COHOMOLOGY OF ~-RINGS
DONALD YAU
Abstract.A cohomology theory for ~-rings is developed. This is then appl*
*ied
to study deformations of ~-rings.
1.Introduction
The notion of a ~-ring was introduced by Grothendieck to study algebraic ob-
jects endowed with operations that act like exterior powers. Since its introduc*
*tion
in the 1950s, ~-rings have been shown to play important roles in several areas *
*of
mathematics. For example, in Algebraic Topology, the unitary K-theory of a topo-
logical space is a ~-ring. When X is a finite CW complex, the ~-operations on
K(X) are induced by exterior powers of vector bundles on X. Similarly, the com-
plex representation ring R(G) of a group G is a ~-ring with ~-operations given *
*by
the exterior powers of representations. There is also an abundant supply of ~-r*
*ings
from Algebra itself. If R is a commutative ring with unit, it can be shown that*
* its
universal Witt ring W(R) is always a ~-ring [5].
The purposes of this note are (i) to introduce a cohomology theory for ~-rings
and (ii) to use this to study ~-ring deformations along the lines of Gerstenhab*
*er's
theory [3].
This note is organized as follows. The following section contains a brief acc*
*ount
of the basics of ~-rings and their Adams operations. In Section 3, we define fo*
*r a
given ~-ring R a cochain complex F* (see (3.4)) whose cohomology groups, denoted
H*~(R), are the ~-ring cohomology groups of R. Several basic observationsPare
made. First, the differential dn for n 1 in F* is an alternating sum (-1)i@*
*i.
There are öc degeneracy" maps oei:Fn ! Fn+1 for n 2 such that the @i and oei
satisfy the cosimplicial identities in dimensions n 2. In fact, F* is a subco*
*mplex
of a certain Hochschild cochain complex ~F*, defined in x3.3, which coincides w*
*ith
F* in dimensions 2 and above. The cosimplicial identities in F* come from the
cosimplicial abelian group that gives rise to the Hochschild complex ~F*. Moreo*
*ver,
there is a composition product on F* that induces a product on cohomology, maki*
*ng
H*~(R) a graded, associative, unital algebra (Corollary 3.7). The section ends *
*with
interpretations of H0~and H1~and the computation of these cohomology groups for
the ~-ring Z (x3.8).
Section 4 is devoted to studying algebraic deformations of ~-rings, making use
of the ~-ring cohomology in Section 3. In particular, the infinitesimal deforma*
*tion
is a 1-cocycle in F* whose cohomology class is well-defined by the equivalence
class of the deformation (Proposition 4.1). It follows that the vanishing of H1*
*~(R)
____________
Date: April 7, 2004.
1
2 DONALD YAU
implies that R is rigid (Corollary 4.3), meaning that every deformation of R is
equivalent to the trivial deformation. The question of extending a 1-cocycle to*
* a
deformation, or "integrability" in the terminology of Gerstenhaber [3], is stud*
*ied
next. Given a 1-cocycle, the obstruction to extending it to a deformation is a
sequence of 2-cocycles (Theorem 4.5). This means that the simultaneous vanishing
of their cohomology classes is equivalent to the extendibility of the given 1-c*
*ocycle
to a deformation (Corollary 4.6). It follows, in particular, that extendibility*
* of a
1-cocycle is automatic if H2~(R) is trivial (Corollary 4.7). The question of wh*
*en two
extensions are equivalent is also considered (Proposition 4.8).
One thing that is clearly missing in ~-ring cohomology is naturality. A ~-ring
map does not in general induce a map in ~-ring cohomology. There is one excepti*
*on,
which is when the map is a ~-ring self-map. This is due to the fact that the al*
*gebra
of linear endomorphisms is used in the definition of ~-ring cohomology. A map of
rings, or even of ~-rings, does not in general induce a map on the algebras of *
*linear
endomorphisms. So we only have naturality in the category whose sole object is
the ~-ring under consideration and whose morphisms are its ~-ring self-maps. Ev*
*en
in this restricted category, the induced map is only a map of graded groups, as*
* it
does not preserve the composition product.
2.~-rings and Adams operations
In preparation for studying ~-ring cohomology in the next two sections, in th*
*is
section we briefly review some basic definitions about ~-rings and Adams opera-
tions. For more discussions about ~-rings, consult Atiyah and Tall [1] or Knuts*
*on
[6]. The author's articles [9, 10] contain some recent results on ~-rings which*
* might
also be of interest to the reader.
2.1. ~-rings. By a ~-ring we mean a unital, commutative ring R endowed with
functions
~i:R ! R (i 0),
called ~-operations, which satisfy the following conditions. For any integers i*
*, j 0
and elements r and s in R:
o ~0(r) = 1.
o ~1(r) = r.
o ~i(1) = 0 forPi > 1.
o ~i(r + s) = ik=0~k(r)~i-k(s).
o ~i(rs) = Pi(~1(r), . .,.~i(r); ~1(s), . .,.~i(s)).
o ~i(~j(r)) = Pi,j(~1(r), . .,.~ij(r)).
The Pi and Pi,jare some universal polynomials with integer coefficients. See the
references mentioned above for the exact definitions of these polynomials. Note*
* that
what we call a ~-ring here is sometimes called a "special" ~-ring in the litera*
*ture.
n
For example, the ring of integers Z is a ~-ring with ~i(n) = i . In this cas*
*e, all
the Adams operations (to be reviewed below) are equal to the identity map on Z.
This is the only ~-ring structure on Z.
COHOMOLOGY OF ~-RINGS 3
One important property of a ~-ring is that it must have characteristic 0. This
can be seen from the linear map
n X o
~t:R ! 1 + tR[[t]] = aiti:ai2 R, a0 = 1
defined by X
~t(r) = ~i(r)ti.
Here the additive group structure on 1 + tR[[t]] is given by the usual multipli*
*cation
of power series. The image of n under ~t is (1 + t)n, which is nonzero in 1 + t*
*R[[t]]
for any n. In particular, for any positive integer n and any prime p, the equat*
*ion
np n (mod pR)
holds. This will be used in the next section when we study the 0th ~-ring cohom*
*ol-
ogy group.
2.2. Adams operations. The ~-operations are sometimes hard to work with,
since they are neither additive nor multiplicative. One can extract ring maps f*
*rom
the ~-operations, obtaining the so-called Adams operations
_n :R ! R (n 1).
More precisely, they are defined by the Newton formula:
_n(r) - ~1(r)_n-1(r) + . .+.(-1)n-1~n-1(r)_1(r) + (-1)nn~n(r) = 0.
The Adams operations satisfy the following properties:
o All the _n are ring maps.
o _1 = Id.
o _m _n = _mn = _n_m .
o _p(r) rp (mod pR) for each prime p and element r in R.
Suppose given a unital, commutative ring R with self ring maps _n :R ! R
satisfying the above four properties of Adams operations. One can ask if it is *
*pos-
sible to use the Newton formula to go backward and to produce a ~-ring structure
on R. This is, in fact, possible provided that R is Z-torsionfree. More explici*
*tly,
a theorem of Wilkerson [8] says that if R is as stated in the first sentence of*
* this
paragraph and is Z-torsionfree, then there exists a unique ~-ring structure on R
whose Adams operations are exactly the given _n.
We note that a ring R with self ring maps _n :R ! R such that _1 = Idand
_m _n = _mn is sometimes called a "weight system" in the literature. See, for
example, Bar-Natan [2].
3.Cohomology of ~-rings
The main purpose of this section is to introduce our ~-ring cohomology groups.
This is done in 3.1. After that, we will discuss its connections with Hochschi*
*ld
cohomology in 3.3 and its product structure in 3.5. The section closes with a
discussion of the 0th and the 1st ~-ring cohomology groups.
Throughout this section, R will denote a ~-ring with ~-operations ~i(i 0) a*
*nd
Adams operations _n (n 1).
4 DONALD YAU
3.1. The complex F* and ~-ring cohomology. To define the complex F* =
F*(R) that gives rise to ~-ring cohomology, we first need to establish some not*
*ations.
Denote by End (R) the (non-commutative) algebra of Z-linear endomorphisms
of R, in which the product is given by composition. To make it clear that we are
composing two endomorphisms f and g, we will sometimes write f O g instead_of
just fg. We also need the following subalgebra of End (R). Denote by End (R) the
subalgebra of End (R) consisting of those linear endomorphisms f of R that sati*
*sfy
the condition,
(3.1) f(r)p f(rp) (mod pR),
for every prime p and each element r 2 R. We will use the symbol T to denote the
set of positive integers.
We are now ready to define the_complex F* = F*(R). Define F0 to be the
underlying additive group of End (R) and F1 to be the set of functions
f :T ! End (R)
satisfying the condition, f(p)(R) pR for every prime p. (The definitions of F0
and F1 might seem a little bit strange at first sight. The reason for defining *
*them
as such will become apparent when we discuss deformations of ~-ring in the next
section.) For n 2, Fn is simply defined to be the set of functions
f :T n ! End (R).
Each Fn (n 1) inherits the obvious additive group structure from End (R).
Namely, if f and g are elements of Fn, then
(f + g)(m1, . .,.mn)(r) = f(m1, . .,.mn)(r) + g(m1, . .,.mn)(r)
for (m1, . .,.mn) 2 T nand r 2 R.
For n 0, the differential dn :Fn ! Fn+1 is defined by the formula
(3.2)
Xn
dnf(m0, . .,.mn) = _m0 Of(m1, . .,.mn) + (-1)if(m0, . .,.mi-1mi, . .,.mn)
i=1
+ (-1)n+1f(m0, . .,.mn-1) O _mn .
The dn are clearly additive group maps, and the only thing that we have to make
sure is that the image of d0 does lie in F1. To see that this is the case, let *
*f be an
element of F0. Then for any prime p and element r in R, we have that
d0f(p)(r)= _p(f(r)) - f(_p(r))
f(r)p - f(rp) (mod pR)
0 (mod pR).
This shows that d0 is well-defined.
Lemma 3.2. For each n 0, we have dn+1dn = 0.
Proof.The identity d1d0 = 0 can be checked directly by writing out all six term*
*s.
COHOMOLOGY OF ~-RINGS 5
P n+1
For n 1, we can write dn as i=0(-1)i@i, where @i:Fn ! Fn+1 is the linear
map given by
8
>><_m0 O f(m1, . .,.mn) ifi = 0
(3.3) @if(m0, . .,.mn) = > f(m0, . .,.mi-1mi, . .,.mn)if1 i n
>: m
f(m0, . .,.mn-1) O _ n ifi = n + 1.
Using the property, _n_m = _mn , of the Adams operations, the öc simplicial
identities"
@j@i = @i@j-1 (i < j)
can then be verified by direct inspection. This implies, as usual, that dn+1dn*
* =
0.
Note that in this proof, we could have written d0 formally as @0 - @1 just as
above. However, @0 and @1 do not necessarily have images in F1.
The lemma gives us the cochain complex F* = F*(R) of abelian groups,
(3.4) 0 ! F0 d0-!F1 d1-!F2 d2-!. . .
with Fn in dimension n.
Definition 3.5. The nth cohomology group of F* = F*(R) is called the nth ~-ring
cohomology group of R, denoted by Hn~(R).
The differentials dn look a lot like those in Hochschild cohomology theory. T*
*here
is, in fact, a close relationship between the complex F* and Hochschild theory,*
* to
which we now turn.
3.3. Connections with Hochschild cohomology. Recall that T denotes the set
of positive integers and that R is a ~-ring . We will compare the complex F* wi*
*th
a certain Hochschild cochain complex. For general discussions about Hochschild
theory, refer to, for example, Weibel [7].
With the usual multiplication of integers, we can consider T as a multiplicat*
*ive,
commutative monoid. Then the underlying additive group of the algebra End (R)
is a bimodule over the monoid-ring Z[T ] via the action
T x End (R) x T ! End (R)
(m, f, n)7! _m O f O _n,
extended linearly to all of Z[T ]. Therefore, we can consider the Hochschild co*
*chain
complex C* = C*(Z[T ], End(R)) of the monoid-ring Z[T ] with coefficients in the
bimodule End (R) and with ground ring Z. The nth cohomology group of C*,
denoted by Hn(Z[T ], End(R)), is called the nth Hochschild cohomology group of
Z[T ] with coefficients in End (R).
There is a canonical ring isomorphism
Z[T ] n ~=Z[T n],
where the multiplication on the monoid T nis defined coordinatewise. Moreover, a
Z-linear map
Z[T n] ! End (R)
6 DONALD YAU
determines and is determined by a function
T n ! End (R).
Therefore, for n 2, there is a canonical bijection
Cn = Hom Z(Z[T n], End(R)) ~= Fn
which, as one can check directly, respects the additive group structures. Likew*
*ise,
for n = 0 and 1, one can identify Fn canonically as a subgroup of Cn. It is
also straightforward to see from (3.2)that, under the above identifications, the
differentials in F* correspond to those in C*. This allows us to identify F* a*
*s a
subcomplex of C*, and the two complexes coincide from dimension 2 onward. In
particular, we have the following result.
Proposition 3.4. There exist a canonical isomorphism
Hn~(R) ~= Hn(Z[T ], End(R))
for each n 3 and a canonical surjection
H2~(R) i H2(Z[T ], End(R)).
It is well-known that the cochain complex C* arises from a cosimplicial abeli*
*an
group C* with Cn = Cn for all n. In the proof of Lemma 3.2, the maps @i:Fn !
Fn+1 for n 2 are, under the identification F* C*, exactly the coface maps of
C*. There are also öc degeneracy äm ps
oei:Fn+1 ! Fn (i = 0, 1, . .,.n)
defined by
oeif(m1, . .,.mn) = f(. .,.mi, 1, mi+1, . .)..
The maps @i and oei satisfy the usual cosimplicial identities in dimensions 2 a*
*nd
above. Once again, under the identification of F* as a subcomplex of C*, these *
*are
the codegeneracy maps of C*.
3.5. Composition product. The purpose of this subsection is to observe that the
~-ring cohomology H*~(R) of R is a graded ring.
Theorem 3.6. Given a ~-ring R, there is an associative, bilinear pairing
- O - :Fn Fk ! Fn+k (n, k 0)
on the complex F* = F*(R) with IdR2 F0 as a two-sided unit. This pairing satisf*
*ies
the Leibnitz identity,
d(f O g) = (df) O g + (-1)|f|f O (dg),
where |f| is the dimension of f.
We call the pairing the composition product. The complex F* with the compo-
sition product is a differential graded algebra. The Leibnitz identity implies *
*that
the product descends to cohomology with [f] O [g] = [f O g], where [f] denotes *
*the
cohomology class of a cocycle.
Corollary 3.7. The composition product on F* induces a product on H*~(R), mak-
ing it into a graded, associative, unital algebra.
COHOMOLOGY OF ~-RINGS 7
Proof of Theorem 3.6.The pairing is defined as follows. Given f 2 Fn, g 2 Fk,
and (m1, . .,.mn+k) 2 T n+k, we set
(f O g)(m1, . .,.mn+k) = f(m1, . .,.mn) O g(mn+1, . .,.mn+k),
where the O on the right-hand side of the equation denotes composition of linear
endomorphisms of R. Associativity and bilinearity are straightforward to check,*
* as
is the assertion that IdR acts as a two-sided identity.
As for the Leibnitz identity, let f and g be as above and let (m0, . .,.mn+k)*
* be in
T n+k+1. Then d(f Og)(m0, . .,.mn+k) is the sum ofPn+k +2 linear endomorphisms
of R, n+k of which come from the alternating sum (-1)i(f Og)(. .,.mi-1mi, . .*
*)..
Using the fact that
(f O g)(. .,.mi-1mi, . .).
(
f(. .,.mi-1mi, . .,.mn) O g(mn+1, .i.).f1 i n
=
f(m0, . .,.mn-1) O g(. .,.mi-1mi, .i.).fn + 1 i n + k,
one observes that the terms for 1 i n (resp. n + 1 i n + k) correspond
to the n (resp. k) terms in (df) O g (resp. (-1)|f|f O (dg)) involving the alte*
*rnating
sum. It follows easily from this observation that the Leibnitz identity holds.
We remark that the composition product can also be defined on the Hochschild
cochain complex C*, and it has the same properties there. Moreover, the subcom-
plex inclusion F* C* is a map of differential graded algebras, and the induced
map on cohomology is a map of graded algebras.
3.8. H0~and H1~. The purpose of this subsection is to discuss some basic proper*
*ties
of the 0th and the 1st ~-ring cohomology groups. We will also compute these gro*
*ups
for the only ~-ring structure on Z.
____
Recall that End (R) is the group of linear endomorphisms f of R that satisfy *
*the
condition, f(r)p f(rp) (mod pR), for each prime p and each element r 2 R. The
following result, which describes H0~explicitly, is immediate from the definiti*
*on of
d0.
Proposition 3.9. For any ~-ring R, we have that
____ n n
H0~(R) = {f 2 End (R) :f_ = _ f for all}n.
Since a ~-ring R must have characteristic 0, for any integer k and any element
r 2 R, the congruence relation
(kr)p k(rp) (mod pR)
holds for each prime p._This_implies that the multiplication-by-k endomorphism,
fk: r 7! kr, lies in End (R). It is also clear that this map commutes with _n f*
*or
any n. In particular, we have the following consequence of the proposition.
Corollary 3.10. For any ~-ring R, H0~(R) contains Z as a canonical subgroup,
which consists of the multiplication-by-k endomorphisms of R.
8 DONALD YAU
Recall that the ring of integers Z has a unique ~-ring structure given by ~i(*
*n) =
n m
i with _ = Id for all m. Since any linear endomorphism f of Z sends n to
f(1)n, a special case of the above corollary is
Corollary 3.11. The ~-ring Z has H0~(Z) ~=Z.
We now turn to the group H1~.
From the definition of d1, the kernel of d1 consists of those functions f 2 F*
*1 such
that
f(mn) = _m O f(n) + f(m) O _n
for all m and n. Due to the similarity of this property with the defining prope*
*rty
for derivations, we call these maps ~-derivations (of R). On the other hand, the
image of d0 consists of those functions T ! End (R) of the form
[_*, g]: n 7-! _n O g - g O _n
____
for some g 2 End (R). _In_other words, they are just the functions obtained by
"twistingä g 2 F0 = End (R) by _*. Because of this, we call these maps ~-inner
derivations (of R).
In particular, we have
Proposition 3.12. H1~(R) is the quotient of the group of ~-derivations by the g*
*roup
of ~-inner derivations.
In the case of the ~-ring Z, the Adams operations _n are all equal to the ide*
*n-
tity, so the only ~-inner derivation is 0. On the other hand, identifying a li*
*near
endomorphism of Z with its image at 1, one observes that a ~-derivation of Z is*
* a
function
f :T ! End (Z) = Z
such that f(p) 2 pZ for each prime p and that
f(mn) = f(m) + f(n)
for all m, n 1. This second property simply means that, if k has the prime
factorization pe11. .p.ellwith ei 1, then
f(k) = e1f(p1) + . .+.elf(pl).
In other words, the function f is determined by the f(p) 2 pZ for p primes via *
*this
last equation.
Summarizing this discussion, we have
Q Q
Corollary 3.13. H1~(Z) ~= ppZ ~= pZ, where the product is taken over the set
of all primes.
COHOMOLOGY OF ~-RINGS 9
4. Deformations of ~-rings
The purpose of this section is to study algebraic deformations of ~-rings alo*
*ng the
path initiated by Gerstenhaber [3], making use of the ~-ring cohomology develop*
*ed
in the previous section.
We remind the reader that T denotes the set of positive integers and R will
always be an arbitrary ~-ring with Adams operations _n.
Let us motivate the definition of a deformation of R as follows. Recall that
the Adams operations are ring endomorphisms with the properties that _1 = Id,
_mn = _m _n, and _p(r) rp (mod pR) for all primes p and r 2 R. We would
like to deform R with respect to these properties.
Now let
(4.1) _*t= _*0+ t_*1+ t2_*2+ . . .
be a formal power series, in which each _*iis a function
_*i:T ! End (R)
with _*0= _* (i.e. _n0= _n). We will write _*i(k) as _ki. Then, in order for _**
*tto
be a deformation of R, it should have the following properties:
o _1t= Id, meaning that
(4.2) _1i= 0 (i 1).
o _mnt= _mt_nt, meaning that
X i
(4.3) _mni = _mjO _ni-j
j=0
for all i 0 and m, n 1.
o For each prime p, _pt(r) rp (mod pR), which means that
(4.4) _pi(R) pR (i 1).
In other words, _*i2 F1(R).
Observe that in (4.3), if one takes m = n = i = 1, then the fact that _1 = Id
implies that _11= 0. By an induction argument, still with m = n = 1, it follows
that _1i= 0 for all i 1. In other words, (4.3)implies (4.2), and we may disre*
*gard
the latter. We, therefore, define a deformation of the ~-ring R to be a formal *
*power
series _*tas in (4.1)with each _*i(i 1) in F1(R), satisfying the identity (4.*
*3).
Following Gerstenhaber [3], the function _*1is called the infinitesimal deforma*
*tion
of _*t. In the rest of this section, we consider the following standard issues*
* in
algebraic deformation theory.
(1) Identify the infinitesimal deformation with an appropriate cohomology
class.
(2) Obtain rigidity result from the previous step.
(3) Describe cohomological obstructions to extending a cocycle to a deforma-
tion.
10 DONALD YAU
(4) Describe cohomological obstructions to two such extensions being equiva-
lent to each other.
To do all this, we first need a suitable notion of equivalence of deformation*
*s.
Define a formal automorphism of the ~-ring R to be a formal power series
OEt = 1 + tOE1 + t2OE2 + . .,.
____
in which each OEi belongs to End (R) with 1 denoting the identity map on R. Two
deformations _*tand ~_*tare said to be equivalent if there exists a formal auto*
*mor-
phism OEt such that
(4.5) ~_*t= OE-1t_*tOEt.
____
This equation is to be understood in the following sense: if f and g are in End*
* (R),
then f_*ig is the function T ! End (R) given by
(f_*ig)(n) = f O _niO g.
It is straightforward to verify that if _*tis a deformation and if OEt is a for*
*mal
automorphism, then ~_*tdefined by (4.5)is also a deformation.
Now we can identify the infinitesimal deformation with a 1-cocycle in F1(R).
Proposition 4.1. The infinitesimal deformation _*1is a 1-cocycle in the complex
F*(R) (see (3.4)), and its cohomology class is well-defined by its equivalence *
*class.
Proof.The fact that _*1is a 1-cocycle follows directly from (4.3)(when i = 1). *
*If
~_*tis a deformation that is equivalent to _*t, then the difference ~_*1- _*1is*
* of the
____
form [_*, OE] for some OE 2 End (R), and this is a 1-coboundary.
Suppose that in the deformation _*t, one has _*1= . .=._*l-1= 0 (i.e. _ij= 0 *
*for
all i 1 and j = 1, . .,.l - 1). Then one observes from (4.3)that _*lis a 1-co*
*cycle.
Theorem 4.2. Suppose that _*t= _*+ tl_*l+ tl+1_*l+1+ . .i.s a deformation of a
~-ring R. If _*lis a 1-coboundary in F1(R), then _*tis equivalent to a deformat*
*ion
of the form ~_*t= _* + tl+1~_*l+1+ tl+2~_*l+2+ . ...
____
Proof.By assumption _*l= [_*, OEl] for some OEl 2 End (R). Using the formal
automorphism OEt= 1 - tlOEl, we see that _*tis equivalent to the deformation
_~*t= OE-1t_*tOEt
(1 + tlOEl)(_* + tl_*l)(1 - tlOEl) (mod tl+1)
_* + tl(_*l- [_*, OEl]) (mod tl+1)
_* (mod tl+1).
This finishes the proof.
An immediate consequence of this result (and its proof) is a cohomological cr*
*i-
terion for the rigidity of the ~-ring R.
Corollary 4.3. If H1~(R) = 0, then every deformation of R is equivalent to _*.
COHOMOLOGY OF ~-RINGS 11
It was established in Proposition 4.1 that the infinitesimal deformation is a*
* 1-
cocycle in F*. This raises the question: Given a 1-cocycle, is it the infinites*
*imal
deformation of a deformation? To what extent is this deformation unique? We will
break each one of these questions into a sequence of "smaller" questions, which*
* we
then approach from an obstruction-theoretic view point.
Fix a ~-ring R. Following Gerstenhaber and Wilkerson [4], we define, for each
N 1, a deformation of order N to be a formal power series
_*t= _* + t_*1+ . .+.tN _*N
with each _*iin F1(R), satisfying the identity (4.3)modulo tN+1 . This last req*
*uire-
ment simply means that
_mt_nt = _mnt (mod tN+1 )
for all m, n 1. One can think of a deformation as a deformation of order 1. A
formal automorphism is defined just as before, and two deformations of order N *
*are
said to be equivalent if there exists a formal automorphism for which (4.5)holds
modulo tN+1 . We say that _*textends to order N + 1 if there exists an element
_*N+12 F1(R) such that the formal power series
(4.6) ~_*t= _*t+ tN+1 _*N+1
is a deformation of order N + 1. We call ~_*tan order N + 1 extension of _*t.
Let _*tbe a deformation of order N. Consider the function
Obs (_*t): T 2 ! End (R)
defined by
NX
Obs (_*t)(m, n) = - _miO _nN+1-i.
i=1
Lemma 4.4. The element Obs (_*t) 2 F2(R) is a 2-cocycle.
P 3
Proof.Recall that we can write d2: F2 ! F3 as i=0(-1)i@i (see (3.3)). For any
triple (m0, m1, m2) 2 T 3, we have
X
(@0 - @1)(Obs (_*t))(m0, m1, m2)= _m0iO _m1jO _m2k
i+j+k=N+1
i,k>0
= -(@2 - @3)((Obs (_*t))(m0, m1, m2).
It follows that d2Obs (_*t) = 0.
Now suppose that _*N+1is an element of F1(R). Consider the formal power series
~_*tdefined by (4.6). It is an order N + 1 extension of _*tif and only if (4.3)*
*holds
when i = N + 1. This is true, since the identities for i N in (4.3)automatica*
*lly
hold, as they only involve _*ifor i N. Collecting the three terms in (4.3)(wi*
*th
i = N + 1) involving _*N+1, (4.3)can be rewritten as
(4.7) (d1_*N+1)(m, n) = Obs (_*t)(m, n).
In other words, _*textends to order N +1 if and only if Obs (_*t) is a 2-coboun*
*dary.
12 DONALD YAU
Summarizing, we have determined the obstructions to extending a deformation
of order N to a deformation of one higher order. We record it as follows.
Theorem 4.5. Let _*tbe a deformation of order N. Then it extends to order N +1
if and only if the 2-cocycle Obs (_*t) is cohomologous to 0.
Starting with a 1-cocycle, we obtain the obstructions to extending it to a de*
*for-
mation by applying this theorem repeatedly.
Corollary 4.6. Let _*12 F1(R) be a 1-cocycle. Then there exists a sequence of
(obstruction) classes !i (i = 1, 2, . .).in H2~(R), where !n is defined if and *
*only if
!1, . .,.!n-1 are all defined and equal to 0. Moreover, the deformation of orde*
*r 1,
_*t= _* + t_*1, extends to a deformation if and only if !i is defined and equal*
* to 0
for each i = 1, 2, . ...
In particular, we have the following cohomological condition that guarantees *
*the
existence of extensions.
Corollary 4.7. If H2~(R) = 0, then every deformation of order N 1 extends to
a deformation.
Finally, we consider the question of whether two extensions are equivalent. L*
*et
_*tbe a deformation of order N and let ~_*tand ~_*tbe two order N + 1 extensions
of _*t. Then it follows from the way the obstruction class is defined that
d1_~*N+1= Obs (_*t) = d1_~*N+1.
In particular, ~_*N+1- ~_*N+1is a 1-cocycle.
Proposition 4.8. If the 1-cocycle ~_*N+1- ~_*N+1is cohomologous to 0, then the
two deformations ~_*tand ~_*tof order N + 1 are equivalent.
The proof of this proposition is almost identical to the argument for Theorem*
* 4.2,
and we will omit it. The author is not sure whether the converse of this propos*
*ition
is true or not.
References
[1]M. F. Atiyah and D. O. Tall, Group representations, ~-rings and the J-homom*
*orphism,
Topology 8 (1969), 253-297.
[2]D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423-472.
[3]M. Gerstenhaber, On the deformation of rings and algebras, Ann. Math. (2) 7*
*9 (1964), 59-103.
[4]M. Gerstenhaber and C. W. Wilkerson, On the deformation of rings and algebr*
*as, V: Defor-
mation of differential graded algebras, Higher homotopy structures in topolo*
*gy and mathe-
matical physics (Poughkeepsie, NY, 1996), 89-101, Contemp. Math., 227, Amer.*
* Math. Soc.,
Providence, RI, 1999.
[5]M. Hazewinkel, Formal groups and applications, Pure and Applied Mathematics*
* 78, Academic
Press, New York-London, 1978.
[6]D. Knutson, ~-rings and the representation theory of the symmetric group, L*
*ecture Notes in
Math. 308, Springer-Verlag, Berlin-New York, 1973.
[7]C. A. Weibel, An introduction to homological algebra, Cambridge studies in *
*advanced math-
ematics 38, Cambridge Univ. Press, Cambridge, UK, 1994.
COHOMOLOGY OF ~-RINGS 13
[8]C. Wilkerson, Lambda-rings, binomial domains, and vector bundles over CP(1)*
*, Comm.
Algebra 10 (1982), 311-328.
[9]D. Yau, Moduli space of filtered lambda-ring structures over a filtered rin*
*g, Int. J. Math.
Math. Sci., accepted for publication.
[10]D. Yau, On adic genus and lambda-rings, Trans. Amer. Math. Soc., accepted f*
*or publication.
E-mail address: dyau@math.uiuc.edu
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W.
Green Street, Urbana, IL 61801 USA