ON THE DEFORMATION OF RINGS AND ALGEBRAS, V:
DEFORMATION OF DIFFERENTIAL GRADED ALGEBRAS
Murray Gerstenhaber
and
Clarence Wilkerson
In this paper we consider the deformation theory of differential graded modu*
*les (DGM's)
and differential graded algebras (DGA's), where only the differential varies, t*
*he underlying
module or algebra structure remaining fixed. At the outset we consider only in*
*dividual
modules or algebras and afterwards we examine deformations of sheaves. In most *
*respects
the theory parallels closely that developed in [2], [3], [4], [5], and [6]. Th*
*ere is a natu-
ral concept of infinitesimal deformation. These infinitesimals are elements of *
*a homology
group of degree or dimension 1 and have obstructions in a group of degree 2. Th*
*ere is also
a natural concept of rigidity. In the module case, the vanishing of all infinit*
*esimals implies
rigidity. For algebras this certainly holds if they are defined over Q, the rat*
*ionals, but at
one time it seemed that it need not hold generally. In analogy with the older t*
*heory, even
a rigid DGA may appear as a member of a family of algebras parameterized by wha*
*t may
be viewed as "continuous" parameters. For example, in the category of commutati*
*ve alge-
bras, separable extensions are rigid but algebraic function fields of a fixed g*
*enus over C may
apppear to vary continuously because the analytic structure does. The paradox i*
*s partially
resolved by considering sheaves, or more generally "diagrams" of algebras (pres*
*heaves of
not-necessarily-commutative algebras over a small category) rather than single *
*algebras.
All the infinitesimal aspects of the deformation theory of complex analytic man*
*ifolds can,
for example, be captured in this way, cf. [7]
An important example of a sheaf of DGA's whose deformation theory we conside*
*r is the
"de Rham" sheaf of a C1 manifold, where over each open set U one takes the de *
*Rham
complex. The degree 0 part of this sheaf is just the sheaf of germs of C1 func*
*tions and
the differential is exterior differentiation. Analogously, in the complex anal*
*ytic case we
consider the holomorphic de Rham sheaf whose degree 0 part is the sheaf of germ*
*s of
holomorphic functions and whose differential is exterior differentiation with r*
*espect to the
complex coordinates. In the C1 case the deformation theory is trivial, both lo*
*cally and
globally. In the complex analytic case it is locally trivial but globally the t*
*heory is identical
(at least in its formal aspects) to the Froehlicher-Nijenhuis, Kodaira-Spencer *
*theory of [1],
[9]. In particular, the infinitesimal deformations of this de Rham sheaf are th*
*e elements
of H1(X; ) where is the sheaf of germs of holomorphic tangent vectors. Questi*
*ons
of obtaining true deformations from formal ones are discussed under certain fin*
*iteness
conditions.
To emphasize the analogy with [2], we will speak of "cohomology" exclusively.
1. Deformations of a DGM and Rigidity.
Typeset by AM S-*
*TEX
1
2 MURRAY GERSTENHABER AND CLARENCE WILKERSON
We begin by defining a deformation of the differential of a DGM in a "naive"*
* way by
power series, analogous to [2]. Throughout this section, M will denote either *
*a graded
differential module (graded either by Z or Z=2) or an ungraded one of character*
*istic 2 over
a ring . The "differential", d, of M is simply an endomorphism with d2 = 0 whic*
*h, in the
graded case, is assumed to have degree +1. That is, if M = i2ZMi, then dMi Mi+*
*1,
while if M = M0 M1 is Z=2 graded, then dM0 M1 and dM1 M0. The cohomology
group of M (strictly of M and d), (kerd)=dM, is denoted as usual by H(M) and in*
* the
graded case is itself a graded module with Hi(M) = ker(d|Mi)=dMi-1.
A formal "one-parameter family of deformations" or briefly a formal deformat*
*ion of d
will mean a formal power series dt = d+td1+t2d2+: : :(tacitly d = d0) in which *
*each di is
an endomorphism of M, having in the graded case degree +1, such that formally o*
*ne has
d2t= 0. This dt is a differential on M[[t]], the [[t]]-module of power series w*
*ith coefficients
in M. If we have only a polynomial dt = d + td1 + . .+.tndn such that d2t 0 mo*
*d tn+1,
then dt will be called an "approximate deformation of order n". ThisPmay be vie*
*wed as a
differential on M[t]=tn+1. In either case one has equivalently, i+j=kdidj = 0*
*, either for
all k orPfor all k 5 n. Transposing to the right those terms with either i = 0 *
*or j = 0, and
writing 0for a summation in which the range of indices is strictly positive, *
*one has
X
(1) 0didj = -(ddk + dkd)
i+j=k
Now in a graded algebra, if a and b are homogeneous elements of degrees m an*
*d n,
respectively, then their graded commutator is defined by [a; b] = ab - (-1)mn b*
*a. Here,
the subalgebra of End M generated by the homogeneous endomorphisms, which we sh*
*all
denote by End *M, is graded. Denoting the homogeneous part of degree i by End i*
*M, we
have dk 2 End1 M for all k. (If only finitely many of the homogeneous component*
*s Mi of
M are different from zero, then, of course, End M = End* M; if the characterist*
*ic is 2 one
can disregard all discussion of the grading.) The right side of (1) can therefo*
*re be written
as -[d; dk] or -(ad d)dk.
Lemma 1. Let d be a graded endomorphism of degree 1 of a graded module M, or of*
* an
ungraded one of characteristic 2. Then d2 = 0 implies (ad d)2 = 0.
Proof. In the graded case, it is sufficient to show that for a homogeneous endo*
*morphism
' one has [d; [d; ']] = 0. If ' has odd degree, then the left side is [d; d' + *
*'d] = d(d' +
'd) - ('d + d')d = 0. The computation is equally trivial if ' has even degree o*
*r if the
characteristic is 0.
Note that in general, d2 = 0 implies only that (ad d)3 = 0.
Corollary. With the preceding notation ad d is a differential on End *M, which,*
* in the
graded case has degree 1.
Now (1), which we may rewrite as
X
(2) 0 didj = -(ad d)dk;
i+j=k
shows that d1 2 ker(ad d), since for k = 1 the sum on the left is empty, and mo*
*re generally,
if d1 = . .=.dk-1 = 0, then dk 2 ker(ad d). That is, the first non-zero dk is *
*a cocycle
DEFORMATION OF DIFFERENTIAL GRADED ALGEBRAS 3
of End *M relative to the differential d, and in the graded case it is a 1-cocy*
*cle (i.e. has
degree 1). We should like, if possible, to interpret the cohomology class of dk*
* rather than
dk itself as the infinitesimal of the deformation dt. To this end, define a fo*
*rmal "one-
parameter family of (linear) automorphisms" (briefly, formal automorphism) of M*
* to be
a formal power series 't = 1 + t'1 + t2'2 + : :w:here 1(= '0) stands for the id*
*entity
morphism of M, and where each 'i is an endomorphism of M which in the graded ca*
*se
has degree 0. Two one-parameter families of deformations, dt = d + td1 + t2d2 +*
* : :a:nd
d0t= d+d0t+t2d02+: : :will be called equivalent if there is a 't such that d0t=*
* '-1tdt't, and
dt is called trivial if it is equivalent to d itself. We make the same definiti*
*on if dt and d0tare
merely approximate deformations of order n. Since '-1t= 1 - t'1 + : :t:he equiv*
*alence
implies that d01= d1 + [d; '1] which shows that passing to an equivalent deform*
*ation
replaces d1 by a cohomologous cocycle. If the deformation dt begins with d + tk*
*dk+ higher
terms, then choosing 't of the form 1 + tk'k shows that the first non-zero dk i*
*n dt can be
replaced by any cohomologous cocycle by passing to an equivalent deformation. W*
*e call d
rigid if every deformation of d is trivial.
Theorem 1. If H1(End *M) = 0, then M is rigid.
Proof. Let Dt = 1 + td1 + t2d2 + : : :be any deformation and suppose that for s*
*ome
k 0 we have found endomorphisms 1; : :;: k of M, all of degree 0, such that s*
*etting
t = (1 + t 1) : :(:1 + tk k) we have -1tdt t = d + tk+1d0k+1+ tk+2d0k+2+ : :.*
*:Then d0k+1
is a cocycle, hence d0k+1= [d; k+1] for some k+1, so setting t = t(1 + tk+1*
* k+1) it
follows that -1tdt t is of the form d + tk+2d00k+2+ : :.:Continuing we can fin*
*d 1; 2; : : :
such that setting 't = (1 + t 1)(1 + t2 2) : :w:e have '-1tdt't = d.
Here is a simple example of a DGM with H1(End *M) = 0 and which is therefore*
* rigid.
Let V be a finite dimensional vector space over a field F , let V0 and V1 be tw*
*o copies of V
and on M = V0 V1 define d by sending every element of V0 to the corresponding e*
*lement
of V1 and by setting dV1 = 0. Then every endomorphism f of degree 1 of M is a c*
*ocycle.
For df = fd = 0 so [d; f] = 0. On the other hand, letting f0 be the endomorphis*
*m of M
sending V1 to 0 and sending every v 2 V0 to the element whose image under d is *
*fv, we
have df0 = f and f0d = 0 so f = [d; f0]. Thus H1(End M) = 0 and M is rigid.
We wish next to define the "infinitesimal" of a deformation dt.
Lemma 2. Suppose that a deformation of the form dt = d + tkdk + tk+1dk+1 + : :i*
*:s
equivalent to another of the form d;t= d + t`d0`+ t`+1d0`+1+ : :,:with ` k. If*
* ` > k, then
dk is a coboundary, while if ` = k, then dk is cohomologous to d0`.
Proof. Let d;t= '-1tdt't with 't = 1 + t'1 + t2'2 + : :a:nd write 't in the for*
*m (1 +
t 1)(1 + t2 2) : :.:A simple induction then shows that 1; 2; : :;: k-1 commut*
*e with d
and therefore that d0t= d + tk(dk + [d; k])+ higher terms.
Theorem 2. If dt is a deformation, then either
i) for every k > 0, dt is equivalent to a deformation of the form 1+tkd0k+tk*
*+1d0k+1+: :,:
in which case dt is trivial, or
ii) there is a largest ` > 0 such that dt is equivalent to a deformation of *
*the form
1 + t`d00`+ t`+1d00`+1+ : :.:In this case the cohomology class of d00`depends o*
*nly on dt and
is called the infinitesimal of dt.
Proof. In case i) suppose that for some k > 0 we already have found 1; : :;: k*
*-1, such
that setting 't = (1 + t 1) : :(:1 + tk-1 k-1) we have '-1tdt't = 1 + tkd0k+ tk*
*+1d0k+1+ : :.:
4 MURRAY GERSTENHABER AND CLARENCE WILKERSON
Then by the preceding lemma, k0kis a coboundary, say d0k= [d; k+1]. This defin*
*es k+1,
and setting 't = (1 + t 1)(1 + t2 2) : :w:e have '-1tdt't = d, proving that dt *
*is trivial.
Case ii) is covered by the lemma.
The theorem that H1 = 0 implies rigidity will fail where we consider the def*
*ormation
of differential graded algebras except when the algebras are defined over Q. We*
* shall also
be unable, in the general case, to define the infinitesimal of a deformation. T*
*he difficulties
are analogous to those discussed in [2] and [6], and appear mainly in character*
*istic p.
If M is a finite dimensional vector space over a field, then after a choice *
*of basis the
differential, d, may be represented as a matrix. Any deformation dt is then a m*
*atrix with
entries which are formal power series in t whose value at t = 0 is the matrix o*
*f d. If the
field is the real or complex numbers, then we really should like to have conver*
*gent power
series and will say that we have a "true" rather than a "formal" deformation in*
* this case.
This can be achieved in the complex case by considering the matrix of dt as the*
* generic
point of a variety V in complex space of suitable dimension. In this variety we*
* can draw
curves through the point representing the matrix of d. Representing the coordin*
*ates of a
point on such a curve by convergent power series in a parameter T (with T = 0 g*
*iving the
matrix of d) we obtain a "true" deformation of d. More is true: suppose for sim*
*plicity that
dt = d + td1 + : :w:ith d1 not cohomologous to 0. Then d1 represents a tangent *
*vector at
the point representing d in the space of matrices over C which represent differ*
*entials on
M. The variety V contains the point representing d and through that point there*
* must be
a curve on V tangent to the vector representing d1. Therefore we can find a mat*
*rix whose
entries are convergent power series such that the value at T = 0 is the matrix *
*of d and
the derivative at T = 0 is the matrix of d1. For analogous arguments in the def*
*ormation
theory of algebras, cf. [2].
Returning to the general case, if we have a deformation dt then it is reason*
*able to ask
how the cohomology of the [[t]]-module M[[t]] is related to that of M. Followin*
*g an idea
of Griffiths [8], we say [6] that a cocycle u of M is "extendible" if there is *
*a formal power
series ut = u+tu1+t2u2+: : :with all ui2 M, such that dtut = 0. A coboundary is*
* always
extendible, since if u = dv, then for ut we can take dtv. We call u a "jump coc*
*ycle" if it has
an extension to a coboundary. Then it can be shown that the cohomology of M[[t]*
*] under
dt is the quotient module of extendible cocycles module jump cocycles with coef*
*ficients
extended to [[t]]. One can also, as in [6], define "jump" or "pop" deformations*
*, obtaining
analogous results to those of [6]. These may be discussed elsewhere.
2. Obstructions.
The following lemma tells under what conditions it is possible to extend an *
*approximate
deformation of some given order to one of higher order.
Lemma 3. If dt = d + td1 + . .+.tndn is an approximate deformation of order n o*
*f d,
then d1dn + d2dn-1 + . .+.dnd1 is a cocycle of End *M denoted Obs dt. The nece*
*ssary_
and sufficient condition that dt be extendible to an approximate deformation dt*
*= d + td1+
. .+.tndn + tn+1dn+1 is that Obs dt be a coboundary.
*
* P 0
Proof.PTo see that Obs dtis a cocycle, recall that for all k n we have [d; dk]*
* = - i+j=kdidj,
where 0indicates that the indices i, j are strictly positive. As add is a gra*
*ded derivation,
we therefore have
X X
[d; Obsdt] = 0[d; di]dj - 0di[d; dj]:
i+j=n i+j=n
DEFORMATION OF DIFFERENTIAL GRADED ALGEBRAS 5
P 0
This vanishes, since both sums on the right are just - i+j+k=ndidjdk. The re*
*st is
trivial.
In view of the foregoing, it is reasonable to call Obs dt the "obstruction c*
*ocycle" of
the approximate deformation dt, and to call its cohomology class, which we deno*
*te by
obs dt, the "obstruction". The lemma then says that dt is extendible to an appr*
*oximate
automorphism of order n + 1 if and only if its obstruction vanishes. It is easy*
* now to see
Theorem 3. The cohomology class obsdt depends only on the equivalence class of *
*dt.
Suppose now that dt = d+td1+. .+.tndn and dt0= d+td01+. .+.tnd0nare approxim*
*ate
deformations of order n which commute in the graded sense, i.e., are such that *
*[dt; d0t] =
dtd0t+ d0tdt 0 mod tn+1. Then (dt + d0t)2 = 0. Let usPdefine the obstructio*
*n cocycle
of dt+ d0t= 2d + t(d1 + d01) + . .+.tn(dn + d0n) to be 0i+j=n+1(di+ d0i)(dj +*
* d0j). It is
easy to check directly that this is a cocycle relative to the differential d. H*
*owever, we may
also do so without computation by observing that it is a cocycle relative to 2d*
*, hence,
if 2 is invertible, relative to d. Since the result is purely formal, the inve*
*rtibility of 2 is
inessential. With the usual definitions of Obs dt and Obs d0twe then have
Theorem 4. Suppose that [dt; d0t] = dtd0t+d0tdt 0 mod tn+2 (rather than just *
* mod tn+1).
Then Obs (dt+ d0t) = Obs dt+ Obs d0t.
Proof. By definition,
X
Obs (dt+ d0t) = 0(di+ d0i)(dj + d0j) =
i+j=n+1
X X X
didj + d0id0j+ (did0j+ d0jdi):
i+j=n+1 i+j=n+1 i+1=n+1
The last sum vanishes, by our hypothesis, and the first two are Obs dt and Obs *
*d0t, respec-
tively.
It follows a fortiori that Obs (dt + d0t) = Obs dt + Obs d0t, but we do not *
*know what
significance, if any, this has.
3. Deformation of DGA's.
Let A be a differential graded algebra with differential d. Here d in additi*
*on to having
square zero and degree +1 is assumed to be a graded derivation of A. That is, i*
*f a is a
homogeneous element of degree m and b arbitrary in A, then d(ab) = (da)b + (-1)*
*m a(db).
(As before if the characteristic is 2, then we can dispense with the grading.) *
* A formal
deformation dt = d + td1 + t2d2 + : :m:ust then also be formally a graded deriv*
*ation,
which will be the case if and only if each di is. In place of our earlier End **
*M we therefore
now consider the graded Lie ring Der* A generated by all graded derivations of *
*A. As
before, add is a differential on this ring.
The definition of equivalent deformations must be slightly refined: dt and *
*d0t= d +
td01+ t2d02+ : :a:re equivalent if there exists a formal one parameter family o*
*f algebra
automorphisms of degree 0, 't = 1 + t'1+ t2'2+ : :s:uch that d;t= '-1tdtd't. He*
*re each
'i is a linear endomorphism of degree 0, 1 stands as before for the identity mo*
*rphism,
but we now require in addition that 't(ab) = 'ta . 'tb for all a; b 2 A. This *
*implies,
in particular that '1 is a derivation of A, so '1 is in Der* A. Thus, by passi*
*ng to the
6 MURRAY GERSTENHABER AND CLARENCE WILKERSON
equivalent deformation d0t, d1 can be replaced by d1 + [d; '1], as before, but *
*here [d; '1] is
a coboundary in Der*A.
As before, we say that a deformation dt is trivial if it is equivalent to d,*
* and that A is
rigid if every deformation is trivial. However, it is now generally no longer t*
*he case that
the vanishing of H1(Der *A) implies rigidity. The problem is that not every ' 2*
* Der0 A
(the derivations of degree 0) need be the "infinitesimal" of a one parameter-fa*
*mily of
algebra automorphisms. That is, if we choose ' 2 Der0 A and k > 0 there need b*
*e no
formal automorphism of the form 't = 1 + tk' + tk+1'k+1 + : :w:here the 'i are *
*all linear
endomorphisms of degree 0. If A is a Q-algebra, then this difficulty disappears*
*, for etk'
is then a well-defined power series and will serve as 't. By a proof analogous *
*to that of
Theorem 1, we then have
Theorem 5. Let A be a DGA over Q. Then H1(Der *A) = 0 implies that A is rigid.
The analogues of the other results of the preceding sections also hold here *
*for a DGA
over Q; the proofs are virtually identical.
In the matter of getting "true" deformations from formal ones, a finitely ge*
*nerated
DGA over C which is graded by the non-negative integers behaves like a finite d*
*imen-
sional differential graded vector space over C. For every derivation (of any d*
*egree, and
in particular) of degree one is completely determined by its effect on the gene*
*rators. If
the algebra is A = i0 Ai and if the generator of highest degree is of degree n,*
* then a
derivation of degree 1 is completely determined by its restriction to a mapping*
* of the set
of generators into n+1i=0Ai. The latter is a finite dimensional vector space, s*
*o the space of
derivations of degree one has finite dimension and one can then argue substanti*
*ally as in
x1.
If A is a DGA, then H(A) is again a graded algebra from which, using the gra*
*ded
commutator as product, we can derive a graded Lie algebra. (Of course, if A was*
* graded
commutative, then so is H(A), so the graded Lie product is then zero.) In Der*A*
* there is
a graded Lie product which induces a graded Lie product on H(Der *A). One can r*
*eadily
verify the following proposition which gives a relation between these two algeb*
*ras.
Theorem 6. There is a graded Lie algebra morphism H(A) ! H(Der *A) defined by
sending the class of a cycle u 2 A to the class of adu.
As an example of the theory, consider a graded commutative A, that is, suppo*
*se that
if a; b are homogeneous elements of degrees r and s, respectively, then ab = (-*
*1)rsba.
(Equivalently, the graded commutator [a; b] vanishes.) Let A be generated over*
* Q by
two elements, x; y each of degree 2 and a third element z of degree 3 with no r*
*elations
other than the graded commutativity. The subalgebra generated by x and y is iso*
*morphic
to the polynomial ring Q[x; y], every element of this subalgebra commutes with *
*z, and
z2 = 0, so as an algebra, A ' Q[x; y; z]=z2. Now choose any quadratic form q(x*
*; y) =
q0x2 + 2q1xy + q2y2, and set dx = dy = 0, dz = q(x; y). The homology ring of A*
* to
Q[x; y]=q(x; y). What deformations are possible? We shall show that if q is non*
*-singular,
i.e. if the matrix q0q q1 (which we may also denote simply by q) is non-sin*
*gular,
1q2
then H1(Der *A) = 0, so A is rigid, and otherwise A is not. To this end, let d*
*1 be a
derivation of degree 1, and set d1x = ffz, d1y = fiz, d1z = r(x; y) where r is *
*another
quadratic form in x and y. For d1 to be a cocycle of Der* A we must have [d; d*
*1] = 0.
But [d; d1]x = ffg(x; y) and [d; d1]y = fiq(x; y) so if q 6= 0, which we now as*
*sume, then
DEFORMATION OF DIFFERENTIAL GRADED ALGEBRAS 7
we must have ff = fi = 0. The latter conditions also imply [d; d1]z = 0. To p*
*rove that
H1(Der *A) = 0, we must show that there is a derivation ' of degree 0 with [d; *
*'] = d1.
Set 'x = ax + by, 'y = cx + ey, 'z = flz. Then [d; ']x = 0 = d1x, [d; ']y = 0 *
*= d1y
and [d; ']z = d'z - 'dz = flq(x; y) - 'q(x; y). To compute 'q(x; y), it is conv*
*enient to
set xy= x; 'x'y= 'x, and to let ' also stand for the matrix of '. Then q(x; y*
*) = xtqx
(where xt denotes the transpose of x). Since ' is a derivation, we have
'q(x; y) = ('x)tqx + xtq('x) = xt('tq + q')x;
so the matrix of 'q(x; y) is 'tq + q'. If a is any symmetric 2 x 2 matrix, then*
* viewing '
as a variable, we can solve 'tq + q' = a as long as q is non-singular by taking*
* ' = 1_2q-1 a.
Thus, if q is non-singular, then we can even set fl = 0 and we will be able to *
*find ' such
that [d; '] = d1. This proves that H1(Der *A) = 0 and therefore that in this c*
*ase A is
rigid. On the other hand, if q is singular, then we claim that not every quadr*
*atic form
r(x; y) is of the form [d; ']z = flq(x; y) - 'q(x; y) for some '. The matrix of*
* the form on
the right is flq - 'tq - q'. Replacing ' by '-1_2fl . 1 eliminates fl so it is *
*only necessary
to show that for some symmetric matrix r, the equation 'tq + q' = r is not solv*
*able for
'. In fact it is not solvable for r = 1. For such an equation implies that if q*
*b = 0, whence
also btq = 0, then btrb = 0. For r = 1 we have btb = 0 which implies b = 0 sinc*
*e all entries
are real, but if q is singular, then there is a b 6= 0 with qb = 0. Denoting t*
*he algebra
now by Aq we have shown that H1(Der *A) = 0 if q is non-singular and that other*
*wise
H1(Der *Aq) 6= 0 (the case q = 0, which we momentarily set aside, being trivial*
*).
It is also easy to see directly that if q is singular, then Aq is not rigid,*
* which would in
particular imply that H1(Der *Aq) 6= 0. For defining a derivation d1 on the un*
*derlying
algebra A by setting d1x = d1y = 0, d1z = r(x; y) where r is an arbitrary quadr*
*atic form,
it is clear that d + td1 is already a deformation, the effect of which is to re*
*place q by q + tr.
The resulting DGA, Aq+tr, may be viewed as defined either over the polynomial r*
*ing Q[t]
or over the power series ring Q[[t]]. If q is singular but q + tr non-singular,*
* then clearly
Aq+tr is not isomorphic over A[[t]] to Aq so the latter has in fact been deform*
*ed. If q is
non-singular, then the rigidity of Aq implies that Aq+trand Aq are isomorphic o*
*ver A[[t]]
for arbitrary r. This in turn implies (and is equivalent to) the elementary fac*
*t that there
is a 2 x 2 matrix a with coefficients in Q[[t]] such that atqa = q + tr. The ri*
*gidity of Aq
for q non-singular perhaps is not surprising since over the complex numbers all*
* such q are
equivalent to x2 + y2, but over Q there are many inequivalent non-singular form*
*s. The
present example can be extended to characteristic p > 0 but becomes more compli*
*cated if
Q[x; y; z] is truncated.
4. The "sophisticated" definition of a deformation.
Let M . . .Mi Mi+1 : :b:e a module with an exhaustive separated, complete
filtration, i.e., such that [Mi = M, \Mi = 0 and M is complete in the weakest t*
*opology
defined by taking the Mi as neighborhoods of 0. Analogous to the Rees ring and*
*Pthe
definition in [3], we define App M to be the module of all formal power series *
* T iui in
a variable T with coefficients ui 2 Mi and having only finitely many terms with*
* negative
powers of T . If N is a second such module and f : M ! NPa filtration-preser*
*vingP
morphism, then App f : App M ! App N is defined by sending T iui to T i(fui*
*).
The modulePApp M can be filtered in two ways (at least). In the first, used *
*in [3], we set
App jM = { T iui|ui 2 Mi+j}. With this we define the "successive approximatio*
*ns" to
M, App jM = App M= Appj+1 M. In particular, App 0M is just the completed associ*
*ated
8 MURRAY GERSTENHABER AND CLARENCE WILKERSON
graded module, which we denotePby cgrM. For denoting Mi=Mi+1 by griM, the eleme*
*nts
of App 0M are formal series T i_uiwith __ui2 griM. The secondPfiltration wa*
*s tacitly
used in [3] but unfortunately not made explicit. Set F jApp M = { T iui|ui2 M*
*j}. Now
let be an element of the center of the coefficient ring and denote the submod*
*ule of
App MPconsisting of all T -1v - v with v 2 AppPM simply by T -1- . (Note that *
*if
v = T iui is an App M, then so is T -1v = T i-1u.) The quotient App M=(T -1*
*- )
is denoted Def M (M "deformed" by ). If we view it as filtered by the images *
*of
the F jApp M, then Def M is a filtered module with the property that griDef M*
* is
canonically isomorphic to Mi=Mi+1 = griM, whatever may be. Now if = 0, then
Def M actually is graded and is canonically isomorphic to cgrM. On the other *
*hand,
if = 1, then Def1 M is isomorphic as a filtered ring to M in a way that induce*
*s the
identity map on cgrM = cgrDef1 M. In fact, since M is complete,Pif isPany ele*
*ment
of , then there is a surjective mapping App M ! M sending T iui to iui, a*
*nd
this is a module morphism if is in the center of . If = 1, then the kernel o*
*f this
morphism is just T -1- 1 and it induces the desired filtration preservingPisomo*
*rphism
DefP1M ! M. Notice that if is invertible and central, then sending T iui2 Ap*
*p M to
-iui2 M is again an epimorphism whose kernel now is T -1- , so there is (cf.*
* [3]) a
canonical isomorphism Def M ! M whenever is invertible. Taken alone, however,*
* that
is misleading. One should consider the category C of (exhaustive, separated, c*
*omplete)
filtered modules whose (complete) associated grade module is isomorphic to cgrM*
* under
a given fixed isomorphism, so that we may view all the associated graded module*
*s in C
as identical. A morphism in C is by definition a filtration preserving morphism*
* inducing
the identity map on the associated graded module. Then for 6= 1, the given mor*
*phism
Def M ! M, while an isomorphism of modules and also filtration preserving, is *
*not a
morphism of C. For the induced mapping on cgrM(= cgrDef M) is the automorphism
multiplying elements of griM by -i. Thus, although Def M is isomorphic to M whe*
*never
is central and invertible, it is apparent that Def M is twisted relative to c*
*grM and
indeed there generally is no filtration preserving isomorphism Def M ! M which *
*induces
the identity map on cgrM. Nevertheless, it is the case that as varies through *
*the center
of the rings Def M vary through a family containing Def0M which is isomorphic*
* in C
to a cgrM, and containing Def1 M which is isomorphic in C to M itself. In this *
*way we
may view M as a deformation of cgrM. Note that the App iand Def are all functo*
*rs, the
image of a (filtrationPpreserving)Pmorphism f : M ! N being induced by the morp*
*hism
App f sending T iui 2 App M to T i(fui) in App N. One property of Def f is*
* that
it induces the identity morphism cgrM(= cgrDef M) ! cgrN(= cgrDef N). One can
show (cf. [3]) that if and are central elements of , then Def Def = Def De*
*f .
All of the foregoing (as well as what follows in this section) holds equally*
* when instead
of M we have a filtered (exhausted, separated, and complete) algebra A . . .Ai
Ai+1 : :,:where now by definition one has AiAj Ai+j, and the coefficient ring*
* is
assumed to be commutative.
With this in view, let M be a differential graded module over a ring with d*
*ifferential
d. Then M[[t]] is a [[t]]-module with two gradings; the "primary" one inherited*
* from M,
which may be only a mod 2 grading or even be no grading at all if M has chara*
*cteristic
2, and the "secondary" one by powers of t in which M[[t]] is complete. A deform*
*ation dt
of d is then just an endomorphism of M[[t]] which has square zero, preserves th*
*e filtration
associated with the second grading (the secondary filtration) and which has the*
* property
that cgrdt is just the extension of d to M[[t]]. Here cgris meant relative to t*
*he secondary
DEFORMATION OF DIFFERENTIAL GRADED ALGEBRAS 9
filtration. The appropriate generalization is to assume that we have at the out*
*set a module
M with two gradations, the first possibly only mod 2 or even non-existent if *
*M has
characteristic 2, and the second by the non-negative integers (or even by all o*
*f Z). We
assume that M is complete in the second, so M is not a graded module in the usu*
*al sense
but is a topological graded module. By a differential on M we mean now an endom*
*orphism
d which preserves both gradations, has degree 1 in the first, degree 0 in the s*
*econd and
has d2 = 0. A deformation edof M is an endomorphism with ed2= 0 preserving the *
*second
filtration and such that cgred= d. Our previous theory concerns the special cas*
*e where all
homogeneous components of M relative to the second grading are isomorphic and d*
* is the
same on each. One can, in the more general case, define infinitesimals and obst*
*ructions by
copying in detail the technique of [3], but we shall not do that here.
All of the foregoing applies equally to algebras, as long as the coefficient*
* ring is com-
mutative. Note, incidentally, that in our earlier theory, if dt is a deformati*
*on of d, then
Def dt is obtained by replacing t by t.
5. Deformations of sheaves.
Let M be a graded sheaf of modules over some topological space X and d be a *
*differential
on M. We can define a deformation dt = d + dt1 + t2d2 + : :j:ust as before, tog*
*ether with
the concepts of triviality and rigidity, but in addition, cf. [5], we can now e*
*xamine dt
locally. We call dt locally trivial if every point x of X has a neighborhood U*
* in which
the restriction of dt is trivial. This says, in particular, that in M|U , the r*
*estriction of M
to U, there is an endomorphism 'U of degree 0 such that d1|U = [d|U ; 'U ]. Th*
*ere may,
however, be no endomorphism ' of all M with d1 = [d; '], in which case dt globa*
*lly is non-
trivial. Now suppose that 'U is chosen for every U in some covering of X by "tr*
*ivializing
neighborhoods", i.e. ones in which dt is trivial (or at least in which d1 is a *
*boundary). If
U, V are such neighborhoods, let 'U|V denote the restriction of 'U to V (or rat*
*her to the
sheaf M|V ). Then generally 'U|V - 'V|U is not zero but on U \ V it is a cocyc*
*le with
respect to add, and has degree 0. Denoting by Z1 the sheaf germs of such cocycl*
*es, it is
clear that we can now define a 1-cocycle of X with coefficients in Z1 and hence*
* an element
of H1(X; Z1). Thus if dt is locally trivial but not actually trivial to first o*
*rder, then the
infinitesimal of the deformation is an element of H1(X; Z1). All of the foregoi*
*ng holds for
a sheaf of differential graded algebras, A.
We conclude this paper by an important example, where the foregoing local tr*
*iviality
holds. Let X be a manifold, either C1 or complex. In the C1 case, let A be th*
*e de Rham
sheaf whose ith graded part Ai is the sheaf of C1 exterior i forms and d is ex*
*terior
differentiation. In the analytic case, A will be the de Rham sheaf of holomorp*
*hic forms
with complex exterior differentiation. In either case (with thanks to S. Shatz)*
*, Z1 is just
the sheaf of tangent vector fields _ either C1 or holomorphic depending on the*
* case. For
suppose that D is a derivative on the ring of C1 functions on some domain in R*
*n (or on
the ring of analytic functions in some domain of Cn). Let the coordinates be x1*
*; : :;:xn,
let ff = (ff1; : :;:ffn) be a specific point, and for any function f in the rin*
*g, write
X X
f(x) = f(ff) + (xi- ffi)fi(ff) + (xi- ffi)(xj - ffj)gij(x)
i i;j
where fi(x) = @f_@xiand the gijare again functions in the ring. We may write si*
*mply
f(x) = f(ff) + (x - ff)f0(ff) + (x - ff)g(x)(x - ff)t:
10 MURRAY GERSTENHABER AND CLARENCE WILKERSON
Then
Df(x) = Dx . f0(ff) + Dx . g(x)(x - ff)t+ (x - ff)g(x) . (Dx)t:
Setting x = ff gives Df(ff) = (Dx)(ff) . f0(ff), but sincePff is arbitrary, thi*
*s implies that
Df(x) = Dx . f0(x). That is, if Dxi= hi(x) then D is just hi_@_@xi.
In the real case the sheaf Z1 is fine and its cohomology vanishes, as does, *
*therefore, the
deformation theory of the de Rham complex. In the complex case, Z1 is the sheaf*
*, usually
denoted , of germs of holomorphic tangent vectors, and H1(X; ) is precisely the*
* space
of infinitesimal deformations of the analytic structure of X in the Froehlicher*
*-Nijenhuis,
Kodaira-Spencer theory (cf. [1], [10]). It is not difficult to show that the ob*
*struction to an
infinitesimal deformation of the de Rham sheaf is representable by an element o*
*f H2(X; )
and is identical to the obstruction when viewed in the analytic deformation the*
*ory. In fact,
the deformation theories of the de Rham sheaf of X and of the analytic structur*
*es of X
are formally the same. This is as far as we presently carry the theory, leaving*
* the problem
of getting true deformations from formal ones (as in [9]) and the additional di*
*fficulties of
characteristic p (hopefully) for another paper.
References
1.A. Froehlicher and A. Nijenhuis, A theorem on stability of complex structure*
*s, Proc. Nat. Acad. Sci.,
USA 43 (1957), 239-241.
2.M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. 79 *
*(1964), 59-103.
3.______, On the deformation of rings and algebras, II, Ann. of Math. 84 (1966*
*), 1-19.
4.______, On the deformation of rings and algebras, III, Ann. of Math. 88 (196*
*8), 1-34.
5.______, On the deformation of rings and algebras, Global Analysis, Papers in*
* Honor of K. Kodaira,
D. C. Spencer and S. Iyanaga (eds.), University of Tokyo Press and Princeton*
* University Press, 1969.
6.______, On the deformation of rings and algebras, IV, Ann. of Math. 99 (1974*
*), 257-276.
7.M. Gerstenhaber and S. D. Schack, Algebraic cohomology and deformation theor*
*y, Deformation of
Algebras and Structures and Aplications, M. Hazewinkel and M. Gerstenhaber (*
*eds.), Kluwer, 1988,
pp. 11-264.
8.P. A. Griffiths, The extension problem for compact submanifolds of complex m*
*anifolds I, The case of
a trivial normal bundle, Proc. Conf. Complex Analysis (Minneapolis, 1964), 1*
*13-142.
9.K. Kodaira, L. Nirenberg, and D. C. Spencer, On the existence of deformation*
*s of complex analytic
structures, I-II, Ann. of Math. 68 (1958), 450-459.
10.K. Kodaira and D. C. Spencer, (a) On deformations of complex analytic struct*
*ures, I-II, Ann. of
Math. 67 (1958), 328-466; (b) On deformations of complex analytic structures*
*, III, Stability theorems
for complex structures, Ann. of Math. 71 (1960).