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).