T
C 2,0,0
N
···
(2.6)
C 3,1,1
N
···
N
···,
−b
b
N
C 3,2,2
−b
b
−b
b
C 0,0,0
C 1,2,2
..
.
b
−b
b
C 0,1,1
..
.
1−T
C 3,0,0
n
where the morphisms b, −b : C p,q,r → C p,q−1,r are given by b = i=0 (−1)i din , −b =
n−1
i i
n
n
i=0 (−1) dn , T = (−1) tn , N = 1 + Tn + · · · + Tn . Following [5], the homology of the
bicomplex (2.6) gives the cyclic homology group: ᏴCn (M) = Ᏼn (C(M)). Following [3],
if we act by the group Z/2 on the bicomplex (2.6): on the column 2 ( > 0) by means
of the automorphism
(−1)n(n+1)/2+ rn = (−1) Rn ,
where Rn a0 ⊗ · · · ⊗ an = (−1)n(n+1)/2 rn ,
(2.7)
and on the column 2 + 1 by means of the automorphism (−1)n(n−1)/2++1 rn =
(−1) Rn Tn , we get the tricomplex (C p,q,r , δi ), i = 1, 2, 3. The differentials δ1 , δ2 are
defined in (2.5) and δ3 : C p,q,r +1 → C p,q,r is defined by


n = 0 (mod 4),
(−1)n 1 + (−1) Rn ,





(−1)n+1 1 + (−1)+1 R T , n = 1 (mod 4),

n n
(2.8)
δ3 =

n
+1

n = 2 (mod 4),
(−1) 1 + (−1) Rn ,






(−1)n+1 1 + (−1)+1 Rn Tn , n = 3 (mod 4), Rn = (−1)n rn .
Following [4], the dihedral homology of the module M(α ᏴᏰ(M)) is the homology of
the complex (C p,q,r , δi ), i = 1, 2, 3, α = ±1.
432
YASIEN GH. GOUDA
Definition 2.2. The dihedral homology of a k-category A with involution is the dihedral homology of the associated dihedral k-module M={Mn } : α ᏴᏰn (A) =α ᏴᏰn (M),
α = ±1.
Definition 2.3. The reflexive homology of a k-category A with an involution is the
reflexive homology of k-module {Mrefl }:
α
Ᏼ᏾n (A) = α Ᏼ᏾n Mrefl ,
α ± 1,
(2.9)
where Mrefl is the reflexive k-module M = {Mn }. Similarly, if we take the cyclic kmodule Mcycl , we obtain the cyclic homology of the k-category A (see [6]): ᏴCn (A) =
Ᏼ᏾n (Mcycl ). Following [3, 4], the dihedral (reflexive) homology of the dihedral (reflexive) module M can be considered as derived functor
α
op
ᏴᏰn (M) = Tork[∆D]
n
kᏰ · M ·
α
op
Ᏼ᏾n (M) = Tork[∆R]
n
k᏾ · M ·
,
(2.10)
where kᏰ (k᏾ ) is a trivial dihedral (reflexive) k-module, k[∆D]op (k[∆R]op ) is the algebra associated with the dihedral (reflexive) category.
Note that (see [3]) the dihedral (reflexive) homology is considered as the hyperhomology of the group Z/2 with coefficients in Tsygan bicomplex (simplicial (Hochschild)
complex). The relations between the cyclic and the dihedral homology and also the
reflexive and the dihedral homology of pre-additive category are given by the following
assertions.
Theorem 2.4. Let k be a commutative ring and let A be a k-category with an involution. Then there exist the following exact sequences
· · · → −α ᏴᏰn (A) → ᏴCn (A) → +α ᏴᏰn (A) → −α ᏴᏰn−1 (A) → · · · ,
· · · → +α Ᏼ᏾n (A) → +α ᏴᏰn (A) → −α ᏴᏰn−2 (A) → +α Ᏼ᏾n−1 (A) → · · · .
(2.11)
Proof. The proof follows from [3].
Corollary 2.5. Let 1/2 ∈ k. Then there exists the natural isomorphism
ᏴCn (A) −α ᏴᏰn (A) ⊕ +α ᏴᏰn (A).
(2.12)
Note that we can define the reflexive cohomology and the dihedral cohomology of
a pre-additive category in the same manner.
3. The dihedral homology of free product algebras. In this section, we study the
product of the algebras associated with a pre-additive k-category, where k has characteristic zero. Let A, B, and C be arbitrary involutive algebras. The free product of
the algebras A and B with respect to algebra C is denoted by A ∗ B. Following [1], the
algebra A ∗ B is C-bimodule. For the algebras A, B, and C,
c
c
TorCi (A, A) = TorCi (A, B) = TorCi (B, A) = TorCi (B, B) = 0.
(3.1)
433
REFLEXIVE AND DIHEDRAL (CO)HOMOLOGY . . .
Consider also the following diagram of algebras and homomorphisms between them:
j1
i2
i1
j1 ◦ i1 = IdC = j2 ◦ i2 .
B,
C
A
j2
(3.2)
Following [3], let R A and R B be the free involutive resolution of the algebras A and B
over the homomorphisms i1 and i2 , respectively. Consequently, we get the following
diagram:
RA
iA
1
C
iB
2
RB .
(3.3)
Consider the diagrams
iA
1
RA
RB
C
iB
2
(3.4)
C
πB
πA
B
A
Clearly, they are commutative. If we define the homomorphisms j1A and j2B as follows:
j1A
R A →
RB ,
j2B
where j1A = j1 ◦ π A , j2B = j2 ◦ π A ,
R B →
C,
(3.5)
we get the diagram
R
A
j1A
iB
2
C
j2B
iA
1
RB ,
(3.6)
B
B
A
A
A
B
B
A
where j1A ◦ iA
1 = IdC , j2 ◦ i2 = IdC , since j1 ◦ i1 = (j1 ◦ π ) ◦ i1 = j1 ◦ i1 = Idc , j2 ◦ i2 =
B
A
B
B
A
B
A
B
(j2 ◦ π ) ◦ i2 = j2 ◦ i2 = Idc . Suppose that R̄ = ker j1 , R̄ = ker j2 . Then R ∗ R is a
c
C-module given by
R A ∗ R B = C + R̄ A ∗ R̄ B + R A ⊗ R B + R B ⊗ R A + R A ⊗ R B ⊗ R A + · · · .
c
c
c
c
c
c
(3.7)
We define an involution on R A ∗ R B as follows:
∗
p0 q0 p1 q1 · · · pn qn qn+1
c
∗
∗ ∗
= pn+1
qn
pn · · · q0∗ p0∗ ,
pi ∈ R A , 0 ≤ i ≤ n + 1, qi ∈ R B , 0 ≤ i ≤ n.
(3.8)
Remarks. (i) The differential on R A ∗ R B is defined by Leibniz formula for differc
ential graded algebras [5].
(ii) The chain complex R̄ A is a free C-biomodule resolution of the C-biomodule Ā.
(iii) Ā + C = A = H · (R A ) = H · (R̄ A ⊕ C) = H · (R̄ A ) + C, that is, H · (R̄ A ) = Ā, where
H is a hyperhomology of R̄ A . From (3.1), (i), (ii), and (iii), we have
Torci Ā, Ā = Torci Ā, B̄ = Torci B̄, Ā = Torci B̄, B̄ = 0.
(3.9)
434
YASIEN GH. GOUDA
From (3.7) and (3.9), we have A ∗ B = H · (R̄ A ⊗ R̄ B ) (see [1]). Consider the following
c
c
diagram:
RA ∗ RB
c
B
iA
1 ∗i2
c
(3.10)
C = C ∗C
c
A∗B
c
i1 ∗i2
c
Lemma 3.1. The diagram (3.10) is commutative.
Proof. The proof follows from the fact that the differential graded algebra R A ∗ R B
c
is an involutive resolution of the algebra A ∗ B over the inclusion i1 ∗ i2 .
c
c
Consider the complex T (n) (C; R̄ A ⊗ R̄ B ) = C ⊗op ((R̄ A ⊗ R̄ B ) ⊗ · · · ⊗ (R̄ A ⊗ R̄ B )). We
c
c
c⊗c
act on the complex by means of an automorphism γn as follows:
∗
∗
∗
γn p0 ⊗ q0 ⊗ · · · ⊗ pn ⊗ qn = (−1)+ν p0∗ ⊗qn
⊗ pn ⊗qn−1
⊗· · ·⊗ p1∗ ⊗q0∗ ; (3.11)
n
n
where pi ∈ R̄ A , qi ∈ R̄ B , 0 ≤ i ≤ n, = deg p0 i=1 deg pi + j=1 deg qi , ν = α(α − 1)/2,
n
α = i=1 deg pi + deg qi , α = ±1. Consider the chain complex homomorphism
T (n) C; R A ∗ R B
R A ∗R B
c
→
µ:
,
Im 1 − tn + Im 1 − γn
C + R A + R B + R A ∗ R B , R A ∗ R B + Im 1 − γn
c
c
(3.12)
such that
µ
p0 ⊗ q0 ⊗ · · · ⊗ pn ⊗ qn mod Im 1 − tn + Im 1 − γn
= p0 q0 · · · pn qn mod C + R̄ A + R̄ B + R A ∗ R B , R A ∗ R B + Im 1 − r .
c
(3.13)
c
In the following lemma, we explain the existence of the homomorphism µ and prove
that it is an isomorphism.
Lemma 3.2. A chain complex homomorphism µ is an isomorphism.
Proof. Clearly, in R A ∗ R B , there exists a subcomplex C + R̄ A + R̄ B (but in T (n) (C;
c
R̄ A ⊗ R̄ B ) there is not), and we can factorize R A ∗ R B by this subcomplex. The elements
c
c
in R̄ A ⊗ R̄ B can be compared by modulo with the commutant of the algebra R A ∗ R B with
c
c
elements in R̄ A ⊗ R̄ B since ᏸ ⊗ a + (−1)m (a ⊗ ᏸ − (−1)(deg a)·(deg ᏸ) ᏸ ⊗ a) = (−1)m a ⊗ ᏸ,
c
where
m=


0,
if (deg a) · (deg ᏸ) is even,

1,
if (deg a) · (deg ᏸ) is odd, a ∈ R̄ A , ᏸ ∈ R̄ B .
(3.14)
435
REFLEXIVE AND DIHEDRAL (CO)HOMOLOGY . . .
Note that the elements in R̄ A ⊗ R̄ B ⊗ R̄ A or in R̄ B ⊗ R̄ A ⊗ R̄ B can also be compared by
c
c
c
c
modulo with the commutant of the algebra R A ∗ R B with elements in R̄ A ⊗ R̄ B , at the
c
c
same time;
pqp − pqp −(−1)deg(p·q)·deg p p pq = (−1)deg(pq)·deg p p p q,
p, p ∈ R̄ A , q ∈ R̄ B .
(3.15)
In the complex T (n) (C; R̄ A ⊗ R̄ B )/ Im(1 − tn ) + Im(1 − γn ), we have the following:
c
p0 ⊗ q0 ⊗ · · · ⊗ pn ⊗ qn = (−1)S pn ⊗ qn ⊗ p0 ⊗ q0 ⊗ · · · ⊗ pn−1 ⊗ qn−1 , (3.16)
n−1
where s = deg(pn ⊗ qn ) i=1 deg(pi ⊗ qi ) and (p0 ⊗ q0 ) ⊗ · · · ⊗ (pn ⊗ qn ) = (−1)+ν n
n
∗
∗
∗
(p0∗ ⊗ qn
) ⊗ pn
⊗ qn−1
) ⊗ · · · ⊗ (p1∗ ⊗ q0∗ ); = deg p0 · i=1 deg pi + j=1 deg qj ,
n
ν = α(α − 1)/2, α = i=1 deg pi + deg qi . Clearly, the same relation holds in the complex: (R A ∗ R B )/C + R̄ A + R̄ B + [R A ∗ R B , R A ∗ R B ] + Im(1 − r) since
c
c
p0 q0 p1 q1 · · · pn−1 qn−1
where s = deg(pn qn )
pn qn = (−1)S pn qn p0 q0 p1 q1 · · · pn−1 qn−1 ,
n−1
i=1
c
(3.17)
deg(pi qi ), and since deg(pn qn ) = deg(pn ⊗qn ), s = s and
∗ ∗
p0 q0 p1 q1 · · · pn qn = (−1)ν qn
pn · · · q1∗ p1∗ q0∗ p0∗
∗ ∗ ∗
= (−1)ν+ p0∗ qn
pn qn−1 · · · p2∗ q1∗ p1∗ q0∗ .
(3.18)
This gives the required isomorphism. It is easily seen that the differentials in the
(n)
(C; R̄ A ⊗ R̄ B )/ Im(1−tn )+Im(1− γn ) and (R A ∗ R B )/C + R̄ A + R̄ B +
complexes ⊗∞
n=0 T
c
c
[R A ∗ R B , R A ∗ R B ] + Im(1 − r) coincide. Using the condition Torci (Ā, B̄) = 0, i > 0,
c
c
we find that R̄ A ⊗ R̄ B is a free C-module resolution of the algebra R̄ ⊗ R̄. Then by
c
c
considering the isomorphism µ, we get
Ᏼ·
RA ∗ RB
c
C + R̄ A + R̄ B + R A ∗ R B , R A ∗ R B + Im 1 − r
c
c
Ω ∼ · C, Ā ⊗ B̄
,
=
∞
Im 1− ⊕ γ̃n
(3.19)
n=0
where γ̃n is an automorphism on the graded K-module H · (Z/(n + 1); T (n) (C; R̄ A ⊗
c
R̄ B )) is induced by the automorphism γn on the complex T (n) (C; R̄ A ⊗ R̄ B ). From the
c
isomorphism µ, we get the following isomorphism:




Ω 
∼i C, Ā ⊗ B̄ 
i1
i2
ᏴᏰi C →
A ⊕ ᏴᏰi C →
B ⊕
∞


γ̃

 Im 1 − ⊕
n
n=0
c
= Ᏼ·
RA ∗ RB
C + R̄ A + R̄ B + R A ∗ R B , R A ∗ R B + Im 1 − r
c
c
A ⊕ ᏴᏰi C →
B .
⊕ ᏴᏰi C →
i1
(3.20)
i2
Lemma 3.3. The right-hand side of relation (3.20) is isomorphic to the group
i1 ∗i2
ᏴᏰi (C →
A ∗ B).
c
436
YASIEN GH. GOUDA
Proof. Since
RA ∗ RB
c
C + R A ∗ R B , R A ∗ R B + Im 1 − r
c
c
RA ∗ RB
c
=
C + R A ∗ R B , R A ∗ R B + Im 1 − r + R̄ A + R̄ B
c
c
R̄ B
R̄ A
+ + A A
R̄ , R̄ + Im 1 − rR A
R̄ B , R̄ B + Im 1 − rR B




R A ∗R B
=
 C + R A ∗ R B , R A ∗ R B + Im 1 − r + R̄ A + R̄ B 
c
+
C + RA , RA
c
A
R
+ Im 1 − rR A
+
(3.21)
RB
,
C + R B , R B + Im 1 − rR B
we have



Ω  ∼

i ∗i

i1 i2 i C, Ā ⊗ B̄
1
2
+ ᏴᏰi C →
A∗B =
A ⊕ ᏴᏰi C →
B .
ᏴᏰi C →
∞


c
Im 1 − ⊕ γ̃n 
(3.22)
n=0
Lemma 3.4. The following isomorphism holds:
Ω ∼i C, Ā ⊕ B̄
.
ᏴᏰi A ∗ B ⊕ ᏴᏰi (C) = ᏴᏰi (A) ⊕ ᏴᏰi (B) ⊕
∞
c
Im 1 − ⊕ γ̃n
(3.23)
n=0
Proof. This follows from the fact that
i1
ᏴᏰi (A) = ᏴᏰi C →
A ⊕ ᏴᏰ(C),
i2
ᏴᏰi (B) = ᏴᏰi C →
B ⊕ ᏴᏰ(C),
i1 ∗i2
c
ᏴᏰi A ∗ B = ᏴᏰi C →
A ∗ B ⊕ ᏴᏰ(C).
c
(3.24)
c
Note that the last three relations are obtained from the long exact sequence of relative dihedral homology of algebras [5]. Following [3] (also, see [4]), the automorphisms tn and γn give the representation of the dihedral group Ᏸn+1 on the complex
n
−1 T (n) (C; R̄ A ⊗ R̄ B ), where (tn )n+1 = ( γn )2 = Id, γn tn
= tn
γn . Then if char(k) = 0, we
c
get the following isomorphism:
Ω ∞
∼i C, Ā ⊗ B̄
⊕ H · Ᏸn+1 ; T (n) C; R̄ A ⊗ R̄ B .
∞
c
n=0
Im 1 − ⊕ γ̃n
(3.25)
n=0
From Lemma 3.3 and relation (3.20), we get
ᏴᏰi A ∗ B ⊕ ᏴᏰi (C)
c
∞
= ᏴᏰi (A) ⊕ ᏴᏰi (B) ⊕ ⊕ H · Ᏸn+1 ; T (n) C; R̄ A ⊗ R̄ B
.
n=0
So, we have proved the following theorem.
c
(3.26)
437
REFLEXIVE AND DIHEDRAL (CO)HOMOLOGY . . .
Theorem 3.5. Consider the following diagram of involutive algebras associated with
a pre-additive K-category:
j1
i2
C
A
i1
where j1 ◦ i1 = IdC = j2 ◦ i2 .
B,
j2
(3.27)
If
TorCi (A, A) = TorCi (A, B) = TorCi (B, A) = Torci (B, B) = 0,
i > 0,
(3.28)
α
⊕ H · Ᏸn+1 ; T (n) C; R̄ A ⊗ R̄ B
.
(3.29)
then we have
ᏴᏰi A ∗ B ⊕ ᏴᏰi (C)
c
= ᏴᏰi (A) ⊕ ᏴᏰi (B) ⊕
n=0
c
Let A be a k-category with an involution, and let Mod A be the category of right
A-modules and P (A) be full subcategory in Mod A, consisting of the finite projective
modules. Consider the category Mrt (k) with objects, the k-categories with involution
and morphisms f : A → B are k-factors F : Mod A → Mod B, such that for every X ∈
P (A), f (X) ∈ P (B), f commutes with an involution. We call these morphisms, Moritamorphisms. Evidently, if f is an equivalence, then f is a Morita-morphism. Following
[7] (also, see [8]), the cyclic (co)homology of k-category Morita equivalence. Using this
fact and considering the deep results of [3], the following fact follows.
Theorem 3.6. The reflexive cohomolgy and the dihedral (co)homology of the kcategory A with involution are invariant under Morita equivalence.
Acknowledgement. This paper has been presented in the International Conference “Mathematics Today and Tomorrow” at the University of Central Florida, 13–15
March 1997.
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Yasien Gh. Gouda: Department of Mathematics, Faculty of Science, South Valley
University, Aswan, Egypt
E-mail address: yasien10@hotmail.com