COHOMOLOGY AND DEFORMATION OF MODULE-ALGEBRAS
DONALD YAU
Abstract.An algebraic deformation theory of module-algebras over a bial-
gebra is constructed. The cases of module-coalgebras, comodule-algebras,*
* and
comodule-coalgebras are also considered.
1.Introduction
Let H be a bialgebra and A be an associative algebra. The algebra A is said
to be an H-module-algebra if there is an H-module structure on A such that the
multiplication on A becomes an H-module morphism. There are many important
examples of this structure. For example, an algebra over the Landweber-Novikov
algebra S [11, 14] is an S-module-algebra. In particular, the complex cobordism
MU*(X) of a topological space X, equipped with its commutative ring structure
and the stable cobordism operations, is an S-module-algebra. Likewise, if p is*
* a
prime, then algebras over the Steenrod algebra Ap [4, 12] are Ap-module-algebra*
*s.
In particular, the singular mod p cohomology H*(X; Fp) of a topological space X,
equipped with its commutative Fp-algebra structure and the Steenrod operations,
is an Ap-module-algebra. Other examples from algebraic topology can be found
in [2]. There are also important examples from Lie and Hopf algebras theory. For
instance, the affine plane admits a module-algebra structure over the enveloping
bialgebra of the Lie algebra sl(2) [10, V.6].
The main purposes of this paper are:
(1) Construct the deformation cohomology for any module-algebra A over a
bialgebra H, where the deformation is taken with respect to the H-module
structure, keeping the algebra structure on A unaltered.
(2) Use the deformation cohomology to describe infinitesimal, rigidity, and *
*ex-
tension results concerning algebraic deformations of module-algebras.
We will also consider the related cases of module-coalgebras, comodule-algebras,
and comodule-coalgebras. These three other algebraic structures are important in
the studies of Hopf algebras and quantum groups (see, e.g., [3, 10, 13, 16]).
Some remarks are in order. In [17] the author considered deformations of alge*
*bras
over the Landweber-Novikov algebra, which, as mentioned above, are examples of
module-algebras. The current paper generalizes the constructions and results th*
*ere
to any module-algebra. Moreover, in the study of the obstructions to extending *
*co-
cycles to deformations, we actually obtain a simpler and more conceptual argume*
*nt.
1
2 DONALD YAU
More precisely, in [17, Lemma 5.2], a certain obstruction class was shown to be*
* a
1-cocycle in the deformation complex using a rather computational argument. It
makes heavy use of the composition law and the Cartan formula for the Landweber-
Novikov operations. In the current paper, the corresponding fact is proved by a
much simpler argument, using a certain cup-product in the deformation complex.
This brings us to the next remark.
In most known cases of algebraic deformations, the cochain complex that con-
trols the deformations is a dg Lie algebra (see, e.g., [1, 6, 7]). The Lie brac*
*ket is
usually used when one tries to show that the obstructions to extending cocycles
to deformations are themselves cocycles. On the other hand, the cochain complex
that controls the deformations of a module-algebra is a differential graded alg*
*ebra
(DGA), whose product we denote by a cup-product. Moreover, part of the building
blocks for this DGA is a certain Hochschild cochain complex. The cup-product in
the deformation DGA extends the usual cup-product in this Hochschild cochain
complex. It remains an open question as to whether the cohomology of the defor-
mation DGA admits a Gerstenhaber-algebra structure, like the one on Hochschild
cohomology [5].
There may be a more general deformation theory of the pair (H, A), in which,
say, both the H-module structure and the algebra structure on A are deformed.
The resulting deformation complex Fo(H, A) should contain the deformation DGA
Fo(A) in this paper. The extended deformation complex Fo(H, A) may then be a
dg Lie algebra. This idea was pointed out by the referee.
1.1. Organization. The deformation DGA Fo(A) for a module-algebra A is con-
structed in the next section. In section 3, we use the deformation DGA to study
deformations, infinitesimals, and rigidity of a module-algebra. Infinitesimals*
* are
identified with certain cohomology classes in H1(Fo(A)) (Theorem 3.3). The rigi*
*d-
ity result (Corollary 3.6) states that every deformation is equivalent to the t*
*rivial
one, provided that both H1(Fo(A)) and H2h(A, A) are trivial. Here Hoh(A, A) is *
*the
Hochschild cohomology of A with coefficients in itself. In section 4, the obstr*
*uctions
to extending a 1-cocycle in the deformation DGA to a full-blown deformation are
identified. They are shown to be 2-cocycles (Lemma 4.3). In particular, if H2 of
the deformation DGA is trivial, then such a deformation always exists (Corollary
4.5). In section 5, the deformation DGAs for a module-coalgebra, a comodule-
algebra, and a comodule-coalgebra are constructed and the corresponding results
about deformations are listed.
1.2. Acknowledgment. The author thanks the referee for reading an earlier ver-
sion of this paper and for his/her constructive comments and suggestions.
COHOMOLOGY AND DEFORMATION OF MODULE-ALGEBRAS 3
2.Deformation cohomology for module-algebras
The purpose of this section is to construct the DGA that controls the deforma-
tions of a module-algebra over a bialgebra.
2.1. Notations. Fix a ground field K. Tensor products and Hom will be taken
over K. Also fix a K-bialgebra (H, ~H , H ).
Denote by End(X) the algebra, under composition, of K-linear endomorphisms
of a vector space X. For an algebra (A, ~A ), a derivation on A is a linear map
' 2 End(A) such that ' O ~A = ~A O (IdA ' + ' IdA). The set of derivations on
A is denoted by Der(A).
In a coalgebra (C, ), we use Sweedler's notation [16] for comultiplication:
P P
(x) = (x)x(1) x(2), 2(x) = (x)x(1) x(2) x(3), etc. A coderivation
on C is a linear map ' 2 End(C) such that O ' = (IdC ' + ' IdC) O . The
set of coderivations on C is denoted by Coder(C).
2.2. Module-algebras. Basic information about module-algebras can be found in
many books on Hopf algebras, e.g., [3, 10, 13, 16]. Let (A, ~A ) be an associat*
*ive K-
algebra. Say that A is an H-module-algebra if and only if there exists an H-mod*
*ule
structure ~ 2 Hom (H, End(A)) on A such that ~A :A A ! A is an H-module
morphism. In other words, ~ is required to satisfy:
~(xy) = ~(x) O ~(y),
X
~(x)(ab)= ~(x(1))(a) . ~(x(2))(b),
(x)
for x, y 2 H and a, b 2 A.
For this and the next two sections, A will be an H-module-algebra with H-acti*
*on
map ~.
2.3. Hochschild cohomology. The deformation complex of a module-algebra
uses the Hochschild cochain complex [8], which we now recall. Let (R, ~) be an
algebra and let M be an R-bimodule with left R-action ffL and right R-action ff*
*R .
For integers n 0, the module of Hochschild n-cochains of R with coefficients *
*in
M is Cnh(R, M) = Hom (R n , M). The differential b: Cnh(R, M) ! Cn+1h(R, M) is
given by
Xn
b' = ffL O (IdR ') + (-1)i' O (IdR (i-1) ~ IdR (n-i))
i=1
+ (-1)n+1ffR O (' IdR)
for ' 2 Cnh(R, M). The Hochschild cohomology of R with coefficients in M is the
cohomology of the cochain complex Coh(R, M) and is denoted by Hoh(R, M).
4 DONALD YAU
2.4. Deformation complex. Now we define the deformation complex (Fo(A), d)
of A as an H-module-algebra. Set:
F0(A) = Der(A),
F1(A) = Hom (H, End(A)).
For integers n 2, define
Fn(A) = Fn0(A) Fn1(A),
where
Fn0(A)= Cnh(H, End(A)),
Fn1(A)= Hom (H, Hom (A n , A)).
In Cnh(H, End(A)), we consider End(A) as an (H, ~H )-bimodule via the structure
map
H End(A) H ! End(A)
x f y7! ~(x) O f O ~(y).
Now we define the differentials d: Fn(A) ! Fn+1(A). For ' 2 Der(A), set
d0' = ~ O ' - ' O ~,
where (~ O ')(x) = ~(x) O ' and (' O ~)(x) = ' O ~(x) for x 2 H. For integers n*
* 1,
set
dn = (dn0; dn1),
where dni:Fni(A) ! Fn+1i(A) (i = 0, 1) is defined as follows:
o dn0= b: Cnh(H, End(A)) ! Cn+1h(H, End(A)), the Hochschild coboundary.
P n+1
o dn1= i=0(-1)idn1[i], where
8
>>>P ~(x )(a ) . ' (x )(a . . .a i)fi = 0
< (x) (1) 1 1 (2) 2 n+1
(dn1[i])('1)(x)(a) = > '1(x)(a1 . . .(aiai+1) . .a.n+1) if1 i n
>>:P
(x)'1(x(1))(a1 . . .an) . ~(x(2))(an+1)ifi = n + 1.
Here ' = ('0; '1) 2 Fn(A), x 2 H, and a = a1 . . .an+1 2 A n+1. In these
definitions when n = 1, we think of F10= F11= F1 and '0 = '1 = '.
Proposition 2.5. (Fo(A), d) is a cochain complex
Proof.One can check directly that d1 O d0 = 0. It is clear that di+10O di0= 0 f*
*or
i 1, since di0= b is the Hochschild coboundary. For 0 k < l i + 2, it is
straightforward to check the cosimplicial identities
di+11[l] O di1[k] = di+11[k] O di1[l - 1].
As usual, this implies that di+11O di1= 0.
COHOMOLOGY AND DEFORMATION OF MODULE-ALGEBRAS 5
2.6. Cup-product. The usual cup-product on the Hochschild cochain complex
Coh(H, End(A)) is defined as
(f [ g)(x1 . . .xm+n ) = f(x1 . . .xm ) O g(xm+1 . . .xm+n )
for f 2 Cmh(H, End(A)), g 2 Cnh(H, End(A)), and x1, . .,.xm+n 2 H.
Using this, we define a cup-pairing
- [ -: Fm (A) Fn(A) ! Fm+n (A) (2.6.1)
for integers m, n > 0 as follows. (Note that we do not consider the cases where
m = 0 or n = 0.) Suppose that f = (f0; f1) 2 Fm (A) and g = (g0; g1) 2 Fn(A) for
m, n > 0. Define:
o (f [ g)0 = f0 [ g0, where the [-product on the right-hand side is the one
on Coh(H, End(A)).
o For x 2 H and a = a1 . . .am+n 2 A m+n ,
X
(f [ g)1(x)(a) = f1(x(1))(a1 . . .am ) . g1(x(2))(am+1 . . .am+n ).
(x)
In these definitions, if m = 1, then we think of f0 = f1 = f, and similarly when
n = 1.
Proposition 2.7. The [-pairing in (2.6.1)is associative and satisfies the Leibn*
*iz
identity,
d(f [ g) = (df) [ g + (-1)|f|f [ (dg),
where |f| = m for f 2 Fm (A). In particular, it follows that (F>0(A), d, [) is*
* a
DGA.
Proof.The associativity of the [-product on Coh(H, End(A)) is obvious from the
definition. The associativity of [ on the component Fo1(A) is an immediate cons*
*e-
quence of the coassociativity of H , namely,
X X
x(1) (x(2))(1) (x(2))(2)= 2H(x) = (x(1))(1) (x(1))(2) x(2).
(x)(x(2)) (x)(x(1))
The Leibniz identity can be seen by direct inspection.
This Proposition implies, as usual, that the [-pairing descends to cohomology.
Corollary 2.8. The [-pairing (2.6.1)induces a well-defined product
- [ -: Hm (Fo(A)) Hn(Fo(A)) ! Hm+n (Fo(A))
for m, n > 0, making H 1(Fo(A)) into a graded algebra.
3.Formal deformation and rigidity
The purposes of this section are to (i) define deformations, (ii) identify in*
*finites-
imals with suitable cohomology classes, and (iii) obtain a cohomological criter*
*ion
for rigidity.
6 DONALD YAU
3.1. Deformation. By a deformation of A (as an H-module-algebra), we mean a
P 1
power series t= i=0~iti with ~0 = ~ and each ~i2 F1(A), satisfying
t(xy)= t(x) O t(y), (3.1.1a)
X
t(x)(ab)= t(x(1))(a) . t(x(2))(b) (3.1.1b)
(x)
for x, y 2 H and a, b 2 A. In particular, by linearity, such a t gives (A[[t]]*
*, ~A ) an
H-module-algebra structure, which reduces to the original one when setting t = *
*0.
The linear term ~1 is called the infinitesimal of t.
In order to identify the infinitesimal with a suitable cohomology class, we n*
*eed
an appropriate notion of equivalence.
P 1
3.2. Equivalence. A formal automorphism of A is a power series t= i=0OEiti
with OE0 = IdAand each OEi2 End(A) such that t is multiplicative, i.e.
t(ab) = t(a) t(b) (3.2.1)
for all a, b 2 A.
Note that this is exactly the same definition as in the special case of algeb*
*ras
over the Landweber-Novikov algebra [17, 3.2].
P 1 __ P 1 __
Suppose that t= i=0~itiand t= i=0~itiare deformations of A. We say
__
that t and tare equivalent if and only if there exists a formal automorphism *
* t
of A such that
__ -1
t = t t t. (3.2.2)
On the right-hand side, one considers OEi~jOEk as an element of F1(A) via the f*
*ormula,
(OEi~jOEk)(x) = OEiO ~j(x) O OEk
for x 2 H. It is clear that this is an equivalence relation. Moreover, given a
deformation tand a formal automorphism t, one can define another deformation
__
tusing (3.2.2), and it is automatically equivalent to t.
The following result properly identifies the infinitesimal of a deformation w*
*ith a
cohomology class.
P 1
Theorem 3.3. Let t = i=0~iti be a deformation of A. Then ~1 2 F1(A) is
a 1-cocycle whose cohomology class is determined by the equivalence class of t.
Moreover, if ~1 = . .=.~k = 0, then ~k+1 is a 1-cocycle in F1(A).
Proof.The condition (3.1.1a)is equivalent to the equality
X
~n(xy) = ~i(x) O ~j(y) (3.3.1)
i+j = n
for all n 0 and x, y 2 H. In particular, when n = 1, we obtain
(d10~1)(x y) = ~(x) O ~1(y) - ~1(xy) + ~1(x) O ~(y) = 0.
COHOMOLOGY AND DEFORMATION OF MODULE-ALGEBRAS 7
Likewise, the condition (3.1.1b)can be restated as
X X
~n(x)(ab) = ~i(x(1))(a) . ~j(x(2))(b) (3.3.2)
(x)i+j = n
for all n 0, x 2 H, and a, b 2 A. By a simple rearrangement of terms, the case
n = 1 then states that d11~1 = 0. Therefore, we have d1~1 = 0 2 F2(A). The last
assertion about ~k+1 is proved by essentially the same argument.
__ -1 __
Now suppose that t= t t t for some deformation t and formal automor-
phism t. Then the condition on the coefficients of t is
__
~1 = ~1 + ~ O OE1 - OE1 O ~,
__
i.e.,_~1- ~1 = d0OE1, a 1-coboundary. Therefore, the cohomology classes of ~1 a*
*nd
~1are the same.
3.4. Rigidity. The trivial deformation of A is the deformation t = ~. The H-
module-algebra A is said to be rigid if and only if every deformation of A is e*
*quiv-
alent to the trivial deformation.
The following preliminary result is needed for the cohomological criterion for
rigidity below.
Proposition 3.5. Let t = ~ + ~N tN + O(tN+1 ) be a deformation of A in which
~N = d0OE for some OE 2 F0(A). Suppose that H2h(A, A) = 0. Then there exists a
formal automorphism of A of the form
t = IdA-OEtN + O(tN+1 ) (3.5.1)
such that the deformation defined by
__ -1
t = t t t
satisfies __
~i= 0
for i = 1, . .,.N.
Proof.The existence of a formal automorphism t of the form (3.5.1)is exactly
[17, Corollary 4.4]. Since t IdA (mod tN ), we have
__ N
t t (mod t )
and, therefore, __ __
~1= . .=.~N-1 = 0.
To finish the proof, it suffices to consider the coefficient of tN in -1t t t.*
* This
coefficient is __
~N = ~N + OE O ~ - ~ O OE = ~N - d0OE = 0,
as desired.
Applying Theorem 3.3 and Proposition 3.5 repeatedly, we obtain the following
cohomological criterion for the rigidity of a module-algebra.
8 DONALD YAU
Corollary 3.6. If both H2h(A, A) and H1(Fo(A)) are trivial, then A is rigid.
4.Extending cocycles to deformations
In view of Theorem 3.3, it is natural to ask the question: Given a 1-cocycle
~1 2 F1(A), does there exist a deformation t of A whose infinitesimal is ~1?
Following [6], if such a deformation exists, we say that ~1 is integrable. The *
*purpose
of this section is to develop the obstruction theory for integrability of 1-coc*
*ycles in
F1(A).
4.1. Deformations of finite order. Let N be a positive integer. A polynomial
P N
t = i=0~iti with ~0 = ~ and each ~i 2 F1(A) is said to be a deformation of
order N if and only if it satisfies the definition of a deformation modulo tN+1*
* , i.e.,
(3.3.1)and (3.3.2)for n N. Such a deformation of order N is said to extend to
order N + 1 if and only if there exists a 1-cochain ~N+1 2 F1(A) such that the
polynomial
N+1X
e t= t+ ~N+1 tN+1 = ~iti (4.1.1)
i=0
is a deformation of order N + 1. In this case, we say that e tis an order N + 1
extension of t.
It is easy to see that a 1-cochain ~1 2 F1(A) is a 1-cocycle if and only if t*
*he
linear polynomial ~ + ~1t is a deformation of order 1. Therefore, in order to f*
*ind
the obstructions to integrating ~1, it suffices to find the obstruction to exte*
*nding a
deformation of order N 1 to one of order N + 1.
P N
4.2. Obstruction. Let, then, t = i=0~iti be a deformation of order N 1.
As explained in the proof of Theorem 3.3, the conditions (3.3.1)and (3.3.2)for
n = 1 are equivalent to ~1 2 F1(A) being a 1-cocycle. For each m = 2, . .,.N, t*
*he
conditions (3.3.1)and (3.3.2)for n = m, by a simple rearrangement of terms, are
equivalent to
m-1X
d1~m = - ~i[ ~m-i, (4.2.1)
i=1
where the [-product was introduced in (2.6.1).
Let ~N+1 2 F1(A) be an arbitrary 1-cochain and set e t= t+~N+1 tN+1 . Then
e tis a deformation of order N + 1 if and only if it satisfies (3.3.1)and (3.3.*
*2)for
n = N + 1. As in the previous paragraph, this is equivalent to
NX
d1~N+1 = - ~i[ ~N+1-i. (4.2.2)
i=1
Consider the 2-cochain N
X
Ob = ~i[ ~N+1-i (4.2.3)
i=1
in F2(A) defined by ~1, . .,.~N .
COHOMOLOGY AND DEFORMATION OF MODULE-ALGEBRAS 9
Lemma 4.3. The class Ob 2 F2(A) is a 2-cocycle.
Proof.This is similar to [6, Proposition 2]. In fact, using d1~1 = 0, (4.2.1), *
*and
Proposition 2.7, we have:
XN
d2Ob = d2(~i[ ~N+1-i)
i=1
XN
= (d1~i) [ ~N+1-i - ~i[ (d1~N+1-i)
i=18 9 8 9
XN < i-1X = NX < N-iX =
= - ~j[ ~i-j [ ~N+1-i - ~i[ - ~j[ ~N+1-i-j
i=2: j=1 ; i=1 : j=1 ;
X 0 X 0
= - ~a [ ~b[ ~c+ ~a [ ~b[ ~c
= 0.
X 0
Here is the sum over all integers a, b, c > 0 with a + b + c = N + 1.
Combining (4.2.2), (4.2.3), and Lemma 4.3, we obtain the desired obstruction
for extending an order N deformation.
Theorem 4.4. The deformation t of order N extends to order N + 1 if and only
if the 2-cocycle - Ob is a 2-coboundary.
Since the obstruction is always a class in H2(Fo(A)), we obtain the following
cohomological criterion for integrability.
Corollary 4.5. If H2(Fo(A)) is trivial, then every 1-cocycle in F1(A) is integr*
*able.
5.Deformation cohomology for module-coalgebras and
comodule-(co)algebras
The purpose of this final section is to describe the deformation DGAs and the
corresponding deformation results for module-coalgebras, comodule-algebras, and
comodule-coalgebras. As before, the deformation is taken with respect to the mo*
*d-
ule (or comodule) action, leaving the algebra (or coalgebra) structure unaltere*
*d.
In each case, once the correct deformation DGA is set up, the statements of re-
sults and their arguments are formally similar to the module-algebra case above.
Therefore, we will describe the constructions and statements of results and omit
the arguments. To avoid too much repetitions, we will concentrate on the case of
comodule-coalgebras. At the end of the section, we will indicate what modificat*
*ions
are needed for the cases of module-coalgebras and comodule-algebras.
5.1. Comodule-coalgebras. We still denote by (H, ~H , H ) a bialgebra. Let
(A, A ) be a coalgebra. An H-comodule-coalgebra structure on A consists of an
10 DONALD YAU
H-comodule structure ae: A ! H A on A such that the map A :A ! A A is
an H-comodule morphism, i.e.,
(IdH A ) O ae = (~H IdA 2) O (IdH o IdA) O ae 2 O A(.5.1.1)
Here o :A H ~=H A is the twist isomorphism.
Until otherwise indicated, A will denote an H-comodule-coalgebra with structu*
*re
map ae.
5.2. Deformations of comodule-coalgebras. A deformation of A is a power
P 1
series Rt = i=0aeiti with ae0 = ae and each aei 2 Hom (A, H A), satisfying *
*the
following two conditions:
(IdH Rt) O Rt= ( H IdA) O Rt, (5.2.1a)
(IdH A ) O Rt= (~H IdA 2) O (IdH o IdA) O Rt2 O A .(5.2.1b)
As before, maps are extended linearly to include modules of power series wherev*
*er
appropriate. The linear coefficient ae1 is called the infinitesimal of Rt.
P 1
A formal automorphism of A is a power series t = i=0OEiti with OE0 = IdA
and each OEi2 End(A) that is comultiplicative, i.e.,
A O t = t2 O A .
__
Two deformations Rtand Rt of A are equivalent if and only if there exists a for*
*mal
automorphism of A such that
__ -1
Rt = (IdH t ) O RtO t. (5.2.2)
5.3. Deformation complex for a comodule-coalgebra. It is the cochain com-
plex (Focc(A), dcc) defined as follows:
F0cc(A)= Coder(A),
F1cc(A)= Hom (A, H A),
Fncc(A)= Fncc,0(A) Fncc,1(A) (n 2),
where
Fncc,0(A)= Hom (A, H n A),
Fncc,1(A)= Hom (A, H A n ).
Now we define the differentials.
d0ccOE= ae O OE - (IdH OE) O ae,
dncc= (dncc,0; dncc,1) (n 1).
COHOMOLOGY AND DEFORMATION OF MODULE-ALGEBRAS 11
The component maps dncc,i:Fncc,i(A) ! Fn+1cc,i(A) (i = 0, 1) are defined by:
Xn
(dncc,0'0)= (IdH n ae) O '0 + (-1)i(IdH (n-i) H IdH (i-1) A) O '0
i=1
+ (-1)n+1(IdH '0) O ae
(dncc,1'1)= (~H IdA (n+1)) O (IdH o IdA n) O (ae '1) O A
Xn
+ (-1)i(IdH A (i-1) A IdA (n-i)) O '1
i=1
+ (-1)n+1(~H IdA (n+1)) O (IdH on IdA) O ('1 ae) O A .
In the last line, the map on :A n H ~=H A n is the twist isomorphism
on(a x) = x a.
5.4. Cup-product on Focc(A). There is a [-product
- [ -: Fmcc(A) Fncc(A) ! Fm+ncc(A) (m, n > 0) (5.4.1)
that is defined as follows. For f 2 Fmcc(A) and g 2 Fncc(A), the components of *
*f [ g
are:
(f [ g)0= (IdH n f0) O g0,
(f [ g)1= (~H IdA (m+n)) O (IdH om IdA n) O (f1 g1) O A .
Theorem 5.5. (Focc(A), dcc) is a cochain complex. Moreover, (F>0cc(A), dcc, [) *
*is a
DGA, and (H>0(Focc(A)), [) is a graded algebra.
5.6. Hochschild coalgebra cohomology. The rigidity result for a comodule-
coalgebra (and also a module-coalgebra) uses Hochschild coalgebra cohomology
[9, 15], which we now recall. The Hochschild coalgebra cohomology of a coalgebra
A with coefficients in an A-bicomodule M (with left A-coaction _l and right A-
coaction _r) is defined as follows. For n 1, the module of n-cochains is defi*
*ned
to be
Cnc(M, A) = Hom (M, A n ),
with differential
Xn
fficoe= (IdA oe) O _l + (-1)i(IdA (i-1) IdA (n-i))O oe
i=1
+ (-1)n+1(oe IdA) O _r
for oe 2 Cnc(M, A). Set C0c(M, A) 0. The cohomology of the cochain complex
(Coc(M, A), ffic) is denoted by Hoc(M, A).
For example, we can consider A as an A-bicomodule with coaction maps _l =
_r = A .
12 DONALD YAU
5.7. Results about deformations of comodule-coalgebras. The trivial defor-
mation of A is the deformation Rt= ae. Rigidity is defined as in the module-alg*
*ebra
case.
P 1
Theorem 5.8. Let Rt = i=0aeiti be a deformation of A as an H-comodule-
coalgebra. Then the infinitesimal ae1 is a 1-cocycle in F1cc(A) whose cohomolo*
*gy
class is determined by the equivalence class of Rt. Moreover:
(1) If H2c(A, A) and H1(Focc(A)) are both trivial, then A is rigid.
(2) If H2(Focc(A)) is trivial, then every 1-cocycle in F1cc(A) is the infini*
*tesimal
of some deformation of A.
5.9. Module-coalgebras. Next we consider the case of module-coalgebras. An H-
module-coalgebra structure on the coalgebra A consists of an H-module structure
~ 2 Hom (H, End(A)) on A such that the map A :A ! A A becomes an H-
module morphism, i.e.,
X
A (~(x)(a)) = ~(x(1))(a(1)) ~(x(2))(a(2))
(a)(x)
for all x 2 H and a 2 A.
P 1
A deformation t = i=0~iti of A is defined as in the module-algebra case,
except that (3.1.1b)is replaced by
X
A ( t(x)(a)) = t(x(1))(a(1)) t(x(2))(a(2)).(5.9.1)
(a)(x)
A formal automorphism of A is defined as in the comodule-coalgebra case (x5.2*
*),
and equivalence is defined as in the module-algebra case (x3.2).
The deformation DGA (Fomc(A), dmc, [) of A (as an H-module-coalgebra with
structure map ~) has the same general form as in the cases of module-algebra and
comodule-coalgebra:
F0mc(A)= Coder(A),
F1mc(A)= Hom (H, End(A)),
Fnmc,0(A)= Fn0(A) (n 2),
Fnmc,1(A)= Hom (H, Hom (A, A n )) (n 2).
The differentials are defined as follows:
d0mc'= ~ O OE - OE O ~,
dnmc,0= dn0 (n 1),
X
(dnmc,1'1)(x)(a)= ~(x(1))(a(1)) '1(x(2))(a(2))
(a)(x)
Xn
+ (-1)i(IdA (i-1) A IdA (n-i))('1(x)(a))
i=1 X
+ (-1)n+1 '1(x(1))(a(1)) ~(x(2))(a(2)) (n 1).
(a)(x)
COHOMOLOGY AND DEFORMATION OF MODULE-ALGEBRAS 13
The [-product is defined by:
(f [ g)0= f0 [ g0 (inC>0(H, End(A))),
X
(f [ g)1(x)(a)= f1(x(1))(a(1)) g1(x(2))(a(2)).
(a)(x)
With these definitions, we have the module-coalgebra analogue of Theorem 5.5 and
Theorem 5.8, where (Focc(A), dcc) is replaced by (Fomc(A), dmc).
5.10. Comodule-algebras. Finally, we consider the case of comodule-algebras.
Here let (A, ~A ) be an algebra. An H-comodule-algebra structure on A consists *
*of
an H-comodule structure ae: A ! H A on A such that the map ~A :A A ! A
becomes an H-comodule morphism, i.e.,
ae O ~A = (~H ~A ) O (IdH o IdA) O ae 2.
A deformation of A (as an H-comodule-algebra with structure map ae) is defined
as in the case of comodule-coalgebras (x5.2), except that the condition (5.2.1b*
*)is
replaced by
RtO ~A = (~H ~A ) O (IdH o IdA) O Rt2.
A formal automorphism of A is defined as in the module-algebra case (x3.2), a*
*nd
equivalence is defined as in the case of comodule-coalgebras (5.2.2).
The deformation DGA (Foca(A), dca, [) of A as an H-comodule-algebra takes the
following form:
(F0ca(A), d0ca)= (Der(A), d0cc)
F1ca(A)= Hom (A, H A)
Fnca(A)= Fnca,0(A) Fnca,1(A) (n 2)
(Fnca,0(A), dnca,0)= (Fncc,0(A), dncc,0) (n 1)
Fnca,1(A)= Hom (A n , H A) (n 2)
dnca,1'1= (~H ~A ) O (IdH o IdA) O (ae '1)
Xn
+ (-1)i'1 O (IdA (i-1) ~A IdA (n-i))
i=1
+ (-1)n+1(~H ~A ) O (IdH o IdA) O ('1 ae) (n 1).
The [-product is given by:
(f [ g)0= (IdH n f0) O g0
(f [ g)1= (~H ~A ) O (IdH o IdA) O (f1 g1).
With these definitions, we have the comodule-algebra analogue of Theorem 5.5 and
Theorem 5.8, where (Focc(A), dcc) and H2c(A, A) are replaced by (Foca(A), dca) *
*and
H2h(A, A), respectively.
14 DONALD YAU
References
[1]D. Balavoine, Deformations of algebras over a quadratic operad, Contemp. Ma*
*th. 202 (1997),
207-234.
[2]J. M. Boardman, Stable operations in generalized cohomology, in: Handbook o*
*f algebraic
topology, 585-686, North-Holland, Amsterdam, 1995.
[3]S. D~as~alescu, et. al., Hopf algebra: An introduction, Monographs and Text*
*books in Pure and
Applied Math. 235, Marcel Dekker, Inc., NY, 2001.
[4]D. B. A. Epstein and N. E. Steenrod, Cohomology operations, Annals of Math.*
* Studies 50,
Princeton Univ. Press, 1962.
[5]M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math*
*. 78 (1963),
267-288.
[6]M. Gerstenhaber, On the deformation of rings and algebras, Ann. Math. 79 (1*
*964), 59-103.
[7]V. Hinich, Deformations of homotopy algebras, Comm. Alg. 32 (2004), 473-494.
[8]G. Hochschild, On the cohomology groups of an associative algebra, Ann. Mat*
*h. 46 (1945),
58-67.
[9]D. W. Jonah, Cohomology of coalgebras, Mem. Amer. Math. Soc. 82, 1968.
[10]C. Kassel, Quantum groups, Grad. Texts in Math. 155, Springer-Verlag, 1995.
[11]P. S. Landweber, Cobordism operations and Hopf algebras, Trans. Amer. Math.*
* Soc. 129
(1967), 94-110.
[12]J. Milnor, The Steenrod algebra and its dual, Ann. Math. 67 (1958), 150-171.
[13]S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conf*
*erence Series
in Math. 82, Amer. Math. Soc., Providence, RI, 1993.
[14]S. P. Novikov, Methods of algebraic topology from the point of view of cobo*
*rdism theory,
Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 855-951.
[15]B. Parshall and J. P. Wang, On bialgebra cohomology. Algebra, groups and ge*
*ometry. Bull.
Soc. Math. Belg. S'er. A 42 (1990), 607-642.
[16]M. Sweedler, Hopf algebras, W. A. Benjamin, Inc., NY, 1969.
[17]D. Yau, Deformation of algebras over the Landweber-Novikov algebra, J. Alge*
*bra 298 (2006),
507-523.
Department of Mathematics, The Ohio State University Newark, 1179 University
Drive, Newark, OH 43055, USA
E-mail address: dyau@math.ohio-state.edu