DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA DONALD YAU Abstract. An algebraic deformation theory of algebras over the Landweber-Novikov

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1 DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA DONALD YAU Absrac. An algebraic deformaion heory of algebras over he Landweber-Novikov algebra is obained. 1. Inroducion The Landweber-Novikov algebra S was originally sudied by Landweber [8] and Novikov [10] as a cerain algebra of sable cohomology operaions on complex cobordism MU ( ). In fac, i is known ha every sable cobordism operaion can be wrien uniquely as an MU -linear combinaion in he Landweber-Novikov operaions. The Landweber-Novikov operaions ac sably, and hence addiively, on he cobordism MU () of a space. However, hey are, in general, no muliplicaive on MU (). Insead, heir acions on a produc of wo elemens in MU () saisfy he so-called Caran formula, analogous o he formula of he same name in ordinary mod 2 cohomology. This srucure - he S-module srucure ogeher wih he Caran formula on producs - makes MU () ino wha is called an algebra over he Landweber-Novikov algebra, or an S-algebra for shor. These S- algebras are herefore of grea imporance in algebraic opology. The algebra S has also appeared in oher seings. For example, Bukhshaber and Shokurov [2] showed ha he Landweber-Novikov algebra is isomorphic o he algebra of lef invarian differenials on he group Diff 1 (Z). Denoing he group of formal diffeomorphisms on he real line by Diff 1 (R), he group Diff 1 (Z) is he subgroup generaed by he formal diffeomorphisms wih ineger coefficiens. This heme ha he Landweber- Novikov algebra is an operaion algebra" is echoed in Wood's paper [12]. Wood consruced S as a cerain algebra of differenial operaors on he inegral polynomial ring on counably infiniely many variables. There are even connecions beween he Landweber-Novikov algebra and physics, as he work of Morava [9] demonsraes. The purpose of his paper is o sudy algebras over he Landweber- Novikov algebra from he specific view poin of algebraic deformaions. We would like o deform an S-algebra A wih respec o he Landweber-Novikov operaions on A, keeping he algebra srucure on A unalered. The resuling 1

2 2 DONALD YAU deformaion heory is described in cohomological and obsrucion heoreic erms. The original heory of deformaions of associaive algebras was developed by Gersenhaber in a series of papers [3, 4, 5, 6]. I has since been exended in many differen direcions, and many kinds of algebras now have heir own deformaion heories. The Landweber-Novikov algebra is acually a Hopf algebra, as he referee poined ou. Therefore, wha we are considering in his paper is really an insance of deformaion of a module algebra over a Hopf algebra. I would be nice o exend he resuls in he curren paper o his more general seing. A descripion of he res of he paper follows. The following secion is preliminary in naure. We recall he Landweber- Novikov algebra S and algebras over i. Our deformaion heory depends on he cohomology of a cerain cochain complex F, which is consruced in he nex secion as well. In Secion 3 we inroduce he noions of a formal deformaion and of a formal auomorphism. The laer is used o defined equivalence of formal deformaions. The main poin of ha secion is Theorem 3.5, which idenifies he infiniesimal" of a formal deformaion wih an appropriae cohomology class in H 1 (F ). Inuiively, he infiniesimal" is he iniial velociy of he formal deformaion. Secion 4 begins wih a discussion of formal auomorphisms of finie order and how such objecs can be exended o higher order ones. See Theorem 4.3 and Corollary 4.4. These resuls are needed o sudy rigidiy. An S-algebra A is called rigid if every formal deformaion of A is equivalen o he rivial one. The main resul here is Corollary 4.7, which saes ha an S-algebra A is rigid, provided ha boh H 1 (F ) and HH 2 (A) are rivial. Here HH 2 (A) denoes he second Hochschild cohomology of A, as an algebra over he ring of inegers, wih coefficiens in A iself. In Secion 5, we idenify he obsrucions o exending a 1-cocycle in F o a formal deformaion. This is done by considering formal deformaions of finie orders and idenifying he obsrucions o exending such objecs o higher order ones. This obsrucion urns ou o be in H 2 (F ); see Theorem 5.3. As a resul, he vanishing of H 2 (F ) implies ha every 1-cocycle occurs as he infiniesimal of a formal deformaion (Corollary 5.5). The paper ends wih Theorem 5.7, which shows ha he vanishing of a cerain class in H 1 (F ) implies ha wo order m +1 exensions of an order m formal deformaion are equivalen.

3 DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA 3 2. The Landweber-Novikov algebra and he complex F The purpose of his preliminary secion is o recall he Landweber-Novikov algebra S and he noion of an algebra over i. Then we consruc a cochain complex F associaed o an algebra over he Landweber-Novikov algebra. This complex will be used in laer secions o sudy algebraic deformaions of S-algebras The Landweber-Novikov algebra. References for his subsecion are [8, 10], where he Landweber-Novikov algebra was firs inroduced. Wood's paper [12] has an alernaive descripion of i as a cerain algebra of differenial operaors. The book [1, Ch. I] by Adams is also a good reference. The Landweber-Novikov algebra S is generaed by cerain elemens s ff, he Landweber-Novikov operaions, indexed by he exponenial sequences, which we firs recall. An exponenial sequence is a sequence ff = (ff 1 ;ff 2 ;:::) of non-negaive inegers in which all bu a finie number of he ff i are 0. When ff and fi are wo exponenial sequences, heir sum ff + fi is defined componenwise. Denoe by E he se of all exponenial sequences. For each exponenial sequence ff, here is a sable cobordism cohomology operaion s ff : MU ( )! MU 0 ( ); called a Landweber-Novikov operaion. The composiion s ff s fi of any wo Landweber-Novikov operaions saisfies he produc formula, which expresses i uniquely as a finie Z-linear combinaion, (2.1.1) s ff s fi = n fl s fl : fl2p (ff;fi) See, for example, Adams [1, xx5,6] or Wood [12, Thm. 4.2] for a proof of his fac. For our purposes, we do no really need o know wha he inegers n fl are, as long as hey saisfy he obvious associaiviy condiion. Thus, using he produc formula (2.1.1), he free abelian group (2.1.2) S = M ff2e Zs ff can be equipped wih an algebra srucure wih composiion as he produc. This is wha we referred o as he Landweber-Novikov algebra. The cobordism MU () of a space or a specrum is auomaically an S-module. In fac, i is known ha every sable cobordism cohomology operaion can be wrien uniquely as an MU -linear combinaion of he

4 4 DONALD YAU Landweber-Novikov operaions. (Here MU denoes MU (p) as usual.) If is a space, hen MU () is no jus a group bu an algebra as well. On a produc of wo elemens, he Landweber-Novikov operaions saisfy he Caran formula, (2.1.3) s ff (ab) = s fi (a)s fl (b): fi+fl=ff Here ff 2 E and a; b 2 MU (). We, herefore, make he following definiion. Definiion 2.2. By an algebra over he Landweber-Novikov algebra, or an S- algebra for shor, we mean a commuaive ring A wih 1 ha comes equipped wih an S-module srucure such ha he Caran formula (2.1.3) is saisfied for all ff 2 E and a; b 2 A. Given a commuaive ring A, le End(A) denoe he algebra of addiive self-maps of A, where produc is composiion of self maps. If f and g are in End(A), hen fg always means he composiion f ffi g. An S-algebra srucure on A is equivalen o a funcion (2.2.1) s : E! End(A); assigning o each exponenial sequence ff an addiive operaion s ff on A, which saisfies he produc formula (2.1.1) on composiions and he Caran formula (2.1.3) on producs in A The complex F. The algebraic deformaion heory of S-algebras discussed in laer secions depends on a cerain cochain complex F, which we now consruc. Firs we need some noaions. Le A be a commuaive ring. Denoe by Der(A) he abelian group of derivaions on A. Recall ha a derivaion on A is an addiive self-map f : A! A which saisfies he condiion f(ab) = af(b) + f(a)b for all a; b 2 A. For a posiive ineger n, we use he shorer noaion A Ωn o denoe he ensor produc over he ring of inegers of A wih iself n imes. The group of addiive maps A Ωn! A is wrien Hom(A Ωn ;A). When n =1, we also wrie End(A) for Hom(A; A). Each se Hom(A Ωn ;A) has a naural abelian group srucure. Namely, given f;g: A Ωn! A and b 2 A Ωn, he sum (f + g) sends b o f(b)+g(b). Le Se be he caegory of ses. Given wo ses C and D, he se of funcions from C o D is wrien Se(C; D). If D is an abelian group, hen, jus as in he previous paragraph, he se Se(C; D) is naurally equipped wih an abelian group srucure as well. Le A be an algebra over he Landweber-Novikov algebra S (see Definiion 2.2). Consider he Landweber-Novikov operaions on A as a funcion s as in

5 DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA 5 (2.2.1). Recall ha E is he se of all exponenial sequences. Also recall he ses P (ff; fi) from he produc formula (2.1.1). Denoe by E n he Caresian produc E E (n facors). Jus as in he case of s,iff is a funcion wih domain E, we will wrie f ff insead of f(ff) for ff 2 E. We are now ready o define he cochain complex F = F (A) of abelian groups. We make he following definiions. ffl F 0 (A) = Der(A). ffl F 1 (A) =Se(E; End(A)). ffl For inegers n 2, se where Now we define he differenials. F n (A) = F n 0(A) F n 1(A); F n 0 (A) = Se(E n ; End(A)); F n 1 (A) = Se(E; Hom(A Ωn ;A)): ffl d 0 (') =s ' 's for ' 2 F 0 (A). ffl For n 1, d n =(d n 0 ;dn 1 ), where d n 0 : F0 n (A)! F0 n+1 (A); d n 1 : F n 1 (A)! F n+1 1 (A) are defined as follows. (Here we have F0 1(A) = F1 1 (A) = F1 (A).) Suppose ha f = (f 0 ;f 1 ) is an elemen of F n (A) for some n 2 (or jus f when n = 1), x = (ff 1 ;:::;ff n+1 ) 2 E n+1, ff 2 E, and a = a 1 Ω Ωa n+1 2 A Ω(n+1). Then we se (2.3.1) (d n 0 f 0)(x) = s ff1 f 0 (ff 2 ;:::;ff n+1 ) (2.3.2) + n» ( 1) i fi2p (ffi;ffi+1) + ( 1) n+1 f 0 (ff 1 ;:::;ff n )s ffn+1 ; n fi f 0 ( ;ff i 1 ;fi;ff i+2 ;:::) where P ( ; ) is as in he produc formula (2.1.1), and (d n 1 f 1)(ff)(a) = fi+fl=ff + n s fi (a 1 )f 1 (fl)(a 2 Ω Ωa n+1 ) ( 1) i f 1 (ff)( Ωa i 1 Ω (a i a i+1 ) Ω a i+2 ) + ( 1) n+1 fi+fl=ff f 1 (fi)(a 1 Ω Ωa n )s fl (a n+1 ): In hese definiions, when n =1,boh f 0 and f 1 are inerpreed as f.

6 6 DONALD YAU Proposiion 2.4. (F ;d ) is a cochain complex of abelian groups. Proof. I is sraighforward o check ha d 1 d 0 =0by direc inspecion. For i =0; 1 and n P 1, he formulas above allow one o rewrie d n i as an alernaing sum d n n+1 i = j=0 ( i n [j]; corresponding o he n + 2 erms in he respecive formulas. I is sraighforward o check ha, for n 1, he cosimplicial n+1 i [l] n i [k] n+1 i [k] n i [l 1] (0» k<l» n +2) hold. Therefore, i follows ha F (A) is a cochain complex. From now on, whenever we speak of cochains," cocycles," coboundaries," and cohomology classes," we are referring o he cochain complex F =(F ;d ), unless saed oherwise. The ih cohomology group of F will be denoed by H i (F ). 3. Formal deformaions and auomorphisms The purposes of his secion are (1) o inroduce formal deformaions and auomorphisms of an S-algebra and (2) o idenify he infiniesimal" of a formal deformaion wih an appropriae cohomology class in he complex F Formal deformaions. Throughou his secion, le A be an arbirary bu fixed S-algebra. Recall ha he S-algebra srucure on A can be characerized as a funcion s as in (2.2.1) from he se E of exponenial sequences o he algebra of addiive self maps of A. In addiion, his funcion saisfies he Caran formula (2.1.3) and he produc formula (2.1.1). We will deform he Landweber-Novikov operaions on A wih respec o hese wo properies. We define a formal deformaion of A o be a formal power series in he indeerminae, (3.1.1) ff = s + s s s 3 + ; in which each s i 2 F 1 (A) (i 1), i.e. is a funcion E! End(A), saisfying he following wo properies (wih s 0 = s and s i (ff) =s i ff): ffl The Caran formula: For every ff 2 E and a; b 2 A, he equaliy (3.1.2) ffff(ab) = fffi(a)ff fl(b) of power series holds. fi+fl=ff

7 DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA 7 ffl The Produc formula: For ff; fi 2 E, he equaliy (3.1.3) ffff ff fi = n fl fffl of power series holds. fl2p (ff;fi) We pause o make a few remarks. Firs, he superscrip i in s i is an index, no an exponen, whereas i is he ih power of he indeerminae. Second, sums and producs of wo power series are aken in he usual way, wih commuing wih every erm in sigh. The coefficiens in (3.1.2) and (3.1.3) are in A and End(A), respecively, and heir calculaions are done in he appropriae rings. In paricular, muliplying ou he righ-hand side of he equaion, one observes ha he Caran formula (3.1.2) is equivalen o he equaliy (3.1.4) s n ff(ab) = n s i fi(a)s n i fl (b) i=0 fi+fl=ff in A for all n 0, ff 2 E, and a; b 2 A. Similarly, unwrapping he lefhand side of he equaion, one observes ha he Produc formula (3.1.3) is equivalen o he equaliy (3.1.5) n i=0 s i ff sn i fi = fl2p (ff;fi) n fl s n fl in End(A) for all n 0 and ff; fi 2 E. When n = 0, (3.1.4) and (3.1.5) are jus he original Caran formula (2.1.3) and produc formula (2.1.1), respecively, for s = s 0. Seing = 0 in he formal deformaion ff, we obain ff 0 = s. So we can hink of ff as a one-parameer curve wih s a he original. We, herefore, call s 1 he infiniesimal, since i is he iniial velociy" of he formal deformaion ff Formal auomorphisms. In order o idenify he infiniesimal as an appropriae cohomology class in F, we need a noion of equivalence of formal deformaions. By a formal auomorphism on A, we mean a formal power series (3.2.1) Φ = 1 + ffi ffi ffi 3 + ; where 1 = Id A and each ffi i 2 End(A), saisfying muliplicaiviy, (3.2.2) Φ (ab) = Φ (a)φ (b) for all a; b 2 A.

8 8 DONALD YAU The same rules of dealing wih power series apply here as well. In paricular, muliplicaiviy is equivalen o he equaliy (3.2.3) ffi n (ab) = n i=0 ffi i (a)ffi n i (b) for all n 0 and a; b 2 A, in which ffi 0 =1. The condiion when n =0is rivial, as i only says ha he ideniy map on A is muliplicaive. When n = 1, he condiion is (3.2.4) ffi 1 (ab) = affi 1 (b) + ffi 1 (a)b; which is equivalen o say ha ffi 1 is a derivaion on A. More generally, if ffi 1 = ffi 2 = = ffi k = 0, hen ffi k+1 is a derivaion on A. I is an easy exercise in inducion o see ha a formal auomorphism Φ has a unique formal inverse (3.2.5) Φ 1 = 1 ffi (ffi 2 1 ffi 2 )+ 3 ( ffi 3 1 +ffi 1 ffi 2 +ffi 2 ffi 1 ffi 3 )+ ; for which The coefficien of n in Φ 1 Φ Φ 1 = 1 = Φ 1 Φ : is an inegral polynomial in ffi 1 ; ;ffi n. Moreover, he muliplicaiviy of Φ implies ha of Φ 1 a; b 2 A, wehave. Indeed, for elemens ab = (Φ Φ 1 (a))(φ Φ 1 (b)) = Φ (Φ 1 (a)φ 1 (b)); which implies ha Φ 1 is muliplicaive. We record hese facs as follows. Lemma 3.3. Le Φ =1+ffi ffi 2 + be a formal auomorphism on A. Then he firs non-zero ffi i (i 1) is a derivaion on A. Moreover, he formal inverse Φ 1 of Φ is also a formal auomorphism on A. Now if s n is a funcion E! End(A), i.e. a 1-cochain, and if f and g are in End(A), hen we have a new 1-cochain fs n g wih (fs n g)(ff) = fsn ff g in End(A) for ff 2 E. Therefore, i makes sense o consider he formal power series Φ 1 ffφ whenever Φ is a formal auomorphism. Proposiion 3.4. Le ff and Φ be, respecively, a formal deformaion and a formal auomorphism of A. Then he formal power series Φ 1 ffφ is also a formal deformaion of A.

9 DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA 9 Proof. We need o check he Caran formula (3.1.2) and he Produc formula (3.1.3) for ~ff =Φ 1 ffφ. For he Caran formula, we have ~ff ff(ab) = (Φ 1 ffff)(φ (a)φ (b)) = Φ 1 = fi+fl=ff fi+fl=ff (Φ 1 (ff fiφ (a))(ff flφ (b)) fffiφ (a))(φ 1 ffflφ (b)): We have used he Caran formula for ff and he muliplicaiviy of boh Φ and Φ 1, exended o power series. The Produc formula is equally easy o verify. Given wo formal deformaions ff and ~ff of A,say ha hey are equivalen if and only if here exiss a formal auomorphism Φ on A such ha (3.4.1) ~ff = Φ 1 ff Φ : By Lemma 3.3 his is a well-defined equivalence relaion on he se of formal deformaions of A. Here is he main resul of his secion, which idenifies he infiniesimal wih an appropriae cohomology class in H 1 (F ). Recall ha cocycles, coboundaries, cochains, and cohomology classes are all aken in he cochain complex F = F (A). Theorem 3.5. Le ff = P n n s n be a formal deformaion of A. Then he infiniesimal s 1 is a 1-cocycle, i.e. d1 s 1 =0. Moreover, he cohomology class [s 1 ] is an invarian of he equivalence class of ff. More generally, if s 1 = = s k = 0 for some posiive ineger k, hen is a 1-cocycle. s k+1 Proof. To show ha s 1 is a 1-cocycle, we need o prove ha d 1 i s1 = 0 for i =0; 1. When n = 1 he Produc formula (3.1.5) saes ha and so s ff s 1 fi + s 1 ff s fi = (d 1 0 s1 )(ff; fi) = s ff s 1 fi fl2p (ff;fi) fl2p (ff;fi) n fl s 1 fl ; n fl s 1 fl + s 1 ff s fi = 0: Similarly, he Caran formula (3.1.4) when n = 1 saes ha which implies ha s 1 ff(ab) = fi+fl=ff (s fi (a)s 1 fl(b) + s 1 fi (a)s fl(b)); (d 1 1 s1 ) ff (a Ω b) = 0; as desired. This shows ha s 1 is a 1-cocycle.

10 10 DONALD YAU If s 1 = = s k = 0, hen he same argumen as above shows ha s k+1 =0. P Now suppose ha ~ff = n n ~s n is a formal deformaion of A ha is equivalen o ff. This means ha here exiss a formal auomorphism Φ such ha ~ff = Φ 1 ffφ s + (s 1 + s ffi 1 ffi 1 s ) (mod 2 ) s + (s 1 + d 0 ffi 1 ) (mod 2 ): In paricular, he 1-cocycle (~s 1 s1 ) 2 F 1 is a 1-coboundary d 0 ffi 1. (Remember ha ffi 1 is a derivaion on A, which is herefore a 0-cochain.) Thus, he cohomology classes, [~s 1 ] and [s 1 ], are equal in H 1 (F ), as assered. In view of his Theorem, i is naural o ask wheher a given cohomology class in H 1 (F ) is he infiniesimal of a formal deformaion. This quesion will be deal wih in Secion Exending formal auomorphisms and rigidiy As in he previous secion, A will denoe an S-algebra wih Landweber- Novikov operaions s and F = F (A). The main purpose of his secion is o obain cohomological condiions under which A is rigid, ha is, every formal deformaion is equivalen o he rivial deformaion s. To do ha, we firs have o consider how runcaed formal auomorphisms can be exended Formal auomorphisms of finie order. Le m be a posiive ineger. Inspired by Gersenhaber-Wilkerson [7], we define a formal auomorphism of order m on A o be a formal power series Φ = 1 + ffi ffi m ffi m wih 1 = Id A and each ffi i 2 End(A), saisfying muliplicaiviy (3.2.2) modulo m+1, i.e. he equaliy (3.2.3) holds for 0» n» m and all a; b 2 A. One can hink of a formal auomorphism as a formal auomorphism of order 1. Given such a Φ, we say ha i exends o a formal auomorphism of order m + 1 if and only if here exiss ffi m+1 2 End(A) such ha he power series (4.1.1) ~Φ = Φ + m+1 ffi m+1 is a formal auomorphism of order m+1. Sucha~Φ is said o be an exension of Φ o order m +1.

11 DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA 11 The quesion we would like o address here is his: Given a formal auomorphism Φ of order m, can i be exended o a formal auomorphism of order m +1? I urns ou ha he obsrucion o he exisence of such an exension lies in Hochschild cohomology, which we now recall. Consider A as a unial algebra over he ring of inegers Z, he Hochschild cochain complex C (A; A) of A wih coefficiens in A iself is defined as follows. The nh dimension is he group The differenial is given by he alernaing sum C n (A; A) = Hom(A Ωn ;A): b n 1 : C n 1 (A; A)! C n (A; A) (4.1.2) (b n 1 f)(a 1 Ω Ωa n ) = a 1 f(a 2 Ω Ωa n )+ n 1 ( 1) i f( a i 1 Ω a i a i+1 Ω a i+2 )+( 1) n f(a 1 Ω Ωa n 1 )a n : The reader can consul, for example, Weibel [11] for more deailed discussions abou Hochschild cohomology. The nh Hochschild cohomology group of A over Z wih coefficiens in A iself is denoed by HH n (A). We now reurn o he quesion of exending a formal auomorphism Φ of order m o one of order m + 1. Consider he Hochschild 2-cochain given by Ob(Φ ): A Ω2 Ob(Φ )(a Ω b) = m! A ffi i (a)ffi m+1 i (b) for all a; b 2 A. Call Ob(Φ ) he obsrucion class of Φ. Lemma 4.2. The obsrucion class Ob(Φ ) is a Hochschild 2-cocycle. Proof. We calculae as follows: (b 2 Ob(Φ ))(a Ω b Ω c) = a = = 0 m ffi i (b)ffi m+1 i (c) + m i+j+k = m+1 i; k>0 m ffi i (a)ffi m+1 i (bc) + c ffi i (a)ffi j (b)ffi k (c) ffi i (ab)ffi m+1 i (c) m ffi i (a)ffi m+1 i (b) i+j+k = m+1 i; k>0 ffi i (a)ffi j (b)ffi k (c):

12 12 DONALD YAU Here we used he muliplicaiviy of Φ, namely, (3.2.3) for ffi i (ab) and ffi m+1 i (bc). Also, ffi 0 = 1 as usual. This shows ha he obsrucion class of Φ isahochschild 2-cocycle. We are now ready o show ha he obsrucion o exending Φ o a formal auomorphism of order m +1 is exacly he cohomology class [Ob(Φ )] 2 HH 2 (A). Theorem 4.3. Le Φ be a formal auomorphism of order m on A. Then exends o a formal auomorphism of order m +1 if and only if he Φ obsrucion class Ob(Φ ) is a Hochschild 2-coboundary. Proof. The exisence of an order m+1 exension ~Φ as in (4.1.1) is equivalen o he exisence of a ffi m+1 2 End(A) for which (3.2.3) holds for n = m +1. This las condiion can be rewrien as Ob(Φ )(a Ω b) = ffi m+1 (ab) + affi m+1 (b)+bffi m+1 (a) = (b 1 ffi m+1 )(a Ω b); and b 1 ffi m+1 is a Hochschild 2-coboundary. An immediae consequence of his Theorem is ha, saring wih a derivaion ffi, he vanishing of he Hochschild cohomology HH 2 (A) implies ha he formal auomorphism Φ = 1+ m ffi of order m 1 can always be exended o a formal auomorphism. This will be useful when we discuss rigidiy below. Corollary 4.4. Le m be aposiive ineger and le ffi be a derivaion on A. Assume ha HH 2 (A) =0. Then here exiss a formal auomorphism on A of he form Φ = 1 + m ffi + m+1 ffi m+1 + m+2 ffi m+2 + : Proof. Using he hypohesis ha ffi is a derivaion, i is sraighforward o verify ha he formal power series Φ = 1 + m ffi is a formal auomorphism of order m, i.e. i saisfies muliplicaiviy (3.2.3) for n» m. As HH 2 (A) = 0, by Theorem 4.3 he obsrucions o exending Φ o a formal auomorphism vanish, as desired Rigidiy. Following he erminology in [3], an S-algebra A is said o be rigid if every formal deformaion of A is equivalen o he rivial deformaion s. Using he resuls above, we will be able o obain cohomological crierion for he rigidiy of A. Firs we need he following consequence of Corollary 4.4.

13 DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA 13 Corollary 4.6. Le k be a posiive ineger and le ff = s + k s k + k+1 s k+1 + be a formal deformaion of A in which s k 2 F1 is a 1-coboundary. Suppose ha HH 2 (A) =0. Then here exiss a formal auomorphism of he form (4.6.1) Φ = 1 k ffi k + k+1 ffi k+1 + such ha (4.6.2) Φ 1 ff Φ s (mod k+1 ): Proof. By he hypohesis on s k, here exiss a derivaion ffi k 2 Der(A) such ha d 0 ffi k = s ffi k ffi k s = s k : Since ffi k is also a derivaion on A, Corollary 4.4 implies ha here exiss a formal auomorphism Φ as in (4.6.1). Compuing modulo k+1,wehave This proves he Corollary. Φ 1 ff Φ (1 + k ffi k )(s + k s k )(1 k ffi k ) s + k (s k + ffi k s s ffi k ) s : By Proposiion 3.4, he formal power series Φ 1 ffφ is acually a formal deformaion ha is equivalen off. Therefore, by applying his Corollary repeaedly, we obain he following cohomological crierion for A o be rigid. Corollary 4.7. If HH 2 (A) =0and H 1 (F (A)) = 0, hen A is rigid. We should poin ou ha in he deformaion heory of some oher kinds of algebras, rigidiy is usually guaraneed by he vanishing of jus one cohomology group, usually an H 1 or an H Exending cocycles As before, A will denoe an arbirary bu fixed S-algebra, and F = F (A). The Landweber-Novikov operaions on A are given by a funcion s as in (2.2.1). Recall from Theorem 3.5 ha he infiniesimal of a formal deformaion is a 1-cocycle in F. The purpose of his secion is o answer he quesions: (1) Wha is he obsrucions o exending a 1-cocycle o a formal deformaion? (2) Wha is he obsrucions o wo such exensions being equivalen? We will break hese quesions ino a sequence of smaller quesions, each of which is deal wih in an obsrucion heoreic way.

14 14 DONALD YAU 5.1. Formal deformaions of finie order. Firs we need some definiions. Le m be a posiive ineger. As in he previous secion, a formal deformaion of order m of A is a formal power series (5.1.1) ff = s + s m s m ; in which each s i (i 1) is in F 1 such ha he Caran formula (3.1.2) and he Produc formula (3.1.3) are saisfied modulo m+1. In oher words, (3.1.4) and (3.1.5) are saisfied for all n» m. We say ha a formal deformaion ff of order m exends o a formal deformaion of order M >mif and only if here exis 1-cochains s m+1 ;:::;s M such ha he power series (5.1.2) ~ff = ff + m+1 s m M s M is a formal deformaion of order M. Call ~ff an order M exension of ff. One can hink of a formal deformaion as a formal deformaion of order 1. Given a 1-cocycle s 1, he problem of exending he formal deformaion ff = s + s 1 of order 1 o a formal deformaion can be hough of as exending ff o order 2, hen order 3, and so forh. We will do his by idenifying he obsrucion o exending an order m formal deformaion o one of order m +1. Le ff be a formal deformaion of order m as in (5.1.1). Consider he 2-cochain Ob(ff ) = (Ob 0 (ff ); Ob 1 (ff )) 2 F 2 = F 2 0 F 2 1 ; whose componens are defined by he condiions, (5.1.3) Ob 0 (ff )(ff; fi) = and (5.1.4) Ob 1 (ff ) ff (aωb) = m m We call Ob(ff ) he obsrucion class of ff. s i ff sm+1 i fi (ff; fi 2 E) s i fi (a)sm+1 i fl (b) (ff 2 E; a;b2 A): fi+fl=ff Lemma 5.2. The obsrucion class Ob(ff ) is a 2-cocycle. Proof. We need o show ha d 2 i Ob i(ff ) = 0 for i = 0; 1. Since ff is he only formal deformaion we are dealing wih, we abbreviae Ob i (ff )oob i.

15 DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA 15 For i = 0, le ff; fi; fl be exponenial sequences. Then we have (5.2.1) (d 2 0 Ob 0 )(ff; fi; fl) = s ff m ffi2p (ff;fi) s i fi sm+1 i fl "2P (fi;fl) n " m s i ff sm+1 i " + m n ffi ffi2p (ff;fi) m + s i ffi sm+1 i fl s i ff sm+1 i fi s fl : This is a sum of four erms. Using he Produc formula (3.1.5), he second erm can be rewrien as m m i n ffi s i ffi s m+1 i fl = s l ff si l fi sm+1 i fl : l=0 In paricular, he m summands in i corresponding o l = 0 cancel ou wih he firs erm in (5.2.1). Consequenly, he sum of he firs wo erms in (5.2.1) becomes (5.2.2) i+j+k = m+1 i;k>0 s i ff sj fi sk fl : An analogous argumen applied o he hird and fourh erms in (5.2.1) shows ha heir sum is exacly (5.2.2) wih he opposie sign. I follows ha d 2 0 Ob 0 = 0, as desired. The proof of d 2 1 Ob 1 =0is quie similar o he argumen above and he proof of Lemma 4.2. Indeed, (d 2 1 Ob 1) ff (a Ω b Ω c) (see (??)) is again he sum of four erms. Using he Caran formula (3.1.4), is second erm can be rewrien as (5.2.3) (Ob 1 ) ff (ab Ω c) = = m s i fi(ab)s m+1 i fl (c) fi+fl=ff m fi+fl=ff i j=0»+ =fi s j»(a)s i j (b) s m+1 i fl (c): The summands corresponding o j = 0 cancel ou wih he firs of he four erms in (d 2 1 Ob 1) ff (a Ω b Ω c). In paricular, he sum of he firs wo erms in (d 2 1 Ob 1) ff (a Ω b Ω c) is (5.2.4) s i fi(a)s j fl(b)s k ffi (c): The summaion is aken over all riples (i; j; k) of non-negaive inegers and all riples (fi;fl;ffi) of exponenial sequences for which and i + j + k = m + 1 (i; k > 0) fi + fl + ffi = ff:

16 16 DONALD YAU A similar argumen applies o he las wo erms in (d 2 1 Ob 1) ff (a Ω b Ω c), showing ha heir sum is exacly (5.2.4) wih he opposie sign. I follows ha d 2 1 Ob 1 = 0, as expeced. This finishes he proof of he Lemma. As he obsrucion class of ff is a 2-cocycle, i represens a cohomology class in H 2 (F ). We are now ready o presen he main resul of his secion, which idenifies he obsrucion o exending a formal deformaion of order m o one of order m +1. Theorem 5.3. Le ff be a formal deformaion of order m of A. Then ff exends o a formal deformaion of order m+1 if and only if he cohomology class [Ob(ff )] 2 H 2 (F (A)) vanishes. Proof. Indeed, he exisence of an order m+1 exension ~ff of ff, as in (4.1.1) wih M = m + 1, is equivalen o he exisence of a 1-cochain s m+1 2 F 1 for which he Caran formula (3.1.4) and he Produc formula (3.1.5) boh hold for n = m +1. Simply by rearranging erms, he Caran formula (3.1.4) when n = m + 1 can be rewrien as (Ob 1 (ff)) ff (a Ω b) = (d 1 1 sm+1 ) ff (a Ω b): Similarly, he Produc formula (3.1.5) when n = m + 1 is equivalen o (Ob 0 (ff))(ff; fi) = (d 1 0 sm+1 )(ff; fi): These wo condiions ogeher are equivalen o (5.3.1) Ob(ff ) = d 1 s m+1 ; i.e., he obsrucion class is a 2-coboundary. Since Ob(ff ) is a 2-cocycle (Lemma 5.2), he Theorem is proved. Applying his Theorem repeaedly, we obain he obsrucions o exending a 1-cocycle o a formal deformaion. Corollary 5.4. Le s 1 2 F1 (A) be a1-cocycle. Then here exis a sequence of classes! 1 ;! 2 ;::: 2 H 2 (F (A)) for which! n (n>1) is defined if and only if! 1 ;:::;! n 1 are defined and equal o 0. Moreover, he formal deformaion ff = s + s 1 of order 1 on A exends o a formal deformaion if and only if! i is defined and equal o 0 for each i =1; 2;:::. Since hese obsrucions lie in H 2 (F (A)), he rivialiy of his group implies ha exensions always exis. Corollary 5.5. If H 2 (F (A)) = 0, hen every formal deformaion of order m 1 of A exends o a formal deformaion.

17 DEFORMATION OF ALGEBRAS OVER THE LANDWEBER-NOVIKOV ALGEBRA Equivalence of formal deformaions of finie order. Le ff be a formal deformaion of A of order m 1, and le ~ff and μff be wo order m+1 exensions of i. Are he wo exensions equivalen? This is he quesion ha we would like o address in his final secion. Firs, we need a definiion of equivalence. Two formal deformaion ff and ~ff of order m of A are said o be equivalen if and only if here exiss a formal auomorphism Φ of order m such ha he equaliy (3.4.1) holds modulo m+1. This is a well-defined equivalence relaion on he se of formal deformaions of order m of A. In fac, i is easy o see ha a formal auomorphism Φ of order m has a formal inverse Φ 1 as in (3.2.5) which is also a formal auomorphism of order m. Now le ff = s + + m s m be a formal deformaion of order m of A, and le ~ff = ff + m+1 ~s m+1 and μff = ff + m+1 μs m+1 be wo order m +1 exensions of ff. Then (5.3.1) in he proof of Theorem 5.3 ells us ha d 1 ~s m+1 = Ob(ff ) = d 1 μs m+1 : I follows ha (~s m+1 μs m+1 ) 2 F 1 is a 1-cocycle, and i makes sense o consider he cohomology class in H 1 (F ) represened by i. Theorem 5.7. If he cohomology class [~s m+1 μs m+1 ] 2 H 1 (F ) vanishes, hen he formal deformaions, ~ff and μff,oforder m +1 are equivalen. Proof. The argumen is quie similar o he proofs of Theorem 3.5 and Corollary 4.6. In fac, by hypohesis, here exiss a derivaion ffi on A such ha We have he formal auomorphism ~s m+1 μs m+1 = d 0 ffi = s ffi ffis : Φ = 1 + m+1 ffi of order m +1onA. Compuing modulo m+2,wehave Φ 1 as desired. μff Φ (1 m+1 ffi)(ff + m+1 μs m+1 )(1 + m+1 ffi) ff + m+1 (μs m+1 + s ffi ffis ) ff + m+1 ~s m+1 ~ff ; This obsrucion heoreic resul is less saisfacory han he analogous Theorem 4.3 and Theorem 5.3 in ha he auhor is no sure wheher he equivalence of ~ff and μff would imply he vanishing of he class [~s m+1 μs m+1 ].

18 18 DONALD YAU Acknowledgemen The auhor hanks he referee for reading an earlier version of his paper. References [1] J. F. Adams, Sable homoopy and generalised homology, Chicago Lecures in Mahemaics, Universiy of Chicago Press, Chicago, Ill.-London, [2] V. M. Bukhshaber and A. V. Shokurov, The Landweber-Novikov algebra and formal vecor fields on he line, Funksional. Anal. i Prilozhen. 12 (1978), no. 3, 1-11, 96. [3] M. Gersenhaber, On he deformaion of rings and algebras, Ann. Mah. (2) 79 (1964), [4] M. Gersenhaber, On he deformaion of rings and algebras. II, Ann. Mah. 84 (1966), [5] M. Gersenhaber, On he deformaion of rings and algebras. III, Ann. Mah. (2) 88 (1968), [6] M. Gersenhaber, On he deformaion of rings and algebras. IV, Ann. Mah. (2) 99 (1974), [7] M. Gersenhaber and C. W. Wilkerson, On he deformaion of rings and algebras, V: Deformaion of differenial graded algebras, Higher homoopy srucures in opology and mahemaical physics (Poughkeepsie, NY, 1996), , Conemp. Mah., 227, Amer. Mah. Soc., Providence, RI, [8] P. S. Landweber, Cobordism operaions and Hopf algebras, Trans. Amer. Mah. Soc. 129 (1967), [9] J. Morava, On he complex cobordism ring as a Fock represenaion, Homoopy heory and relaed opics (Kinosaki, 1988), , Lecure Noes in Mah., 1418, Springer, Berlin, [10] S. P. Novikov, Mehods of algebraic opology from he poin of view of cobordism heory, Izv. Akad. Nauk SSSR Ser. Ma. 31 (1967), [11] C. A. Weibel, An inroducion o homological algebra, Cambridge sudies in advanced mahemaics 38, Cambridge Univ. Press, Cambridge, UK, [12] R. M. W. Wood, Differenial operaors and he Seenrod algebra, Proc. London Mah. Soc. 75 (1997), address: dyau@mah.ohio-sae.edu Deparmen of Mahemaics, The Ohio Sae Universiy Newark, 1179 Universiy Drive, Newark, OH 43055