DEFORMATION OF ALGEBRAS OVER THE
LANDWEBER-NOVIKOV ALGEBRA
DONALD YAU
Abstract. An algebraic deformation theory of algebras over the
Landweber-Novikov algebra is obtained.
1. Introduction
The Landweber-Novikov algebra S was originally studied by Landweber
[8] and Novikov [10] as a certain algebra of stable cohomology operations
on complex cobordism MU*(-). In fact, it is known that every stable
cobordism operation can be written uniquely as an MU*-linear combination
in the Landweber-Novikov operations. The Landweber-Novikov operations
act stably, and hence additively, on the cobordism MU*(X) of a space X.
However, they are, in general, not multiplicative on MU*(X). Instead,
their actions on a product of two elements in MU*(X) satisfy the so-called
Cartan formula, analogous to the formula of the same name in ordinary mod
2 cohomology. This structure - the S-module structure together with the
Cartan formula on products - makes MU*(X) into what is called an algebra
over the Landweber-Novikov algebra, or an S-algebra for short. These S-
algebras are therefore of great importance in algebraic topology.
The algebra S has also appeared in other settings. For example,
Bukhshtaber and Shokurov [2] showed that the Landweber-Novikov alge-
bra is isomorphic to the algebra of left invariant differentials on the group
Diff1(Z). Denoting the group of formal diffeomorphisms on the real line
by Diff1(R), the group Diff1(Z) is the subgroup generated by the formal
diffeomorphisms with integer coefficients. This theme that the Landweber-
Novikov algebra is an "operation algebra" is echoed in Wood's paper [12].
Wood constructed S as a certain algebra of differential operators on the
integral polynomial ring on countably infinitely many variables. There are
even connections between the Landweber-Novikov algebra and physics, as
the work of Morava [9] demonstrates.
The purpose of this paper is to study algebras over the Landweber-
Novikov algebra from the specific view point of algebraic deformations. We
would like to deform an S-algebra A with respect to the Landweber-Novikov
operations on A, keeping the algebra structure on A unaltered. The resulting
1
2 DONALD YAU
deformation theory is described in cohomological and obstruction theoretic
terms.
The original theory of deformations of associative algebras was developed
by Gerstenhaber in a series of papers [3, 4, 5, 6]. It has since been extended
in many different directions, and many kinds of algebras now have their own
deformation theories.
The Landweber-Novikov algebra is actually a Hopf algebra, as the referee
pointed out. Therefore, what we are considering in this paper is really an
instance of deformation of a module algebra over a Hopf algebra. It would be
nice to extend the results in the current paper to this more general setting.
A description of the rest of the paper follows.
The following section is preliminary in nature. We recall the Landweber-
Novikov algebra S and algebras over it. Our deformation theory depends
on the cohomology of a certain cochain complex F*, which is constructed in
the next section as well.
In Section 3 we introduce the notions of a formal deformation and of a
formal automorphism. The latter is used to defined equivalence of formal de-
formations. The main point of that section is Theorem 3.5, which identifies
the "infinitesimal" of a formal deformation with an appropriate cohomology
class in H1(F*). Intuitively, the "infinitesimal" is the initial velocity of the
formal deformation.
Section 4 begins with a discussion of formal automorphisms of finite order
and how such objects can be extended to higher order ones. See Theorem 4.3
and Corollary 4.4. These results are needed to study rigidity. An S-algebra
A is called rigid if every formal deformation of A is equivalent to the trivial
one. The main result there is Corollary 4.7, which states that an S-algebra A
is rigid, provided that both H1(F*) and HH2(A) are trivial. Here HH2(A)
denotes the second Hochschild cohomology of A, as an algebra over the ring
of integers, with coefficients in A itself.
In Section 5, we identify the obstructions to extending a 1-cocycle in F*
to a formal deformation. This is done by considering formal deformations
of finite orders and identifying the obstructions to extending such objects to
higher order ones. This obstruction turns out to be in H2(F*); see Theorem
5.3. As a result, the vanishing of H2(F*) implies that every 1-cocycle occurs
as the infinitesimal of a formal deformation (Corollary 5.5). The paper
ends with Theorem 5.7, which shows that the vanishing of a certain class
in H1(F*) implies that two order m + 1 extensions of an order m formal
deformation are equivalent.
DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA 3
2. The Landweber-Novikov algebra and the complex F*
The purpose of this preliminary section is to recall the Landweber-Novikov
algebra S and the notion of an algebra over it. Then we construct a cochain
complex F* associated to an algebra over the Landweber-Novikov algebra.
This complex will be used in later sections to study algebraic deformations
of S-algebras.
2.1. The Landweber-Novikov algebra. References for this subsection
are [8, 10], where the Landweber-Novikov algebra was first introduced.
Wood's paper [12] has an alternative description of it as a certain alge-
bra of differential operators. The book [1, Ch. I] by Adams is also a good
reference.
The Landweber-Novikov algebra S is generated by certain elements sff,
the Landweber-Novikov operations, indexed by the exponential sequences,
which we first recall.
An exponential sequence is a sequence
ff = (ff1, ff2, . .).
of non-negative integers in which all but a finite number of the ffi are 0.
When ff and fi are two exponential sequences, their sum ff + fi is defined
componentwise. Denote by E the set of all exponential sequences.
For each exponential sequence ff, there is a stable cobordism cohomology
operation
0
sff:MU*(-) ! MU* (-),
called a Landweber-Novikov operation. The composition sffsfi of any
two Landweber-Novikov operations satisfies the product formula, which ex-
presses it uniquely as a finite Z-linear combination,
X
(2.1.1) sffsfi = nflsfl.
fl2P(ff,fi)
See, for example, Adams [1, xx5,6] or Wood [12, Thm. 4.2] for a proof of this
fact. For our purposes, we do not really need to know what the integers nfl
are, as long as they satisfy the obvious associativity condition.
Thus, using the product formula (2.1.1), the free abelian group
M
(2.1.2) S = Zsff
ff2E
can be equipped with an algebra structure with composition as the product.
This is what we referred to as the Landweber-Novikov algebra.
The cobordism MU*(X) of a space or a spectrum X is automatically
an S-module. In fact, it is known that every stable cobordism cohomology
operation can be written uniquely as an MU*-linear combination of the
4 DONALD YAU
Landweber-Novikov operations. (Here MU* denotes MU*(pt) as usual.) If
X is a space, then MU*(X) is not just a group but an algebra as well. On
a product of two elements, the Landweber-Novikov operations satisfy the
Cartan formula,
X
(2.1.3) sff(ab) = sfi(a)sfl(b).
fi+fl=ff
Here ff 2 E and a, b 2 MU*(X). We, therefore, make the following defini-
tion.
Definition 2.2. By an algebra over the Landweber-Novikov algebra, or an S-
algebra for short, we mean a commutative ring A with 1 that comes equipped
with an S-module structure such that the Cartan formula (2.1.3)is satisfied
for all ff 2 E and a, b 2 A.
Given a commutative ring A, let End (A) denote the algebra of additive
self-maps of A, where product is composition of self maps. If f and g are
in End (A), then fg always means the composition f O g. An S-algebra
structure on A is equivalent to a function
(2.2.1) s*: E ! End(A),
assigning to each exponential sequence ff an additive operation sffon A,
which satisfies the product formula (2.1.1)on compositions and the Cartan
formula (2.1.3)on products in A.
2.3. The complex F*. The algebraic deformation theory of S-algebras dis-
cussed in later sections depends on a certain cochain complex F*, which we
now construct.
First we need some notations. Let A be a commutative ring. Denote by
Der(A) the abelian group of derivations on A. Recall that a derivation on
A is an additive self-map f :A ! A which satisfies the condition
f(ab) = af(b) + f(a)b
for all a, b 2 A. For a positive integer n, we use the shorter notation A n to
denote the tensor product over the ring of integers of A with itself n times.
The group of additive maps A n ! A is written Hom (A n , A). When n = 1,
we also write End (A) for Hom (A, A). Each set Hom (A n , A) has a natural
abelian group structure. Namely, given f, g :A n ! A and b 2 A n , the
sum (f + g) sends b to f(b) + g(b).
Let Set be the category of sets. Given two sets C and D, the set of
functions from C to D is written Set(C, D). If D is an abelian group, then,
just as in the previous paragraph, the set Set(C, D) is naturally equipped
with an abelian group structure as well.
Let A be an algebra over the Landweber-Novikov algebra S (see Definition
2.2). Consider the Landweber-Novikov operations on A as a function s* as in
DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA 5
(2.2.1). Recall that E is the set of all exponential sequences. Also recall the
sets P (ff, fi) from the product formula (2.1.1). Denote by En the Cartesian
product E x . .x.E (n factors). Just as in the case of s*, if f is a function
with domain E, we will write fffinstead of f(ff) for ff 2 E. We are now
ready to define the cochain complex F* = F*(A) of abelian groups.
We make the following definitions.
o F0(A) = Der(A).
o F1(A) = Set(E, End(A)).
o For integers n 2, set
Fn(A) = Fn0(A) x Fn1(A),
where
Fn0(A)= Set (En, End(A)),
Fn1(A)= Set (E, Hom (A n , A)).
Now we define the differentials.
o d0(') = s*' - 's* for ' 2 F0(A).
o For n 1, dn = (dn0, dn1), where
dn0:Fn0(A) ! Fn+10(A),
dn1:Fn1(A) ! Fn+11(A)
are defined as follows. (Here we have F10(A) = F11(A) = F1(A).)
Suppose that f = (f0, f1) is an element of Fn(A) for some n 2
(or just f when n = 1), x = (ff1, . .,.ffn+1) 2 En+1, ff 2 E, and
a = a1 . . .an+1 2 A (n+1). Then we set
(2.3.1)
(dn0f0)(x) = sff1f0(ff2, . .,.ffn+1)
Xn ~ X ~
+ (-1)i nfif0(. .,.ffi-1, fi, ffi+2, . .).
i=1 fi2P(ffi,ffi+1)
+ (-1)n+1f0(ff1, . .,.ffn)sffn+1,
where P (-, -) is as in the product formula (2.1.1), and
(2.3.2)
X
(dn1f1)(ff)(a) = sfi(a1)f1(fl)(a2 . . .an+1)
fi+fl=ff
Xn
+ (-1)if1(ff)(. . .ai-1 (aiai+1) ai+2. .).
i=1 X
+ (-1)n+1 f1(fi)(a1 . . .an)sfl(an+1).
fi+fl=ff
In these definitions, when n = 1, both f0 and f1 are interpreted as
f.
6 DONALD YAU
Proposition 2.4. (F*, d*) is a cochain complex of abelian groups.
Proof.It is straightforward to check that d1d0 = 0 by direct inspection.
For i = 0, 1 and n P 1, the formulas above allow one to rewrite dnias an
alternating sum dni= n+1j=0(-1)j@ni[j], corresponding to the n + 2 terms in
the respective formulas. It is straightforward to check that, for n 1, the
cosimplicial identities,
@n+1i[l] O @ni[k] = @n+1i[k] O @ni[l - 1] (0 k < l n + 2)
hold. Therefore, it follows that F*(A) is a cochain complex.
From now on, whenever we speak of "cochains," "cocycles," "cobound-
aries," and "cohomology classes," we are referring to the cochain complex
F* = (F*, d*), unless stated otherwise. The ith cohomology group of F* will
be denoted by Hi(F*).
3. Formal deformations and automorphisms
The purposes of this section are (1) to introduce formal deformations and
automorphisms of an S-algebra and (2) to identify the "infinitesimal" of a
formal deformation with an appropriate cohomology class in the complex
F*.
3.1. Formal deformations. Throughout this section, let A be an arbi-
trary but fixed S-algebra. Recall that the S-algebra structure on A can be
characterized as a function s* as in (2.2.1)from the set E of exponential se-
quences to the algebra of additive self maps of A. In addition, this function
satisfies the Cartan formula (2.1.3)and the product formula (2.1.1). We will
deform the Landweber-Novikov operations on A with respect to these two
properties.
We define a formal deformation of A to be a formal power series in the
indeterminate t,
(3.1.1) oet*= s* + ts1*+ t2s2*+ t3s3*+ . . .,
in which each si*2 F1(A) (i 1), i.e. is a function E ! End (A), satisfying
the following two properties (with s0*= s* and si*(ff) = siff):
o The Cartan formula: For every ff 2 E and a, b 2 A, the equality
X
(3.1.2) oetff(ab) = oetfi(a)oetfl(b)
fi+fl=ff
of power series holds.
DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA 7
o The Product formula: For ff, fi 2 E, the equality
X
(3.1.3) oetffoetfi= nfloetfl
fl2P(ff,fi)
of power series holds.
We pause to make a few remarks. First, the superscript i in si*is an
index, not an exponent, whereas ti is the ith power of the indeterminate t.
Second, sums and products of two power series are taken in the usual way,
with t commuting with every term in sight. The coefficients in (3.1.2)and
(3.1.3)are in A and End (A), respectively, and their calculations are done in
the appropriate rings. In particular, multiplying out the right-hand side of
the equation, one observes that the Cartan formula (3.1.2)is equivalent to
the equality
Xn X
(3.1.4) snff(ab) = sifi(a)sn-ifl(b)
i=0fi+fl=ff
in A for all n 0, ff 2 E, and a, b 2 A. Similarly, unwrapping the left-
hand side of the equation, one observes that the Product formula (3.1.3)is
equivalent to the equality
Xn X
(3.1.5) siffsn-ifi= nflsnfl
i=0 fl2P(ff,fi)
in End (A) for all n 0 and ff, fi 2 E. When n = 0, (3.1.4)and (3.1.5)
are just the original Cartan formula (2.1.3)and product formula (2.1.1),
respectively, for s* = s0*.
Setting t = 0 in the formal deformation oet*, we obtain oe0*= s*. So
we can think of oet*as a one-parameter curve with s* at the original. We,
therefore, call s1*the infinitesimal, since it is the "initial velocity" of the
formal deformation oet*.
3.2. Formal automorphisms. In order to identify the infinitesimal as an
appropriate cohomology class in F*, we need a notion of equivalence of
formal deformations.
By a formal automorphism on A, we mean a formal power series
(3.2.1) t = 1 + tOE1 + t2OE2 + t3OE3 + . . .,
where 1 = IdA and each OEi2 End (A), satisfying multiplicativity,
(3.2.2) t(ab) = t(a) t(b)
for all a, b 2 A.
8 DONALD YAU
The same rules of dealing with power series apply here as well. In partic-
ular, multiplicativity is equivalent to the equality
Xn
(3.2.3) OEn(ab) = OEi(a)OEn-i(b)
i=0
for all n 0 and a, b 2 A, in which OE0 = 1. The condition when n = 0 is
trivial, as it only says that the identity map on A is multiplicative. When
n = 1, the condition is
(3.2.4) OE1(ab) = aOE1(b) + OE1(a)b,
which is equivalent to say that OE1 is a derivation on A. More generally, if
OE1 = OE2 = . .=.OEk = 0, then OEk+1 is a derivation on A.
It is an easy exercise in induction to see that a formal automorphism t
has a unique formal inverse
(3.2.5) -1t = 1 - tOE1 + t2(OE21-OE2) + t3(-OE31+OE1OE2+OE2OE1-OE3) + . .,.
for which
t -1t = 1 = -1t t.
The coefficient of tn in -1tis an integral polynomial in OE1, . .,.OEn. More-
over, the multiplicativity of t implies that of -1t. Indeed, for elements
a, b 2 A, we have
ab = ( t -1t(a))( t -1t(b)) = t( -1t(a) -1t(b)),
which implies that -1tis multiplicative.
We record these facts as follows.
Lemma 3.3. Let t = 1 + tOE1 + t2OE2 + . .b.e a formal automorphism on
A. Then the first non-zero OEi (i 1) is a derivation on A. Moreover, the
formal inverse -1tof t is also a formal automorphism on A.
Now if sn*is a function E ! End (A), i.e. a 1-cochain, and if f and g are
in End (A), then we have a new 1-cochain fsn*g with
(fsn*g)(ff) = fsnffg
in End (A) for ff 2 E. Therefore, it makes sense to consider the formal power
series -1toet* t whenever t is a formal automorphism.
Proposition 3.4. Let oet*and t be, respectively, a formal deformation and
a formal automorphism of A. Then the formal power series -1toet* t is also
a formal deformation of A.
DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA 9
Proof.We need to check the Cartan formula (3.1.2)and the Product formula
(3.1.3)for "oet*= -1toet* t. For the Cartan formula, we have
"oetff(ab)= ( -1toetff)( t(a) t(b))
` X '
= -1t (oetfi t(a))(oetfl t(b))
fi+fl=ff
X
= ( -1toetfi t(a))( -1toetfl t(b)).
fi+fl=ff
We have used the Cartan formula for oet*and the multiplicativity of both t
and -1t, extended to power series. The Product formula is equally easy to
verify.
Given two formal deformations oet*and "oet*of A, say that they are equivalent
if and only if there exists a formal automorphism t on A such that
(3.4.1) "oet*= -1toet* t.
By Lemma 3.3 this is a well-defined equivalence relation on the set of formal
deformations of A.
Here is the main result of this section, which identifies the infinitesi-
mal with an appropriate cohomology class in H1(F*). Recall that cocycles,
coboundaries, cochains, and cohomology classes are all taken in the cochain
complex F* = F*(A).
P
Theorem 3.5. Let oet*= n tnsn*be a formal deformation of A. Then the
infinitesimal s1*is a 1-cocycle, i.e. d1s1*= 0. Moreover, the cohomology class
[s1*] is an invariant of the equivalence class of oet*.
More generally, if s1*= . . .= sk*= 0 for some positive integer k, then
sk+1*is a 1-cocycle.
Proof.To show that s1*is a 1-cocycle, we need to prove that d1is1*= 0 for
i = 0, 1. When n = 1 the Product formula (3.1.5)states that
X
sffs1fi+ s1ffsfi = nfls1fl,
fl2P(ff,fi)
and so
X
(d10s1*)(ff, fi) = sffs1fi- nfls1fl+ s1ffsfi = 0.
fl2P(ff,fi)
Similarly, the Cartan formula (3.1.4)when n = 1 states that
X
s1ff(ab) = (sfi(a)s1fl(b) + s1fi(a)sfl(b)),
fi+fl=ff
which implies that
(d11s1*)ff(a b) = 0,
as desired. This shows that s1*is a 1-cocycle.
10 DONALD YAU
If s1*= . . .= sk*= 0, then the same argument as above shows that
sk+1*= 0.
P
Now suppose that "oet*= n tn"sn*is a formal deformation of A that is
equivalent to oet*. This means that there exists a formal automorphism t
such that
o"et*= -1toet* t
s* + t(s1*+ s*OE1 - OE1s*) (mod t2)
s* + t(s1*+ d0OE1) (mod t2).
In particular, the 1-cocycle ("s1*- s1*) 2 F1 is a 1-coboundary d0OE1. (Remem-
ber that OE1 is a derivation on A, which is therefore a 0-cochain.) Thus, the
cohomology classes, ["s1*] and [s1*], are equal in H1(F*), as asserted.
In view of this Theorem, it is natural to ask whether a given cohomology
class in H1(F*) is the infinitesimal of a formal deformation. This question
will be dealt with in Section 5.
4. Extending formal automorphisms and rigidity
As in the previous section, A will denote an S-algebra with Landweber-
Novikov operations s* and F* = F*(A).
The main purpose of this section is to obtain cohomological conditions
under which A is rigid, that is, every formal deformation is equivalent to the
trivial deformation s*. To do that, we first have to consider how truncated
formal automorphisms can be extended.
4.1. Formal automorphisms of finite order. Let m be a positive inte-
ger. Inspired by Gerstenhaber-Wilkerson [7], we define a formal automor-
phism of order m on A to be a formal power series
t = 1 + tOE1 + t2OE2 + . . .+ tm OEm
with 1 = IdA and each OEi 2 End (A), satisfying multiplicativity (3.2.2)
modulo tm+1 , i.e. the equality (3.2.3)holds for 0 n m and all a, b 2 A.
One can think of a formal automorphism as a formal automorphism of order
1.
Given such a t, we say that it extends to a formal automorphism of
order m + 1 if and only if there exists OEm+1 2 End (A) such that the power
series
(4.1.1) " t= t + tm+1 OEm+1
is a formal automorphism of order m+1. Such a " tis said to be an extension
of t to order m + 1.
DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA 11
The question we would like to address here is this: Given a formal auto-
morphism t of order m, can it be extended to a formal automorphism of
order m + 1? It turns out that the obstruction to the existence of such an
extension lies in Hochschild cohomology, which we now recall.
Consider A as a unital algebra over the ring of integers Z, the Hochschild
cochain complex C*(A, A) of A with coefficients in A itself is defined as
follows. The nth dimension is the group
Cn(A, A) = Hom (A n , A).
The differential
bn-1 :Cn-1 (A, A) ! Cn(A, A)
is given by the alternating sum
(4.1.2) (bn-1f)(a1 . . .an) = a1f(a2 . . .an) +
n-1X
(-1)if(. .a.i-1 aiai+1 ai+2. .).+ (-1)nf(a1 . . .an-1)an.
i=1
The reader can consult, for example, Weibel [11] for more detailed discus-
sions about Hochschild cohomology. The nth Hochschild cohomology group
of A over Z with coefficients in A itself is denoted by HHn (A).
We now return to the question of extending a formal automorphism t
of order m to one of order m + 1. Consider the Hochschild 2-cochain
Ob ( t): A 2 ! A
given by
mX
Ob ( t)(a b) = - OEi(a)OEm+1-i (b)
i=1
for all a, b 2 A. Call Ob ( t) the obstruction class of t.
Lemma 4.2. The obstruction class Ob ( t) is a Hochschild 2-cocycle.
Proof.We calculate as follows:
(b2 Ob( t))(a b c)
mX Xm
= -a OEi(b)OEm+1-i (c) + OEi(ab)OEm+1-i (c)
i=1 i=1
Xm mX
- OEi(a)OEm+1-i (bc) + c OEi(a)OEm+1-i (b)
X i=1 i=1X
= OEi(a)OEj(b)OEk(c) - OEi(a)OEj(b)OEk(c).
i+j+k = m+1 i+j+k = m+1
i, k>0 i, k>0
= 0
12 DONALD YAU
Here we used the multiplicativity of t, namely, (3.2.3) for OEi(ab) and
OEm+1-i (bc). Also, OE0 = 1 as usual. This shows that the obstruction class of
t is a Hochschild 2-cocycle.
We are now ready to show that the obstruction to extending tto a formal
automorphism of order m + 1 is exactly the cohomology class [Ob ( t)] 2
HH2(A).
Theorem 4.3. Let t be a formal automorphism of order m on A. Then
t extends to a formal automorphism of order m + 1 if and only if the
obstruction class Ob ( t) is a Hochschild 2-coboundary.
Proof.The existence of an order m+1 extension " tas in (4.1.1)is equivalent
to the existence of a OEm+1 2 End (A) for which (3.2.3)holds for n = m + 1.
This last condition can be rewritten as
Ob ( t)(a b) = -OEm+1 (ab) + aOEm+1 (b) + bOEm+1 (a) = (b1OEm+1 )(a b),
and b1OEm+1 is a Hochschild 2-coboundary.
An immediate consequence of this Theorem is that, starting with a deriva-
tion OE, the vanishing of the Hochschild cohomology HH2(A) implies that
the formal automorphism t = 1 + tm OE of order m 1 can always be
extended to a formal automorphism. This will be useful when we discuss
rigidity below.
Corollary 4.4. Let m be a positive integer and let OE be a derivation on A.
Assume that HH2(A) = 0. Then there exists a formal automorphism on A
of the form
t = 1 + tm OE + tm+1 OEm+1 + tm+2 OEm+2 + . . ..
Proof.Using the hypothesis that OE is a derivation, it is straightforward to
verify that the formal power series
t = 1 + tm OE
is a formal automorphism of order m, i.e. it satisfies multiplicativity (3.2.3)
for n m. As HH2(A) = 0, by Theorem 4.3 the obstructions to extending
t to a formal automorphism vanish, as desired.
4.5. Rigidity. Following the terminology in [3], an S-algebra A is said to be
rigid if every formal deformation of A is equivalent to the trivial deformation
s*.
Using the results above, we will be able to obtain cohomological criterion
for the rigidity of A. First we need the following consequence of Corollary
4.4.
DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA 13
Corollary 4.6. Let k be a positive integer and let
oet* = s* + tksk*+ tk+1sk+1* + . . .
be a formal deformation of A in which sk*2 F1 is a 1-coboundary. Suppose
that HH2(A) = 0. Then there exists a formal automorphism of the form
(4.6.1) t = 1 - tkOEk + tk+1OEk+1 + . . .
such that
(4.6.2) -1toet* t s* (mod tk+1).
Proof.By the hypothesis on sk*, there exists a derivation OEk 2 Der(A) such
that
d0OEk = s*OEk - OEks* = sk*.
Since -OEk is also a derivation on A, Corollary 4.4 implies that there exists
a formal automorphism t as in (4.6.1). Computing modulo tk+1, we have
-1toet* t (1 + tkOEk)(s* + tksk*)(1 - tkOEk)
s* + tk(sk*+ OEks* - s*OEk)
s*.
This proves the Corollary.
By Proposition 3.4, the formal power series -1toet* t is actually a formal
deformation that is equivalent to oet*. Therefore, by applying this Corollary
repeatedly, we obtain the following cohomological criterion for A to be rigid.
Corollary 4.7. If HH2(A) = 0 and H1(F*(A)) = 0, then A is rigid.
We should point out that in the deformation theory of some other kinds
of algebras, rigidity is usually guaranteed by the vanishing of just one coho-
mology group, usually an H1 or an H2.
5. Extending cocycles
As before, A will denote an arbitrary but fixed S-algebra, and F* = F*(A).
The Landweber-Novikov operations on A are given by a function s* as in
(2.2.1).
Recall from Theorem 3.5 that the infinitesimal of a formal deformation is
a 1-cocycle in F*. The purpose of this section is to answer the questions: (1)
What is the obstructions to extending a 1-cocycle to a formal deformation?
(2) What is the obstructions to two such extensions being equivalent?
We will break these questions into a sequence of smaller questions, each
of which is dealt with in an obstruction theoretic way.
14 DONALD YAU
5.1. Formal deformations of finite order. First we need some defini-
tions. Let m be a positive integer. As in the previous section, a formal
deformation of order m of A is a formal power series
(5.1.1) oet*= s* + ts1*+ . .+.tm sm*,
in which each si*(i 1) is in F1 such that the Cartan formula (3.1.2)and the
Product formula (3.1.3)are satisfied modulo tm+1 . In other words, (3.1.4)
and (3.1.5)are satisfied for all n m.
We say that a formal deformation oet*of order m extends to a formal de-
formation of order M > m if and only if there exist 1-cochains sm+1*, . .,.sM*
such that the power series
(5.1.2) "oet*= oet*+ tm+1 sm+1* + . . .+ tM sM*
is a formal deformation of order M. Call "oet*an order M extension of oet*.
One can think of a formal deformation as a formal deformation of order
1. Given a 1-cocycle s1*, the problem of extending the formal deformation
oet*= s* + ts1*
of order 1 to a formal deformation can be thought of as extending oet*to order
2, then order 3, and so forth. We will do this by identifying the obstruction
to extending an order m formal deformation to one of order m + 1.
Let oet*be a formal deformation of order m as in (5.1.1). Consider the
2-cochain
Ob(oet*) = (Ob 0(oet*), Ob1(oet*)) 2 F2 = F20x F21,
whose components are defined by the conditions,
mX
(5.1.3) Ob 0(oet*)(ff, fi) = - siffsm+1-ifi (ff, fi 2 E)
i=1
and
mX X
(5.1.4)Ob 1(oet*)ff(a b) = - sifi(a)sm+1-ifl(b) (ff 2 E, a, b 2 A).
i=1fi+fl=ff
We call Ob (oet*) the obstruction class of oet*.
Lemma 5.2. The obstruction class Ob (oet*) is a 2-cocycle.
Proof.We need to show that d2iObi(oet*) = 0 for i = 0, 1. Since oet*is the
only formal deformation we are dealing with, we abbreviate Ob i(oet*) to Ob i.
DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA 15
For i = 0, let ff, fi, fl be exponential sequences. Then we have
(5.2.1)
` Xm ' X ` mX '
(d20Ob0)(ff, fi, fl) = -sff sifism+1-ifl + nffi siffism+1-ifl
i=1 ffi2P(ff,fi)i=1
X ` mX ' ` Xm '
- n" siffsm+1-i" + siffsm+1-ifisfl.
"2P(fi,fl) i=1 i=1
This is a sum of four terms. Using the Product formula (3.1.5), the second
term can be rewritten as
Xm ` X ' mX Xi
nffisiffism+1-ifl= slffsi-lfism+1-ifl.
i=1 ffi2P(ff,fi) i=1l=0
In particular, the m summands in it corresponding to l = 0 cancel out with
the first term in (5.2.1). Consequently, the sum of the first two terms in
(5.2.1)becomes
X j
(5.2.2) siffsfiskfl.
i+j+k = m+1
i,k>0
An analogous argument applied to the third and fourth terms in (5.2.1)
shows that their sum is exactly (5.2.2)with the opposite sign. It follows
that d20Ob0 = 0, as desired.
The proof of d21Ob1 = 0 is quite similar to the argument above and the
proof of Lemma 4.2. Indeed, (d21Ob1)ff(a b c) (see (?? )) is again the
sum of four terms. Using the Cartan formula (3.1.4), its second term can
be rewritten as
(5.2.3)
mX X
-(Ob 1)ff(ab c)= sifi(ab)sm+1-ifl(c)
i=1fi+fl=ff
mX X ` Xi X '
= sj~(a)si-j~(b) sm+1-ifl(c).
i=1fi+fl=ffj=0~+~=fi
The summands corresponding to j = 0 cancel out with the first of the four
terms in (d21Ob1)ff(a b c). In particular, the sum of the first two terms
in (d21Ob1)ff(a b c) is
X
(5.2.4) sifi(a)sjfl(b)skffi(c).
The summation is taken over all triples (i, j, k) of non-negative integers and
all triples (fi, fl, ffi) of exponential sequences for which
i + j + k = m + 1 (i, k > 0)
and
fi + fl + ffi = ff.
16 DONALD YAU
A similar argument applies to the last two terms in (d21Ob1)ff(a b c),
showing that their sum is exactly (5.2.4)with the opposite sign. It follows
that d21Ob1 = 0, as expected.
This finishes the proof of the Lemma.
As the obstruction class of oet*is a 2-cocycle, it represents a cohomology
class in H2(F*).
We are now ready to present the main result of this section, which iden-
tifies the obstruction to extending a formal deformation of order m to one
of order m + 1.
Theorem 5.3. Let oet*be a formal deformation of order m of A. Then oet*
extends to a formal deformation of order m+1 if and only if the cohomology
class [Ob (oet*)] 2 H2(F*(A)) vanishes.
Proof.Indeed, the existence of an order m+1 extension "oet*of oet*, as in (4.1.*
*1)
with M = m + 1, is equivalent to the existence of a 1-cochain sm+1*2 F1 for
which the Cartan formula (3.1.4)and the Product formula (3.1.5)both hold
for n = m + 1. Simply by rearranging terms, the Cartan formula (3.1.4)
when n = m + 1 can be rewritten as
(Ob 1(oet*))ff(a b) = (d11sm+1*)ff(a b).
Similarly, the Product formula (3.1.5)when n = m + 1 is equivalent to
(Ob 0(oet*))(ff, fi) = (d10sm+1*)(ff, fi).
These two conditions together are equivalent to
(5.3.1) Ob(oet*) = d1sm+1*,
i.e., the obstruction class is a 2-coboundary. Since Ob (oet*) is a 2-cocycle
(Lemma 5.2), the Theorem is proved.
Applying this Theorem repeatedly, we obtain the obstructions to extend-
ing a 1-cocycle to a formal deformation.
Corollary 5.4. Let s1*2 F1(A) be a 1-cocycle. Then there exist a sequence
of classes !1, !2, . .2.H2(F*(A)) for which !n (n > 1) is defined if and only
if !1, . .,.!n-1 are defined and equal to 0. Moreover, the formal deformation
oet* = s* + ts1*
of order 1 on A extends to a formal deformation if and only if !i is defined
and equal to 0 for each i = 1, 2, . ...
Since these obstructions lie in H2(F*(A)), the triviality of this group
implies that extensions always exist.
Corollary 5.5. If H2(F*(A)) = 0, then every formal deformation of order
m 1 of A extends to a formal deformation.
DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA 17
5.6. Equivalence of formal deformations of finite order. Let oet*be a
formal deformation of A of order m 1, and let "oet*and ~oet*be two order m+1
extensions of it. Are the two extensions equivalent? This is the question
that we would like to address in this final section.
First, we need a definition of equivalence. Two formal deformation oet*
and "oet*of order m of A are said to be equivalent if and only if there exists
a formal automorphism t of order m such that the equality (3.4.1)holds
modulo tm+1 .
This is a well-defined equivalence relation on the set of formal deforma-
tions of order m of A. In fact, it is easy to see that a formal automorphism
t of order m has a formal inverse -1tas in (3.2.5)which is also a formal
automorphism of order m.
Now let oet*= s* + . .+.tm sm* be a formal deformation of order m of A,
and let "oet*= oet*+ tm+1 "sm+1*and ~oet*= oet*+ tm+1 ~sm+1*be two order m + 1
extensions of oet*. Then (5.3.1)in the proof of Theorem 5.3 tells us that
d1"sm+1*= Ob(oet*) = d1~sm+1*.
It follows that ("sm+1*- ~sm+1*) 2 F1 is a 1-cocycle, and it makes sense to
consider the cohomology class in H1(F*) represented by it.
Theorem 5.7. If the cohomology class ["sm+1*- ~sm+1*] 2 H1(F*) vanishes,
then the formal deformations, "oet*and ~oet*, of order m + 1 are equivalent.
Proof.The argument is quite similar to the proofs of Theorem 3.5 and Corol-
lary 4.6. In fact, by hypothesis, there exists a derivation OE on A such that
"sm+1*- s~m+1*= d0OE = s*OE - OEs*.
We have the formal automorphism
t = 1 + tm+1 OE
of order m + 1 on A. Computing modulo tm+2 , we have
-1t~oet* t (1 - tm+1 OE)(oet*+ tm+1 ~sm+1*)(1 + tm+1 OE)
oet*+ tm+1 (~sm+1*+ s*OE - OEs*)
oet*+ tm+1 "sm+1*
"oet*,
as desired.
This obstruction theoretic result is less satisfactory than the analogous
Theorem 4.3 and Theorem 5.3 in that the author is not sure whether the
equivalence of "oet*and ~oet*would imply the vanishing of the class ["sm+1*-~sm*
*+1*].
18 DONALD YAU
Acknowledgement
The author thanks the referee for reading an earlier version of this paper.
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E-mail address: dyau@math.ohio-state.edu
Department of Mathematics, The Ohio State University Newark, 1179 Uni-
versity Drive, Newark, OH 43055