Sem Analysis MATH 607
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Formal Group Laws and Algebraic Topology ecture notes Math 607 Winter 2008 Hal Sadofsky 1 INTRODUCTION These notes are from a one quarter course in one dimensional formal group laws in homotopy theoryl Chapter 1 gives an introduction and covers de nitions Chapter 2 gives a short introduction to stable homotopy for the sole purpose of exposing the link between homotopy theory and one dimensional formal group lawsl Chapter 3 proves Lazardls theorem identifying the universal formal group lawl Chapter 4 sketches a proof of Quillenls theorem which identi es the Lazard ring with the coefficient ring of complex cobordisml Because this is really a result in stable homotopy much of this chapter is homotopy theory including a description of the Steenrod algebra and of the Adams spectral sequence Chapter 5 develops some of the theory of p typical formal group laws and Cartierls theorem that every formal group law over a Zpalgebra is naturally isomorphic to a p typical onel Chapter 6 discusses nite height in characteristic pl We prove Lazardls theorem that height is the unique isomorphism invariant over a separably closed eld and calculate some endomorphism ringsl Finally in chapter 7 we summarize Lubin and Tate s work 11 on starisomorphisms We covered this in detail in the course but we essentially used the original source 11 so see no need to include it here That chapter also has some notes of an expository nature on applications to homotopy theoryl None ofthe material in these notes is original though an occasional proof may be Much of the material and many proofs were taken directly from part ll of l and from the beautifully concise Appendix 2 of 14 There are also some examples and ideas take from In particular chapters 3 and 4 owe much to my recollection of l and 14 and much of chapters 5 and 6 is an expansion of material from Appendix 2 of 14 ll De nitions and Examples De nition 11 A ndimensional formal group law over a commutative 7iug R is a collection ofn power seiies Fzll 1 y1l l l yn E R 11 l l Iny1ulyn such that 1 R0E FE0 E 2 WERE WHEEL Exercise 11 Show that an inverse series automatically exists ie there is a vector of power series ilf E so that FT 0 Examples 11 l Fz y zyl This is called the additive FGL for obvious reasons and welll write it sometimes as Fa z y l 2 Fz y z y uzyl This is called the multiplicative FGL because 1 uFzy 1 uzl uy and well write this as Fm z y which is ambiguous since u is not speci ed 3 Fzy 13 Well use notation like this freelyl There is no concern with convergence of power series here so an expression like 172 1 means I y17 ry WV 1W a a i We havenlt veri ed any of the axioms for these examples They are all easy7 but welll do the hardest one for this example zFyz 7 IL1LZ zyzzy2 lzFyz 71zl lzyzzy2 FIvFyyz By symmetry7 this is clearly the same as FFz7 y De nition 12 A homomorphism of FGLs f F 7gt G is a power series E such that fFE Cff7 This is the right de nition you pretend that and G are formulas for group operations Obviously f is an isomorphism if it is an invertible sequence of power series It is called a strict isomorphism the coe cient off is l ie E E modulo terms of degree greater than 1 note that the constant term of f must be 0 If R is complete with respect to a maximal ideal m7 then a formal group law F of dimension n gives a group structure on m RF If f F 7gt G is a homomorphism of formal group laws over R then f induces a group homomorphism RF 7gt RG De nition 13 A formal group law is called commutative FM 7 Fan We will refer to onedimensional commutative formal group laws as just formal group laws or FGLs for short7 since we will only be concerned with the one dimensional commutative case in this course7 though many of the ideas in this chapter make sense for formal group laws of dimension greater than 1 De nition 14 A logarithm for a formal group law F written logF is an iso morphism from F to the additive FGL An exponential for F written epr is an isomorphism the other way Example 12 Let expz denote the series I 722 e 7 1 For this to exist as a formal power series in RHIH one needs R to contain Q Let logz denote the inverse series I 7 g g 7 lnl Let Fzy z y my 3 Then logFzy log1 z1 y 71 1n1 z1 y ln1 z lnl y logz logy So logz is a logarithm for the multiplicative formal group law when u l7 and expz is the corresponding exponential Exercise 12 Find the log and exp for Fzy z y uzy e 7equot eze72 7 Example 13 For a similar example7 we recall that tanhu and one can verify that tanhu tanhv t h an u l v 1 tanhu tanhv Using this one sees that I tanhil tanh 1z tanh 1y which identi es the log series for this formal group or at least the exp series 3 One can do a similar trick with the addition formula for tanu v and the FGL Fm zy z 7 Actually we can use this example to illustrate the fact that whether two formal group laws are isomorphic depends strongly on what ring they are taken over logFcan is tanz ie the formal integral of the power series his r This series is I3 I5 I7 17 E77Hi which is de ned over Z 2 so Han is isomorphic to the additive formal group over Z On the other hand Fa 1 Fa z 31 0 over ZS and Fumz Fumz 713 do the calculation over Z3i If we had a logarithm for Ear over ZS then IOgFan 13 Fa dogma 17Fa10gFan 17 logpm 1 0 This contradicts the form of a logarithm function logFcan 713 must equal a unit times 713 mod terms of degree 6 and higher So the two formal group laws are not isomorphic over ZS or over Z A similar veri cation concerning isomorphisms is that Fa LE 1 r r r 0 over Zp where p 17s appear while FmzFmz r r r 11 u 1 So Fa and Fm are not isomorphic over any nite e r This is a natural place to introduce a notation we won7t really need for a while If Fz y is a formal group law we Fm FltzFltz i gtgtgt where 1 appears n times By associativity it doesnlt matter exactly how the Fls are inserted and again by associativity it is obvious that I nm In fact with the convention that 0 0 this justi es our convention to use ilz for the formal inverse power series It also suggest that we should de ne in inductively as F7n lz The series p where p is a prime has a special place in the theory of formal groups over a eld of characteristic p The degree of its leading term turns out to be the most important invariant called the height of the formal groupi Exercise 13 Check that plpa 0 ifR has characteristic p and that p 11 if R has characteristic pi Fm1 12 Construction of a universal FGL The last construction we need to make in this section is that of the universal formal group lawi We consider the functor 97 here where 9R is de ned to be the set of formal group laws over the commu tative ring Rf 97 is a covariant functor from Q the category of commutative rings to mi The covariance is supplied by if f R A S and F E 9R can be written I y Eaijziyj then fquot F is the formal group law over S that can be written I y E faijziyji Our universal formal group law will be a ring L together with a FGL over L FL such that homCRL R is naturally isomorphic to Here the isomorphism is by f E homCRL R goes to fquot FL The consmction of L is formali De ne a polynomial ring on countably many generators P Zaijlij 2 1 Let Fzy E be the power series I y E ai iyj Let I be the ideal generated by the coefficients of ziyjzk in FFI7y7z F17Fyyz as ij k run through all triples ij k 2 1 together with the elements aij 7 ajii Then L PI and FL is the image of F in L It is now a triviality to verify that the pair L FL is universal in the desired sense For the purposes of topology and sometimes even for algebra it is appropriate to note that L has a natural gradingi We wish our indeterminate z in the formal group law to have degree two not one since then the graded sign convention requires that 12 0 over most rings hence power series are not very interesting Then our formal group law should be homogeneous of degree 2 so laijl 72239 7 2j 2 13 FGLS over Qalgebras There is an easy proof that onedimensional com mutative formal group laws over Q are not terribly interestingi Given a formal group law Fzy over R 2 Q let a 1 39 F2Iyy 9in179 1ZaijJI y 1 a l 1 0 W dti All the operations here are formali For example F2 t 0 1 2 anti and hence l 39 2 39 3 m 1 anti Zailtl anti 39 quot Integration is just formal integration of power series so an 39 l 7 tz1 m zltZH gt but this is where we7ve used the fact that R contains denominatorsi Now let X lFz 7 X is a power series in two variables and we wish to show X 0i Certainly the constant term is 0 We calculate using the chain rule that 6X 67 l FIyyF2Iyy My Now let 1 l F 7 7 swmmm2 emm Now we take the relation FFzyz 7 FzFyz 0 and differentiate this with respect to 2 at 2 0 We get F2F17 y70 F217Fy70F2y70 0 Noting that F2 zFy 0 F2 zy and dividing by F2y0 F2 Fzy 0 we get 1 l 7 7 F i My 0 F2ltFltzygt0gt 2 M So BXBy 0 By symmetry BXBz 0 It follows that X 0 Exercise 14 Suppose E and fg have no constant termsi Prove that wmm 81 1 implies that Find a counterexample to this assertion as well 5 Exercise 15 l Prove that logarithms are not necessarily unique lie nd a ogarithm for the additive formal group law over Zp besides logz I 2 Prove that if R has no torsion and F is a formal group law over R then logF is unique if it exists We have several goals in this course related to these fundamental de nitions Two of them are to discover 1 when two FGLs are isomorphic 2 the structure of L and FD Welll pursue 2 in section 3 after brie y explaining the place of formal group laws in homotopy theoryl We7ll make a pass at l in section 6 2i GENERALIZED COHOMOLOGY THEORIES AND THE ATIYAH HIRZEBRUCH SPECTRAL SEQUENCE 2L Generalized cohomology theories We denote by EX the topological space obtained from y IXXOXz0gtltzandlXzlgtltzl This is pronounced the suspension of X77 or EXl If X has a basepoint 10 the same symbols will denote the space IXXOXXUlXXUIXIO This ambiguity is excused by the fact that if the inclusion of the basepoint is a co bration both constructions have the same homotopy type If the inclusion of the basepoint isn t a co bration l don7t want to hear about it For this discussion and the discussion below if you don7t know what a co bration is think of the inclusion of a subcomplexl Welll also have occasion to use the construction 9X pronounced loops on Xi77 For this construction X must have a basepoint and then 9X mapxsl X mapllt1o1gt cam the space of continuous pointed say taking 1 E 5391 to 10 E X maps from 5391 to Xi De nition 21 A reduced generalized cohomology theory Equot 7 is afuhctorfrom Spaces t0 Zgmded groups Gers such that l fA A X is a co bmtizm we get a long exact sequence gt E XA A E X A E A A E 1XA A E 1X A l l H 2 ff 239s homotopz39c to y then Equot 3 If Xa is a set of spaces with basepoihts Ia and V Xa denotes the one point union then E V Xa H EXal Remark 21 If X is the cone on A then XA 2 EA so from the rst axiom we get the suspension isomorphim77 E A En1EAl Also from the rst axiom if Equot is a reduced cohomology theory Equot pt 0 An unreduced cohomology theory is then a functor as above only the third axiom is replaced with EuXa H E X0 6 and the rst axiom is applied to the corresponding reduced theory F X cokernelE pt a E X To recover the unreduced theory from the reduced theory EX take EX EB ESoi It is common when E 7 is reduced to abbreviate E 50 by E5 Welll do that here and from context it should not lead to ambiguity When Equot 7 is an unre duced theory Equot usually denotes Equot pti Examples 22 l H 7A cohomology with coef cients in your favorite abelian group or the corresponding reduced theory The reduced theory is recognizable by its values on spheres A k n 0 k f n Welll be interested in Z Q and Zp mostlyi 2 K X K theoryi Here we could deal with real complex or conceivably some other sort of Ktheoryi The example closest to the topics in this course is complex so 1711 brie y describe thati Let X be a space and VectX the set of isomorphism classes of complex vector bundles over Xi VectX is a contravariant functor under pullback and it is easy to prove that it is homotopy invarianti Unfortunately it isn7t a group though it is an abelian monoid under Whitney sumi Given an abelian monoid one can construct a group by for example looking at formal difference of elements We de ne 1 K000 lal bl a bl E VectXlal bl Cl M if L169 4 1769069 Ekl Here it is obvious how to add and subtract The 5k in the relation denotes the trivial vector bundle of dimension kl This bit of the relation gives us the reduced theory if we were to take the same relation without the 5 we would get the corresponding unreduced K theoryi K 1X K0EX tells us how to de ne Ki when i is negative Bott periodicity tells us that K0X K0 22X and this allows us to shift into positive dimensioni So complex K theory is a cohomology theory that has the property along with some others that HnskA n k 7 Z k 7 n even K S7 0 k7nelsei 3 MU X Complex cobordismi It is a little awkward to de ne the coho mology theory MUquot 7 geometrically but see 13 There is an associated homology theory MU X given by MUmX Mm 7gt XlM is almost complex N Here by almost complex I mean that M embeds in some RN in such a way that its normal bundle has a complex structure in other words if 1 M 7gt BON 7 m classi es the normal bundle then 1 lifts to BUN 7 The relation N is that f M A X is equivalent to g z N A X if fg extend to an almost complex cobordism from to One can describe MU X geometrically if X happens to be an n man ifold itself In this case MUmX hmu M HM g X X Rk N k where the normal bundle of 1 has a complex structure and the limit is taken by crossing with Ri As before the relation N is almost complex cobordisms that are submanifolds of X X Rki Welll prove in section 4 that MUquot pt Zzlzgi where 72239 but welll use this fact prematurely for some calculations in section 23 22 The AtiyahHirzebruch Spectral Sequence If we lter a CW complex X by skeleta and apply Equot 7 EX lt7 EX1 lt7 EX2 lt7 J7 EX we get a cohomology spectral sequence converging to an associated graded of Equot X corresponding to the decreasing ltration F5 kerE X A Equot X5i Here F 1 is everything as is F0 if Equot is reduced and X is connected Assuming E is reduced Efquot E X5X5 1 05X E where C is the reduced cellular cochain complex It is easy to verify that d1 is the differential in the cellular cochain complex so H5ltX E where we use H to stand for reduced cohomology and likewise for E i if I Em A Esmirr If the cohomology theory Equot 7 is a ring theory by which I mean Equot X is naturally a ring not necessarily with unit in the reduced case then this spectral sequence is a spectral sequence of algebras and the differentials are derivationsi This means as usual that the E00 term with the algebra structure induced by the E2 term is the associated graded algebra to E X when the spectral sequence converges The spectral sequence is a right halfplane spectral sequence rst quadrant if t lt 0 implies E 0 and convergence can be an issue since it is possible for a speci c group to support in nitely many nonzero differentials For nite cell complexes X the spectral sequence always converges to Equot X and for in nite complexes X it frequently converges to Equot i For MU 7 the coef cients of the cohomology theory are concentrated in non positive degrees the spectral sequence is 4th quadranti For ordinary cohomology the E1 term is the cellular cochain complex and is concentrated along t 0 he spectral sequence collapses at E Remark 23 The Atiyah Hirzebruch spectral sequence can also be constructed by looking at the Postnikov tower for the spectrum representing the cohomology theory Equot 7 instead of the cellular ltration of Xi This is a happier arrangement for proving naturality results and so fort Welll use this spectral sequence to compute MUquot CPm Keep in mind that cohomology theories are taken to be reduced unless we explicitly state otherwise The Egterm for the Atiyah Hirzebruch SS for CP is HCP MU H CP MU Since H CP is a truncated polynomial ring on a generator in degree 2 and MUquot is polynomial on generators in even dimensions the Egterm is concentrated in even bidegreesi Hence the spectral sequence collapses at E2 If we let i be a lift of I l generating E30 to MUquot Cpn then we get a map of ltered rings MUIz 1 A MUquot Cpn by sending z to i This map is an isomorphism on associated graded rings so is an isomorphismi To complete our proof we need to use Milnor7s exact sequence which says that if Equot 7 is a cohomology theory and X0HX1HX2 is a sequence of co brations then there is a short exact sequence 0 411m1E1X a E11mX a 11mEX a 0 Applying this when Equot MUquot and Xn CP we get MU CPm MU Exercise 21 lmitate this proof to show that MUCP gtlt CP X MUM 2 MUCP MUCP 9 where y Trim and 2 761 Here 7r1 and Tr are the projections onto the rst and second factors 23 Formal groups in topology The point of this digression into cohomology theories is to describe the source of FGLs in homotopy theoryi Recall that CP B51 so it is the classifying space for complex line bundlesi Let M CP gtlt CP A CP be the multiplication map classifying tensor product of line bundles that is Mquot 51 52 where 5 is the tautological line bundle and is the tautological line bundle pulled back under projection to the ith factor ma We get a power series in two variables Fyz E by taking Mquot I using the isomorphisms MUCP MUCP gtlt CP MU llIll MU llyyzllA By applying MUquot 7 to the diagrams involving the associativity commutativity and unicity of it one shows that Fzy is a formal group lawi One calculuation we7ll obviously need that we havenlt mentioned for associativity is MUquot CP 3 which is the obvious power series ring on 3 indeterminatesi For example7 because tensor product of line bundles is associative7 ie VEE lt Vwegetnngtlt12n1gtltiasamap CP o gtlt CP gtlt CP A 013 Commutativity involves observing that the commutativity of tensor product implies M o 7 2 n where 739 is the map from CP gtlt CP to itself that interchanges factors Unicity involves expressing the fact that tensoring with a trivial line bundle is the isomorphism as a commutative diagram between products of CI De nition 22 We de ne a ring valued cohomology theory to be complex oriented if ECP A Equot 52 is surjective A complex orientation of Equot 7 is a choice of class I E E2CP that maps to a generator of E2 52 Note that a complex orientation gives a generator not just of the group E2527 but of the graded Emodule Equot 527 which by the suspension isomorphism is free on one generator We could de ne an orientation to have arbitrary degree as long as it generated Equot 52 as an E module7 but it is a bit more convenient here to think about it as degree 2 Exercise 22 lequot 7 is complex oriented and z is an orientation7 then Equot CPm And similarly for two or more factors The same construction as that for MUquot shows that there is a formal group law over Equot corresponding to a complex orientation of Equot We have the following proposition to keep track of how much our choices matter Proposition 23 If 1112 are two complex orientations for Equot 7 then the cor responding formal group laws F1 and F2 are isomorphic Proof By hypothesis7 we can write ECP E llrlll E llwll So 11 where f is a power series having no constant term and with the coef cient of 12 a unit Construct a diagram E CP X L ECP gtlt CP X l l E llxlll Elllylvzlll E llmll E lly2722ll where the vertical maps are isomorphisms7 the horizontal maps are induced by n and yi Trf 2i 7r Then the vertical maps to the bottom row send 71 to f72 where 7 can be Ly or 2 All maps are maps of rings7 so since this diagram commutes we get fF2 yg 22 F1fy27 by following X1 around the bottom square both ways Hence f is an isomorphism from F2 to F1 D Exercise 23 Let Equot 7 be complex K theory7 so Equot 50 Z 1 where 72 Let 5 be the tautological line bundle over CI 1 Show that z B WE 7 l is a complex orientation of K 2 Show the corresponding FGL is Fzy z y 3 LAZARD S THEOREM For this material7 welll primarily follow part 11 of 17 pp 64 ff There are several proofs of this an interesting one om a very categorical point of View is at least implicit in 1 donlt think that is Lazardls original proof though Before embarking on the hard work7 we study L Q L is the ring constructed in section 12 Proposition 31 L Q Qm1m2m3 where if fr IEmizi1 then FL Qlt 79gt f 1fr fy In other words7 the power series f7 made from the generators of the ring L Q is the logarithm for the rationalization of the universal formal group law Consistent with our grading of L7 mi has degree 72239 Proof We can use Qm1m2m3 to classify FGLs over Qalgebras by sending f to the logarithm for the formal group law In other wor s7 HomQ7algQm1 R lt gt 9R by 9 H y 191yy where 91 I 29071011113 On the other hand7 it is clear that L Q also classi es formal group laws over Qalgebras since any map L A R extends uniqely to L Q7 so the usual argument together with uniqueness of logarithms ensures that the resulting maps between these two rings are inverse isomorphisms Proposition 32 Let M Zm1m2 Q Qm1m2 Then the map L A Qml7 m2 factors through M Proof The map factors through the smallest subring of Qm1m2 over which the formal group is de ned That is certainly no bigger than the subring where the logarithm is defined7 which is M D We need some combinatorics De ne a homogeneous degree n polynomial over Z by 1y 1 y ny pf CnltI7y zyn 7 In iyn n pfy The combinatorics we want to do amounts to showing that On I y is de ned over Z7 and that it is not divisible by any integer The rst statement is obvious when n y pf For n pf it is necessary to check that p divides every binomial coef cient pk gt when 1 lt h lt pf It is easy to check this by hand7 but welll want some more systematic results on binomial coef cients Lemma 33 Write a akpkak1pk 1 a110a0 and b bjpj bj71p771 b1p 120 where the coe cients are between 0 and p 7 1 so we ve written padz39c expansions of a and 12 en ltzgt2ltgtltgtmltggt Proof We rst need the fact that ltgt2ltgt 11 mod p This is directly from the fact that Iyp E Ipyp mod p7 so I 1 E I 1 The proof of the lemma is by induction The inductive step is to show that if up c 7 a c eavdltpltbpdgtltbgtltd The binomial coef cient on the left is the coef cient of 1517 in I 1 erC I 1apz 1C 3117 1 z 1 With the conditions on c d the only way to get an 1175 out of the left hand is to take 1171 from the rst factor times Id from the second factor Now you assemble the induction using D Now write 7 b0 pb1 pf 1bf71i We assume 1 S 7 lt pf7 so some bi is f nonzero7 and each bi is between 0 and p 7 1 Then 1 gt has a factor lt gt l f modulo p7 and lt gt is 0 So p divides lt p This shows that Op is de ned overZsince 1 EOmodpiflSTSpfili f f On the other hand lt pl gt pf is a unit mod 4 ifq is prime to p7 and lt pail gt is by direct inspection p times a unit mod p So the greatest common divisor of the coef cients of Op is 1 If n is not a prime power7 let pf be the largest power of p dividing n Then by 27 ltsgt2ltnapfgtw mod p7 which is a unit modulo p and all primes that are not factors of no Repeating this argument for each prime dividing n7 we see that the greatest common factor of the coef cients of On is also 1 when n is not a power of pi Lemma 34 Suppose Fzy and Gzy are two formal group laws over a ring R and that FIyy E CIyy mOd 1707K Then Fz7 y Cz y aCnzy mod zy 1 Proof Let lquotzy be the homogeneous degree n part of Fzy 7 Gzyi So FIyy E CIyy N17 9 111001 17 y 1A Since F and G are commutative formal group laws7 we get 1 Way NEE 2 lquotz0 lquot0z 0 and 3 Way N1 yyz Nay Z How The last statement deserves a proof Write Cz y z y 517 FFIyy7 2 E CFIyy72 FFI7y72 111001 17 M 2V E FIyy 2 5FIyy7 2 N1 yyz mod 17 y 2V E CIyy Fzy 2 EGz yz Fz y 2 CGW yM may N1 y 2 m0d17y72 1A By symmetry we also get F17Fy7 2 C179y72 NEW Z N972 Taking the difference of these two equations and the associativity of F G gives us item 3 above We now wish to de ne the group SnR Fz yllquotsatis es l 2 and 3 Note that Cnz y E SnRi 1 and 2 are free For 3 we look at zy z 7y zyz i zy 2 zy2 iz 7yniznv This or this divided by p is the left hand side of 3 for Cnzyi It is clearly symmetric so is equal to the right hand side of 3 Now let A E SnRi Write Azy Eaiziyn ii We have 1 a0 an 0 2 ai an7i and The rst two follow from symmetry and vanishing on axes The third follows by looking at the coef cient of ziyjzn i j on both sides of May M1 yyz May Z My 2 Our goal is to prove that SnR R Cnzyi If R happens to be Q or a n all The Qalgebra we re done since in this case if A E R al v j coef cient al for the polynomial On is either n or np and hence A alnCn or mmTLC in case n is a power 0 p i This also suf ces for the case R torsion free In this case R embeds into R Q and we have R On C SnR C SnR Q R Q CW If the rst inclusion is not equality there must be an element of SnR that is of the form On where m does not divide 7 in Rf But m must divide Tai for each 239 since we have a polynomial over R This contradicts our choice of On since the greatest common divisor of the ails is just 1 Now we wish to consider R Zp Pick Az y Eaiziyn i E SnRi We7ll distinguish between the two cases n not a power of p and n a power of p In the rst case write n spk k 2 0 s gt 13 0 mod p lfi is not a multiple ofpkwriteicpk7j0ltc s0ltjltpki Then ailt sicpkj gt acpkltck The rst binomial coef cient is 1 and the second one is 0 mod p so ai 0 Hi cpk 0 lt c lt s then we assume WLOG that p c since acpk a5cpk and p doesnlt divide both c and s 7 c 3 Upk Oink k apt lt C 7 Dink acpk pk cap So a k is a multiple of apt independent of A k1 017 Now we tackle the case n p we shift the index just to keep our notation like in the previous case If i is not a multiple of pk proceed exactly as in the previous case lfi cpk0 lt c lt p then again as in the previous case we get 3 Upk 0016 k apt lt C 7 Dink acpk pk cap so acpk is a multiple of apt independent of A In either case apt is a unit for On so every A 6 Sn is a multiple of On with the multiple given by apkAapkcn We now need to check R Zpr Suppose by induction we know SnZ pT 1 ZpT 1Cn Let A E SnZpr ThenA aCnpT 1A But since AaCn 6 Sn so is pT lA Therefore A E SnZp so A E an mod p So A apT 1bCn Now let R be an arbitrary commutative ring and let A 6 Sn The coef cients of A lie in a nitely generated subgroup of R isomorpic to Z1 69 B Zk with each Zk isomorphic to Z or Z pr Because the conditions on elements of Sn only use the additive structure of R we see from the fact that the projection of A to Zk is akCn that A 11 akCn D We will do as much as possible of the rest of our analysis by examining the indecomposables of L De nition 35 If R is an augmented halgebra we are thinking of h Z then the indecomposables of R QR is the hmodulet IRIR IR where IR is the augmentation ideal If R is a graded connected halgebra for example L then R is a quotient of SymMQR Welll study QL to recover L Lemma 36 QL is torsion free Proof QL is 0 except in dimensions 72n for n gt 0 Let R Z 69 QignL with the obvious grading and multiplicative structure Let an be the formal group given by the quotient map L A R By Lemma 34 an z y aCnzy Therefore a divides the image of the aijs in Q72nL but the aijs must generate Q72nL So a generates QLQnL hence QLQnL is cyclic Finally anL E Q Q2nL E Q Q so anL must be Z D Theoreom 37 L ZIl12 In QL Q I can be taken to be mi ifi1 is not a power of p pm l a power of p Proof From Lemma 36 Q72nL Z for all n gt 1 So L is a quotient of Zzlzg l 72i From Proposition 31 we see that there cannot be any relations To identify the zils in L Q note that ZmnZ ai iyjn1 Zmn n 9 ln QL Q we get Zmnr y Z aijziyj Zmnwn on Z aijziyj Zmydzn on i I 0 This shows that aij E 7mij1 lt 2 gt mod decomposables So 1k can be h l chosen to hit the greatest common factor of lt gt t1mes mk l S i S 16 mod decomposables D There are some powerful corollaries to this theorem The rst one follows from the fact that the ring representing 97 is a projective object in the category of commutative rings This corollary is actually equivalent to Lazard s theorem7 that is to say one can prove Lazardls theorem from this corollary together with a little easy analysis of the rational case Corollary 38 fa R A S is a surjection and F5 is a formal group law over S then there is a formal group law over R FR so that aiFR F5 In other words7 97 takes surjections to surjections Next we7ll think about nite approximations to formal group laws This is useful7 because most of the interesting formal group laws are in nite power series De nition 39 An n bud is a polynomial of degree n or the residue of a power series modulo terms of degree greater than n FIyyzyzaijziyj 2an such that Fzy Fy7 I and FIFy7 E FFzy7 2 mod zyz 1 Corollary 310 1 Ln Z11Hrn1 together with the nbud Fn that results from thinking of this as a quotient of L is the universal nbud 2 Every nbud is the nbud of a formal group law By the universal n bud7 we mean that if 9Rn is the set of n buds over R there is a natural isomorphism Hom Ln7 7 A Proof 1 as in our construction of L7 it is clear that the ring Zaijli j S N classi es n buds The relation N is just what it has to be it involves exactly the relations of total degree 2 72n 2 from L It follows that Ln L in dimensions 0 through 72n 2 hence QkLn Z when k is even and 0 lt h 2 72n 2 There is a map Ln A L because we can take a formal group law7 forget that it is a formal group law and think about it as an n bud This gives a natural transformation 9 A 9n where 9Rn is the set of n buds over R Since both functors are representable7 there is a corresponding map of representing objects Since the map is an isomorphism on degrees from 0 to 72n 2 it is an isomor phism on indecomposables in that range Hence Ln7 which must be a quotient of Zzl zn1 is the trivial quotient7 because any relations map to 0 under the map Ln A L 2 We give a map of rings L A Ln by sending 11 to 11 if it is available in the range7 and 0 if it isnt This gives a way to make a formal group from an n bu 15 Note that there were many choices we could have made L A Ln In particular the large mils didn t have to go to 0 This corresponds to the fact that there are many in fact many functorial ways to extend n buds to formal group laws 31 Addendum on other types of FGLS Most of this section is taken slightly altered from Example 31 A noncommutative onedimensional formal group law Let R be a ring with an element a such that a is torsion and nilpotent Let m be the minimal positive integer such that am 0 and take I am l Let n be the minimal positive integer such that nb 0 p a prime factor of n d np Take 5 db Thenc7 0c20pc FIyy zyczyp Notice that FFIyy72 FIyy 26FIyy2p zy2czypcry2p zy2czypzzpyzp ICzypzzpFy72 z Fyz czy 2 z Fyz czy z 6W7quot FzFy 2 We used pc 0 to say Cyp 2p Cy 239quot and c2 0 to say 69 2 Cy 2 cyzp 1711 state a theorem below and sketch a bit of its proof that says this is the only sort of situation in which a onedimensional formal group law is noncommutative l don7t know who to credit this theorem too but it is proved in detail in Theoreom 311 If R is a ring with no elements a that are both torsion and nilpotent then all onedimensional formal group laws over R are commutative The most interesting thing in the proof of this theorem is the following lemma which is elementary Lemma 312 Suppose a F A F is a homomorphism of FGLs over a domain R If F is commutative so is F Exercise 31 Prove this lemma It helps to think about the form of the commu tator77 Czy FzFy F71z 71 which must be sent to 0 under 1 Also you have to use that R is a domain Sketch of proof of Theorem 311 There are two substantive steps 1 Show that if R A R Q is an injection that the theorem holds this case corresponds to no torsion 2 Show that if R is an integral domain no nilpotents the theorem holds Once these steps are accomplished we look at the maps R A Rp for all primes p and R A R Q The elements ai39 7 aji are in the kernels of all these maps therefore are torsion and in the intersection of all primes nilpotent 1711 do the second step since it s interesing and lemma Suppose R is an integral domain De ne HW y FIyFy7l1lI Then H0y y and we write Hzy y for rnz E RHIH with rn 0 0 If F is not commutative some rmz is not 0 Let m be minimal so that rmz 0 Case 1 m 1 Hzy E y1 r1z mod 3 HFI717y H17FI7Fyv ElmW H17HI7yA by the associativity of F One extra fact that gets used here which is easy is that ilslFIyr F1lr 7 1 96 O7 ylt1 nltFltzz gtgtgt 2 Hltzcygtlt1 nltzgtgt 2 ylt1 nltzgtgtlt1 w mod y Therefore 1 r1Fzz 1 r1z 1 r1z in other words r1 is a homo morphism from F to m the multiplicative formal group Case 2 m gt 1 Then Hzy E y rmzym mod ym1 Applying 3 we get y TmFI71ym 2 Hear Tmltzgtym E Y rm 1 mod ym1 So rm gives a homomorphism to the additive formal group In either case an application of Lemma 312 tells us that F must have been commutative Exercise 32 Do the rst step of the proof That is show that if R has no torsion then any 1dimensional FGL over R is commutative Finally llll make some remarks about higher dimensions Formal group laws were invented ca 1940 by Buchner to prove Lie s theorem recovering a Lie group from its Lie algebra The formal group law of a Lie group appears to contain more information than the Lie algebra though it can t really since the Lie group is essentially recoverable from its Lie algebra Obviously this is only interesting in dimensions greater than 1 7 contains many results for higher dimensions In particular there are many noncommutative formal group laws in dimensions greater than 1 even over Over Qalgebras every commutative n dimensional formal group law is isomorphic to the additive one and the analogue of Lazardls ring is also a polynomial algebra 4 QUILLEN7S THEOREM Recall from Section 23 that there is a formal group law which welll denote FMU over MUquot SO MUquot induced by the product on CP that classi es the tensor product of line bundles Because of this we get a map L A MUquot classifying FMU The object of this section is to prove the following theorem Theoreom 41 The map L A MUquot classifying FMU is an isomorphism This gives a connection between the geometry of complex cobordism and the algebra of formal group laws Let me point out that independent of Theorem 41 MUquot E does have an in teresting universal property Proposition 42 Given any complex oriented cohomology theory Equot E there is a unique natural transformation of ringvalued cohomology theories 9 MUE A EE such that 9 FMU FE 17 Welll skip the proof of this it is rather easier than Quillenls theorem relying on a series of cohomology calculuations all of which are straightforward In fact this is essentially the de nition of MU 7 that is the proof relies mainly on the fact that MUquot 7 is made from considering the Thom spaces of the universal complex vector bundles It would be nice to be able to prove Theorem 41 by using this proposition or some other universal property of MU that comes from geometryi 13 seems to do this but I don7t completely understand that The line of proof we will proceed on below involves calculating MU as m MU77 which llll de ne precisely below but essentially involves considering the limit of the homotopy of spaces representing MUquot Then we7ll look at the Hurewicz ho momorphism77 from 7r to H another thing to be de ned below and identify that map as isomorphic to the map A M from the proof of Lazardls theoremi Finally we do a bit of geometry to compute the logarithm of the image of FMU in 4i Generalized cohomology theories again We begin with some conven tionsi We7ll generally assume our spaces have distinguished base points which we won t often call attention to We remind the reader that EXXgtltIzogtltIUXgtlt0UXgtlt1 and that 9X mapSlX where map means continuous topological maps preserving base point and is topol ogized by the compactopen topologyi We de ne by XYXgtltYXgtlty0UzogtltY and we use XY to abbreviate the set of homotopy classes of continuous maps from X to Y We have an important adjointness relationship that follows from the usual mapX X Y Z w mapXmapY Z namely mapX Y Z w map X mapY Since EX 5391 X it follows that lt4 mum Y map on my Finally we assume our cohomology theories are reduced which means Equot 0 and Equot 50 is isomorphic to the corresponding unreduced theory evaluated on a point We can recover the unreduced theory from the reduced theory by using Equot X as our functor where X is X plus a disjoint basepointi We can reduce a theory by taking the cokernel of the map induced by X A i Theoreom 43 Brown IfEquot X is a generalized cohomology theory then Ek X X for some space Ek unique up to homotopy type We won7t prove this theorem but refer the interested reader to The proof is not actually terribly hard One does the more or less obvious thing in the more or less obvious way Anyhow it is a nice paper and I recommend it By Yoneda s lemma it is equivalent to consider either cohomology theories or their representing sequences of spaces There is an important property of the sequence of spaces Ek associated with a cohomology theory Corollary 44 There is a homotopy equivalence Ek A QEIHJ Proof We look at the co ber sequence X A CX A CXX 2 EX Here CX is the cone on X hence is contractible If we apply our cohomology theory to this co ber sequence our long exact sequence becomes an isomorphism Ek X A Ek1EX This is a natural isomorphism lX7Ekl A lEX7Ek1l or equivalently using our adjointness 4 le kl A lX79Ek1lA By Yonedals lemma this gives a map Ek A QEIHI which must give a homotopy equivalence Corollary 45 f is a natural transformation of cohomology theories ox EX 4 FM X commuting with all structure then 9 is induced by maps 9 z E n A Enk such that 99n1 2 97 De nition 46 A spectrum is a sequence of spaces with designated homotopy equivalences a z E a as which will usually be omitted from the notation If Equot is a cohomology theory then the sequence of spaces as described above will be called the spectrum of E and be denoted by We will also use the adjoints of on which give us maps 2E7 A En1 We ll probably abuse notation and call these maps on also if we bother to name them Remark 41 Each spectrum corresponds to a cohomology theory and vice versa There are several ways to de ne spectra and the category in which they live but I won7t attempt to do this seriously here when many better quali ed authors have done such a good job For an informal approach one always thinks of part III of There is also 2 12 10 and a number of more recent constructions My limited account here is only meant to provide the framework for understanding the proof of Quillen7s theorem Example 42 1 Equot 7 H7A for some abelian group A Then the representing spaces which one might want to write as n of Mn are the Eilenberg Mac Lane spaces KAn These are characterized by their homotopy groups which are worm 6 Note that it is possible to construct these spaces with one7s bare hands by building a space beginning with some n cells to generate A and adding n1 hn else 19 cells for relations to kill extra homotopy in degree n Then one has to add higher cells to kill all homotopy in degrees above n Note also that since nkX SEX KMQX nk1X This is enough to prove that QKA n 1 2 KA E7 K 7 K27 Z X EU and K2n1 U Here U is the in nite unitary group th limit of the Un and EU is its classifying space the limit of the BUn See below for a little more information about BUn 3 Equot 7 MU Recall that if C is a topological group there is a classifying space BC together with a principal C bundle EC over BC such that A to A X BC lt gt equivalence classes of principal C bundles over X Here the corresondence sends a map f to the bundle f EC EC is char acterized by being a contractible free C space and then BC is the orbit space of EC under C These spaces were constructed by Milnor In the case C Un we have our principle bundle EUn A BUn Un acts on C of course and we can make the corresponding vector bundle gtltUn C BUn Welll abbreviate this bundle by 7 This bundle gives us the equivalence X BC lt gt equivalence classes of complex n plane bundles over X by using maps X A BC to pull back 7n We de ne a space MUn ThomWn D7nS7n Here the Thom space of a vector bundle 5 is the quotient space of the disk bundle of E D by the sphere bundle of 5 one wants to choose some kind of metric to make these de nitions but this can be avoided if it makes you unhappy Anyhow after these preliminaries I need to admit I won7t describe Mn but the MUn7s are good approximations to MU2n so welll use them instead After some more de nitions we7ll revisit these constructions so I can describe the structure maps and the multiplicative structure that comes with MU 7 De nition 47 Given a spectrum E de ne the corresponding generalized homol ogy theory Ek lim7rkn X n The limit is taken by using the adjoints 0f the structure maps of the spectrum E a 2E7 A En 1 There are a number of functors normally de ned on spaces that we wish to extend to spectra Primarily we7ll be interested in homotopy and homology but also in generalized homology and cohomology theories De nition 48 Let Equot 7 F7 be cohomology theories De ne 7MB Wk7lEn 20 Here n must be chosen so hn 2 0 but k may be negative The structure condition on the spaces En guarantees that this group is independent of n once it is de ned De ne WE g Ewan n The limit is taken over 9 that is EnvEnkl A QEmQEan E F nk71l n47 De ne FME liankEn where the limit is under the adjoints of the structure maps ofE using the suspension isomorphims for generalized homology theories There are various symmetries and isomorphism hidden in these de nitions ln particular7 F E W E E and m EASO E Sol One other thing that should be clear from these de nitions is that the Hurewicz homomorphism still exists h17rEgt 42 The homology of MU So our plan now is to proceed as I rst outlinedl We need to understan hzmMUHHiMU which will require understanding the domain7 range and map The easiest of those three is HiMU7 which we will actually use to compute TriMUl In the process of that computation we will gather the facts we need to calculate the effect of h We begin by asserting that Hi MU7 which by de nition is a limit of Hi MUn can actually be calculated as a limit of the homology of the Thom spaces To calculate HX MUn7 we need to know the Thom isomorphism theoreml Theoreom 49 fE A B is an oriented vector bundle of dimension n there is a natural among bundle maps isomorphism Hi B A HinThomEl between unreduced homology of B and reduced homology of ThomE Proofs of this theorem can be found in many standard algebraic topology refer ences l7 and 8 are two references7 p258 works for both At any rate7 to compute HiMU limn Hi2nMUn we rst need HiBUn and then use the Thom theoreml Welll think of HiBUn as sitting inside HiBU limn HiBUn There are maps Un gtlt Um A Un m by putting an n X n matrix in the upper right and an m X m matrix in the lower left These maps induce maps BUn gtlt BUm A BUn m7 hence a sort of external product on the HiBUn and an internal one on HiBUl We begin by noting that BU1 CP i This can be seen several ways Sm limn S2n 1 is a free contractible 5391 Ul space so the ber bundle EU1 A BU1 can be taken to be S A SmSl CP i Welll write HXCPoo Zlt o li i where the notation stands for the free Zmodule on the Bis and 2239 We get maps in homology from BUG A BUn HBU1 HBU1 HBU1n a HBUni The result is that HBUn is a free Z module with basis monomials of length S n in the ils where 60 is interpreted as 1 This should be thought of as a subgroup of HBU Z817827837 l For a proof we refer the reader to Milnor and Stasheff but 1711 outline one argument erei The calculation is usually done in cohomology and then the results we7ve stated about homology can be deduced from the universal coef cient theoremi One begins with the splitting principle Exercise 41 Show that ifE is a complex vector bundle over X then there is a space and map f P A X such that H is an injection and f splits as a sum of complex line bundlesi To prove this you may nd it helpful to think about the map A X where is the projective bundle of 5 replace each ber with its projective space It also might be nice to know that the cells of BUn are all in even dimensionsi With the splitting principle one is able to show that the map BUG A BUn induces an injection on cohomology for all coef cients Since this map is induced by U1 A Un by including diagonal matrices one sees that the map on co homology lands in H BU1 Squot the part of the cohomology xed by the symmetric groupi Since HBU1 2m n the part xed under the symmetric group is just the symmetric polynomials which gives H BU1 Squot Zali i i an where a is the ith elementary symmetric polynomial in the zjlsi Now a dimension counting argument says that the map H BUOL A Zaluian must be an isomorphismi Now we move on to MUi First we note that MU1 2 CP i To see this note that MU1 Thom71 Den571 Now the disk bundle is homotopy equivalent to the base space and the sphere bundle in this case is also 71 gtltU1 Ul EU1 which is contractiblei So MU1 2 BU1i 22 Exercise 42 Suppose we have two bundles E A B and E A BC Then if we make a bundle E X E A B X B which has dimension equal to dimE dimE ThomE X E ThomE ThomEl Note that Thom spaces come with distinguished base points since they are quotient spacesl So the makes sense This fact gives us a bunch of interesting maps between the MUn7sl Let 77 X Ym Ynm BUn gtlt BUm gt BUnm be the bundle diagram where the bottom horizontal map is the map classi ng the nm bundle 7 X yml By the exercise this induces MUnMUm A MUnml Note that the bottom map can also be described as the map induced by the inclusion Un gtlt Um A Un m discussed above We also get an interesting map from the diagram 81 X M gt 71 X M gt 7M1 9 gtlt BUn gt BU1 gtlt BUn gt BUn 1 Here 51 is the trivial complex line bundle This gives 5 22MUTL SQAMUO L HMU1AMUn HMUn1 where the map S2 A MU1 is the inclusion of the bottom cell The Thom isomorphism gives us an isomorphism H 013 HBU1 A F2MU1 FWQCPKquot Write HMU1 Zltb0b1ulgt so gt gt bi under the Thom isomorphism of course bi is secretly 5H1 and annoyingly 2239 2 but this second problem will get xed in the limit when we get to HMUl Then because of the naturality of the Thom isomorphism together with exer cise 42 we get a commutative diagram in which the vertical maps are isomorphisms HBU1 HBUn F MU1 gt This shows that FMUTL is a free abelian group with basis monomials in the bi of length exactly n The lowest class in dimension 2n is 123 and the map from HBUn sends a monomial in the Bis to the same monomial in the his except that if it is of length less than n the remainder is lled by bolsl Also by the naturality of the Thom theorem the diagram HBUn HBUm gt HBUn m l FMUn FMUm gt FMUnm 23 ensures that the monomials in the his multiply together properly Now remember that H MU limn H2nMUn where the limit is taken under the map 5 7H which is multiplication by 120 on homology 0 becomes 1 in the direct limit and the dimension of bi becomes 2239 So welve just prove Proposition 410 H MU Zb1b2bgi 43 The Adams Spectral sequence We now know H MU but we also need to know 7rMU and what the Hurewicz map does The best device for getting us from homology to homotopy in this case is the Adams spectral sequence Fix a prime number p For the duration of section 43 H7 will mean Hquot 7 F17 and H 7 will mean H 7 F17 Welll let stand for the representing spectrum so K F n i We normally think of H 7 as a functor from Spaces to Gers or to graded rings but we want to notice that it naturally lands in MOdA the category of left Amodules where A is the graded ring of natural transformations of the cohomology theory This is a tautology of course and tells us nothing but by noting that natural transformations ofcohomology are representable and then studying the representing spaces we can compute A This was rst done by Serre Essentially Ak HW lim HnkKFp n Notice that A is a ring since you can add natural transformations between graded abelian groups and you can compose two natural transformations There is some extra structure on A however Because H 7 lands in graded rings we have a natural transformation H X HmX A HnmXi If we take X KFpn gtlt KFpm and take the element 7r1 projection to the rst factor in H X 7r in HmX and take the product of 7r1 and Tr in H KFpn gtlt KFpm we get a map KFpn gtlt KFpm A KFpn The collection of these maps as n and m vary induce maps in cohomology 1r1lKan7 LmH HKFp7n HKFp7m These induce a coproduct that together with the ring structure on A makes A a Hopfalgebra By the universal coefficient theorem A H is the dual Hopf algebra HomGrVSAFp and by the same duality H 7 naturally takes values in Corned the category of left Acomodules It turns out to be slightly more convenient to work with H 7 and comodules so that is what welll do even though modules are psychologically easier 1711 describe A which is equivalent to but slightly easier than describing A HomGrVS A Fp Note that most of the objects welll deal with are nite dimen sional in each degree so duality poses no problem as long as we are working over a eld 14 Fpl517527537ml E739077391773927m 24 as an algebra where 217i 7 2 and 217i 7 1 The coproduct is given by 1673 k 1673 k A i 2516 516 and An ESLk 7 71 1 160 160 50 is to be interpreted as 1 whenever it appears This formula is correct as it stands for all odd primesi For even primes there are no 77s and the dimension of is 2Z 7 1 instead The coproduct formula for the 57s stays the same Theoreom 411 There is a spectral sequence with E31 Eng HTE F17 Emil FINHsE gt 7TL75E Z17 where dr Ejgt 7 E5Ttr7139 There are three confusing things in this statement The spectral sequence is graded in a nonstandard way If 1 had said E25quot Ext2 tFpHE note sign change in s it would be a left halfplane homology spectral sequencer Also there is a lie embedded in the statement It converges to some associated graded to a decreasing ltration on mEi In good cases the completion with respect to this ltration is completion with respect to p but that is not automatic 1n better cases yet like when the homology of E is nitely generated in each degree that completion is the same as RE Zpi Finally I ve written the Egterm two ways The second way is the one we7ll use but it is a bit confusing There Ext means extensions of comodules which is something a bit unfamiliari Extensions of comodules are however equivalent to extensions as modules of the dual modules over the dual ring The idea of the construction is to resolve E by copies of but we won t construct this here See 14 for a simple outline of the construction 44 Hi MU as an AiComodule We continue to have a xed prime p and unless otherwise speci ed homology and cohomology groups will have coef cients in F171 We7ll use the Adams spectral sequence to compute mMU and to compute the Hurewicz homomorphism mMU 7gt HiMUi To compute TRMU with the Adams spectral sequence we need to be able to compute Extif Fp HiMU For this we need some understanding of the structure of HiMU as an Aicomodulei We begin with a device for keeping track of products Proposition 412 Cartan Under the isomorphism HX Y HX E H Y an element 9 E A acts by 6z E y 271l9 ll1l9I 8 My where A9 29 9 Proof Let I E HnX y E Hin Then I is a map I z X 7gt KFpn and yzY 7KFpmi So z y is amap X Y A KFpn KFpmM KFpn m Now let 1 E H KFp n be the identity map from the Eilenberg Mac Lane space KFp n to itself This element can also be described as the dual of the the image of the generator of 7TnKFp n under the Hurewicz mapi 25 By naturality 9z y is Mnn6 lmni But by the de nition of A 91n gtlt 1m 26 1 X 71l9Hl 6 1 So 6z y Zenoz 9 D Proposition 413 Under the isomorphism Hi X Y Hi X E HAY the diagonal of an element I lt8 y is given by A1 8 y Z 1lw7llxll6ij Ii yj where AI 29139 I and Ay 2ng yj Proof Dualize Proposition 412 D Recall H CP00 FIJI and HCP Fplt60761762gt We want to apply this proposition to determine the diagonal on HiMU Fpb1b2 from A Hi CPm 8 A where E HgiCPmi Recall that b 6111 and the product in HiMU is given by HMUn HMUm H MUn MUm 4 HMUn m which takes the tensor product of a monomial of length n and one of length m to one of length n in Therefore A i together with Proposition 413 suf ce to determind the coaction on HiMUi We write Sqi1 ir E A for the dual element in the monomial basis of 14 1T r mew5 Exercise 43 Show Sq0iu0lz 1171 where the l is in the ith place and z E HQCPmi You can assume sqlz 11 We actually know the action and coaction on CP quite precisely but all well really need is the exercise above llll cite some results thoughi These follow from the unstable axiom of the action of the Steenrod algebra and can probably be found in many places 18 is one source but anyplace where the Steenrod algebra is developed will probably also do Lemma 414 Sqr1i rkz 0 except as in the exercise Corollary 415 Sqr1i rkzm 0 unless in r1 Hm in which case the result is IPIIT1quot39Z Proof The lemma above plus Proposition 412 D Corollary 416 771 A pz 5i 615 1 p5 1 pu The result we need which follows from the exercise is Corollary 417 514851 is a term in A517 so Ei l is a term in Abp1 in HiMU We de ne a ring N Fpzili pr 7 l with 2i There is an obvious map 7r HiMU 7gt N that sends bi to 11 or 0 depending on i Recall there is an extension of rings actually of Hopf algebras Fp7gtP 7A7gtE 7F where Pi Fp 1 2 and Ei E7 07 71 Using our formula for Abi and Proposition 413 we see that the map A HiMU 7Ai HiMU factors through Pi HiMU We de ne a map HiMULPi HiMU Jz N Lemma 418 1 EN 0 A is an isomomhism of Ai comodules where the coaction on P E N is given by the diagonal on P A Pquot 7gt P P 7gt A Pi Proof Proposition 413 insures that A is a map of rings 1 7r is obviously a map of rings The composite is an isomorphism on indecomposables7 since bi gt gt 11 if i pr 7 1 mod decomposables and 121751 gt gt 3 mod decomposables We need only check that the map is a map of coalgebras We have the following diagram HiMU A Pi HiMU amp Pi N Al A 1l lml Ai HiMU E Ai Pi HiMU W Ai Pi N 7r The rst square commutes by the associativity ofthe coaction on HiMU7 and the second square obviously commutes So 1 7r 0 A is a map of comodules Next we need a change of rings theorem for extensions A more general version of this elementary theorem can be found in 5 V124 for example Proposition 419 If h7A7A7An7h is an extension of augmented halgebras then ExtAA 8 M7 k ExtAM k Dually7 Proposition 420 If h7C7gtC7gtC7gth is an extension of coaugmented hcoalgebras then Exthc C 8 M ExtckM We apply this with C A7 C P7 C Ei and M N to conclude Ext FpHMU ExtEFpFp 8 N Fph0h1h2 8 N We can immediately draw some conclusions about the Adams spectral sequence converging to TRMU Zp Our polynomial generators are the hi which are in bidegrees 12101 7 l and the zj with the given restrictions on j which are in bidegrees 07 207 7 2 So the Egterm of the Adams spectral sequence is all in even total degree7 and hence there are no differentials 27 Some remarks on mm are in order before we identify it precisely There are two important points rst by the Hurewicz theorem WkM 0 if k is negative and 7T0MU Z Next by a theorem originally due to Serre 15 since HMU is nitely generated in each degree so is 7mm Of course See7s theorem is about spaces not spectra but the extension is routine We would like to identify 7rMU from the E00 term of the Adams spectral se quencesi We begin by observing that the associated graded to Wow is Fph0 It follows that p is represented by ho We map from Zzlzgi to 7rMU by sending z to a lift of z from the Ecoterm of the Adams spectral sequence if it exists in other words when i l is not a power of p and 11171 gets sent to a lift of hi for i gt 0 Then if we lter Zzl 12 i i by the ideal p 1171 11271 i i i we get an isomor phism on associated gradeds and hence an isomorphism between the completions Since this argument was independent of the prime chosen we see that mm Zzl 12 i H i We actually want slightly more information than this Not only do we need to know what 7rMU is but weld also like to understand something about the Hurewicz homomorphism We begin with a fundamental observation about the Hurewicz homomorphism and the Adams spectral sequence An element 7 E Eg t of the Adams spectral sequence for X is also an element of HEX to make it to the Egterm of the Adams spectral sequence 7 must be primitive with respect to the coaction of the dual of the Steenrod algebra Simultaneously if 7 is a permanent cycle it represents in the Ecoterm of the Adams spectral sequence an element of HEX up to ltration The observation is that if g is any lift of 7 to HEX then the image ofg under the Hurewicz map is just 7 E HEX recall this means coef cients in Fp in this section so I really mean the reduction of the Hurewicz map mod p It follows when i1 is not a prime power that the Hurewicz image of z is 12 mod p and decomposables The decomposables come in because the element of homology representing 11 is not exactly 12 but the element l z in P N which is modulo decomposables the image of bi under the isomorphism from HMU to P Ni Since this is true for all p the Hurewicz image of z is bi mod decomposables The situation is a bit more complicated when i l In this case the Hurewicz homomorphism takes I to a unit multiple of bi in H MJ Q where q f p mod decomposables so I can be chosen so that the Hurewicz map takes I to pkbi for some k gt 0 We want to show that in fact k l 1711 sketch a proof here but it requires some familiarity with the Adams spectral sequence Recall that the Egterm of the Adams spectral sequence for mm is given by ExtEFpFp N Fph0h1h2m N where 1771 hj and is represented in the cobar resolution for ExtE F17 F17 by le E Ext1 2p7 1 If we try to lift Tj to a cocycle in the cobar complex for ExtAltFpHMUgt we get le 7 121470 mod 515253PH Now cochains in the cobar complex in homological degree one correspond to elements of H MU E where g f1berS0 A HFP These elements map to the integral homology of MU by lifting to integral homology and then applying the map MUAFHMUAS MU 28 which takes bls to 127s7 and elements of Hi g to 0 except ho gt gt p So the homotopy class that is given by the lift of le from ExtE FpFp E N maps under the Hurewicz map to pbpy at least mod decomposables and higher ltration in the Adams spectral sequence Welve just proven the following proposition Proposition 421 mMU Zzlzg where 2i and Il czm be chosen so that its image under the Hurewicz map mod decomposables is bi 1 is not a power of p and pbi 1 pk Exercise 44 Prove that bi E HiMU is a primitive modulo decomposables directly from our description of HiMU as an Aicomodule when i is not a power 0 p Exercise 45 Fill in the many details I ve left out to show that the Hurewicz image of 11171 is 7101217171 mod decomposables 45 Proof of Quillen s theorem Proposition 421 should look familiar from the statement of Theorem 37 In fact7 the universal property of Lazard s ring gives us most of a diagram L gt mMU l l M H M U Here the horizontal map is induced by classifying the formal group law over TQMU The vertical maps are the factorization L A M Zm1 m2 A Q L and the Hurewicz map respectively Weld like to ll in this square To do this note that M is the smallest extension of L such that the universal formal group law has a logarithm This follows from the formula for the logarithm Emizi1 If we look back at the de nition of the homology groups of a cohomology theory7 there is a visible symmetry H M U M Ui E where I write E for the spectrum representing ordinary integral cohomology theory There are two evident ring maps ending up in this object One is the Hurewicz map mMU A HiMU The other is the MUHurewicz map77 m A MUi To construct this map7 an element of WkX is a map from Sk A X 5k is an almost complex manifold7 and a homotopy between two maps g Sk A X gives a cobordism So this map can be thought of as an element of MUM This gives a natural transformation mX A MUiX Alternatively and less geometrically MUn is 2n 7 1 connected and has 7r2nMUn Z7 so assembling the generators of these homotopy groups together with the structure maps of the spectrum M gives the map SO A MU Welve just nished analyzing the Hurewicz map The MU Hurewicz map is much simpler7 it is just the obvious ring homomorphism Z A Zzlzg rom all this information we get two formal group laws over HiM U the pushforward of the MU formal group law7 FMU7 and the pushforward of the Hformal group law which is the additive one7 Fa Furthermore7 both of these formal group laws are induced by complex orientations on the spectrum EA MU that come from the natural transformations MUquot 7 A H MU 7 and Hquot 7 A H MU7 29 respectively By Proposition 23 the two formal group laws are isomorphic In other words FMU has a logarithm over HMUi This allows us to complete the commutative square 45 above with a map gal L mMU M L H M i We next need to calculate expMUi Denote the complex orientation on E M coming from the natural transformation MUquot 7 A MU7 by EMU and the orientation from H7 by zHi Write EMU gzH in H MUCP Zb1b2i Then since my is the orientation associated to the additive formal group law and XMU is the orientation associated to the formal group law induced from mm 9 is the exponential series for FMU Lemma 422 mg 2 12131 b0 1 Proof We7ll prove this by induction proving the nite statement n71 EMU Z bizgl i0 after restricting to CP K We need to remind ourselves what 1H isi IE1 CP A E2i2H is a generator of H2i2CP i The dual generator of H2i2CP is a map 5H1 S2i1 A CP A i Finally 12 is the map SQiAYQCPm MU g Assume for n 7 1 Then we look at the effect of the map n71 my 7 2mg CP a EQMU Ag i0 Here welll abuse notation because we7ll denote the map CP A a EQMU A A LYMU Ag by the same symbolsi Here M is multiplication in Our inductive hypothesis tells us this is 0 when restricted to Cpn li On homo topy that is as a map from HCP A HMU the EMU takes n to bn1 by def inition of bn1 as does the 2201 bizgl since the Kronecker pairing Now the map OF A E2MU A E is null when restricted to Cpn l A by hypothesis so the map factors through as 3Pn a CP AELSW A a EQMU Ag 01 CP LHSQ LHSWAEHZFMUA i 30 We7ve just checked that the map CP A A EQMU A is 0 on homotopy from which it follows that the composite 52 A EQMU A is null Therefore the map from CP itself is null D This identi es the b s in HMU as the coef cients of the exponential series It is easy to check that if we de ne mi to be the zi1 coefficient of the inverse logarithm series then mi 712 mod decomposablesi It follows that go is an isomorphim mod decomposables since it takes m E M to 7b in HMU mod decomposables and hence go is an isomorphismi Finally by comparing the map L A M with the Hurewicz map mMU A HMU we see that the map L A mMU is also an isomorphism modulo decomposablesi Hence since mMU is a polynomial ring the map is an isomorphismi Exercise 46 CP is a complex manifold so it represents an element of MUquot mMUi Show that under the Hurewicz homomorphism mMU A HMU CP goes to n lmni This is proved in l but you ll nd his argument rather incomplete from our perspective 4 6 Digression on strict isomorphisms of formal group laws and the LandweberNovikov theorem De ne a functor 513 F17f7F2lf 0 17fF1Iyy F2fzfy Here f is supposed to be a power series in one variable over R and the Fi are supposed to be formal group laws over Ri The condition on f 0 is supposed to guarantee that z cizHli Such a triple is completely determined by F1 and f so 517 is represented by LI Lb1 b2 i where L is the Lazard ringi Given a map 9 LI A R F1 is given by restricting 9 to L and z 9bizi1i Now 517 has some extra structure besides being a set It is a groupoz39d since we can compose some triples over R F17f7F2 0 F2797F3 F179 0 fyFs We think of this as a category with objects formal group laws F and morphisms triples like F1 f F2 but it has more structure than an ordinary category because every morphism is invertible F17 f7F2 0 F27f717F1 F1717F17 the identity morphismi Since L and LI represent the objects and morphism of a groupoid valued functor on rings we get structure on L and LI from Yonedals lemma 0 We have 77L L A LI corresponding to domain from morphisms to objects The L in 77L stands for Left not Lazard and 77L is the obvious map L A Lb1b2i o 773 L A LI corresponding to range classi es fF1f 1zf 1y where 77L classi es F1 over LI and z bizi1i o c LI A LI conjugation corresponds to the natural transformation from 517 to itself that takes F1 f F2 to F2f 1F1i Clearly CnLa nRa and 02a a The series 20 Cbi1i1 is inverse to 20 bizi1 so we get recursive formulas for Cbj from Z bjzj1i1 z i0 j0 This also leads to information about 773 77R Q I le17m2pl A le17m2llbiyb27ml can be calculated though 77 is very complicated in terms of the mils gen erating L logF1 z EEO mizi1 and f 1 is an isomorphism from F2 to F1 So logF2 20 mif 1zi1i On the other hand 773mi is the coef cient of 1H1 in the log series for F2 So we get 27712071095 Zmdf W y Z Cbj j 0 There is a way to get a triple from a formal group law F gt gt FzF which gives an augmentation e LI A L that takes all the b s to 0 SOUL EOUR L 0 Finally there is a diagonal induced by composition If we take 91 LI A R and 92 LI A R such that 91 o 773 92 o 77L so 91 and 92 give composable triples then 91 and 92 correpond to a map LI L LI where L acts on the left hand LI by 773 and the right hand LI by UL Composing the triple give a map AzLIHLI LLI The pair LLI with these structure maps is called a Hopf algebroid because it represents a groupoidvalued functor recall a Hopf algebra represents a group valued functor What does this all have to do with MU Recall that 7rMU L Just as in ordinary cohomology Brown representability implies that the ring of stable natural transformations of the functor MUquot 7 is given by MUquot This object has a product given by composition of maps and a coproduct given by the map MUn MUm A MUn m inducing MUMUnm A MUMUnAMUm g MU MUn MUMUMUmi Dualizing which take more justi cation than 1711 give here as does the above paragraph we nd MU MU has a product and coproduct and MU 7 lands in MU MU comodules In fact MU MUMU turns out to be a Hopf algebroid and the beautiful and useful in homotopy theory anyhow extension of Quillen7s theorem due to Landweber and Novikov is that LLI g MUMUMU as Hopf algebroids Welll omit the proof lt isn7t very hard in terms of what we7ve already done and amounts to noticing that MU MU7 has two different complex orientations and expressing one in terms of the other 32 4 A simple minded example of a Hopf algebroid Something similar to the LandweberNovikov theorem above happens with ordinary cohomology with coefficients in Fp but it is much simpler therel Let Fp with the additive formal group law play the role of L and work over the category of Fpalgebrasl Then HomFpR is not very interesting It contains one element which induces the additive formal group law over Rf SHE on the other hand will be triples FafFa where Fa is the additive formal group and z E bizi1 is an automorphism of Far I ll leave the Fa out of the notation for obvious reasons Since we7re in characteristic p we know exactly what the possibilities are for this was an exercise z icizpll i1 So 517 is represented by P Fpcl52l Now in this example things are simple enough so we can write explicit formulas for all the structure mapsl Since the domain and target formal group laws are the same 77L 773 and is just the obvious map Fp A Fpcl02l l Similarly the augmentation which corresponds to Fa gt gt FazFa is also the obvious map taking all c to 0 e diagonal is more interesting It is induced by composition of power series Represent P P by Fpcl 02 i i 51 52 i Then the diagonal takes ck to the 117k coefficient of the power series 20 ciE0 cgzpjy li This coefficient is just the sum 220 54027le Our formula for the diagonal is then k ck gt gt Zci i This should look familiar as it is the twist of the diagonal on the polynomial subal gebra of the dual to the Steenrod algebra or just the dual of the Steenrod algebra when p 2 The formula for the conjugation CCi is as before we have Zcltcigtltzc gt x which gives recursive formulas for Cci If we take our generators of P to be the elements CCi instead of Ci then we get an isomorphism of Hopf algebras with P the polynomial part of the dual of the Steenrod algebra by mapping to Cci In other words this untwists the diagonal above making the formula for the diagonal in terms of the CCi the same as that for the in P 5 p TYPICAL FORMAL GROUP LAWS AND CARTIERls THEOREM Welve done everything well do in terms of understanding global classi cation The rest of the time we will concentrate on understanding formal group laws over various Zpalgebrasl So henceforth an arithmetic prime p will be xed and well work over Zpalgebras instead of arbitrary commutative ringsi 33 De nition 51 A FGL F over a Zpalgebra R is p typical if whenever q is prime to p where Q is a primitive qth root of unity This sum looks like it takes place over rather than just RHIll but since F is commutative each coe cient is symmetric in the roots of unity so the power series is really in RHI Exercise 51 Show that it is equivalent to require that q 11 0 i1F vanishes whenever q is a prime different from p Note the following if R is torsion free7 write logFz z Ezlmiz l over R Q Then 1 logFZ ciz Z ijxj1ltij1 ijzj1 Z ltij1 i1F j i j l lf 4 does not divide j 1 then WD 0 lf 4 does divide j 17 Ei ij1 211 4 So q 00 10gFZ Ti 247711117111 i1F j1 Exercise 52 Use the above to verify that if R is torsion free7 F is p typical if and only if logFz z 2f mpfilzpfl Theoreom 52 Cartier Over a Zpalgebra any formal group law F is canon ically isomorphic to a ptypical one F This isomorphism is the identity F happens to be already ptypical Proof We will write FL for the universal formal group law over the Lazard ring7 and construct an isomorphism from f FL A F1 to a p typical FGL over L Zpl Recall rst that logFLz z EzlmizHl over L E Q Qm1m2u Welll de ne a power series f so that Fp will be fFf 1z7 f ly and logFP IogFltr1ltzgtgt z We remind the reader that we de ned for n an integer at the end of section 11 Suppose n is not divisible by p Then E nz modulo I27 so is an invertible power series over a Z algebra Rl We de ne to be the inverse power series to Hence if F is a FGL over a Z1algebra7 r is de ned for all r E Zpl This actually gives an embedding of Z07 into EndF7 at least when A R is an embedding7 and often even when it isnt Zltpgt Note that logr rlogzl So 11 logo 7F MM 4 logo 7 2771171ij 2mm 7 qu m Hi 1 i j 34 This suggests that an appropriate series for f 1z should be something like I F 2 MIKE Ur q pF i1F though that series itself obviously doesnlt work since its logarithm has for example the term imkz whenever k 1 is a product of exactly two primesi To produce a good formula for f 1z we need the Mobius function n 7 71V if n is the product of 7 distinct primes M 7 0 if n is divisible by a perfect square and take M1 1 Exercise 53 Check that if n is not a power of p An Eq n gng l 71 Now welll de ne q 10711 Z 4ll1 ll Z lt11 421 F i1F Observer that this series is de ned over L Zp since the only denominators we need are prime to p and we have symmetry in the qth roots of unity for each 97 Next we calculuate oo 10gFf 1I 10gFI Z 4 ijqelrjq 1Mgt1 j1 By the exercise7 this is m r logFz 7 Z mk1zk z Empfilzp i kgpf f1 So we de ne the formal group law Fp over L Zp by Fpz7 y fFf 1z7 f lyi logFP logFf 17 which we7ve just computedi Since L Z077 we see that Fp is p typicali To nish the proof of the theorem7 consider the functor on Zpalgebras R gt gt formal group laws over Ri This functor is given by HomLR HomL ZpR since we are only looking at R in Z07 algebrasi Given a map 9 L E Zp A R we naturally get a p typical formal group law over R7 9in together with an isomorphism between 9FL and 9 F17 from 9 f Note that q 9f711 Z 4ll1 ll Z lt11 4 9FL 1111 so if 9 FL is p typical7 9 f 1z If It follows that if we start with a formal group law that is already p typical7 our canonical isomorphism with a p typical formal group law will be the identity D 35 Its interesting to pursue some of the implications of this theorem First note that Fp the p typical formal group law over L Zp is a formal group law7 therefore is classi ed by some map e L A L Z07 llll abuse notation and refer to the induced map from L Z07 as e also Notice that we7ve already calculated the effect of e on L Q since eX logFL logFP f I 7 mi 239 p 71 em 0 ia pfil It follows that e is idempotent7 so we will refer to the image of e from L Z07 as eL abusively again Proposition 53 f9 L A R classi es a ptypz39cal formal group law C over R then 9 factors through e Proof Note as in the proof above that 9 f 1z I so z This shows that 9FL L Fp 9eFL Since G is classi ed by a unique map from L7 we learn that 9 6 o e D Corollary 54 The ring eL Q L Z07 classi es ptypz39cal formal group laws over Zpalgebras Finally we7d like to identify eL a bit more explicitly Note that eL is a summand of L Zp since e is an idempotent So QeL is a summand of QL Z007 and in particular7 QeL Q is a summand of QL From our calculation of the effect of e on L Q7 we know QeL Q is Q in degrees equal to 2pf 7 2 and 0 in other degrees It follows since and hence QL Zp have no torsion7 that QeL is Z07 in degrees equal to 2pf 7 2 and 0 in other degrees Since eL is a subring of L Z07 and there are no relations in that ring7 we get eL Zpezp1ezp21 l and the map eL A eL Q takes Ipfil to Wpfil mod decomposables Henceforth we will denote eL by V There is a piece of topology that goes with this If we consider the cohomology theory X gt gt Zp this cohomology theory has a complex orientation giVing a formal group law Fp that is isomorphic to FMU and is p typical we use Quillenls theorem here If is the isomorphism7 then is a complex orientation over MU Z07 corresponding to the formal group law Fp It follows from Proposition 42 that there is a natural transformation of cohomology theories 0 e MU7 A MU7 Z07 Since the target is a Z1algebra7 this extends to a natural transformation of co homology theories e MU7 Z07 A MU7 Z07 where e induces the map e L E Zp A L Z07 This natural transformation is given by tensoring with e L E Zp A L Z07 and since this is an idempotent7 36 we can show that the image is a cohomology theory It is called BP named after its original discoverers7 E Brown and F P Peterson see 4 and BP Finally we7d like to explore the functor we7ll call pSIR RHF7fyG where F and G are p typical formal group laws and f is a strict isomorphims of p typical formal group laws As in the case where F and C were ordinary FGLs7 this functor is really determined by F and f or f and C so we begin by asking what we can say about Lemma 55 Suppose F is ptypical and f F A G is a strict isomorphism of FGLs G is ptypical i f 1z EZOFtizpl Here to l but the other ti are arbitrary Proof It is no loss of generality to write f 1z EigtOFCiIi Here cl and Cg are just the coef cients of z and 12 respectively in f 1z and Cj is the coef cient of zj in f 1z minus the coef cient of zj in Now since Cz y fFf7117 16719 4 q 16 or Z f 1ltizZ go39sz i1g i1F i39j F But since F is p typical7 ELlFQiCjI 0 for xed j7 so we get 4 oo 16 or Z meww i1G i1F Now this gives G p typical iff EFqFcqlzqi 0 for all 4 prime to p7 which is true iff cl 0 whenever i is not a power of p D Corollary 56 The functor 10517 is represented by Vt1t2t37 Proof Given a map 9 Vt A R we get F from precomposing 9 with the obvious map V A Vt7 and f is determined by f 1z EF9tizpl our convention is to 1 Similarly7 given F7 f7 G F deterines a map from V7 f 1 tells us where to send the tis provided F7 G are p typical D Like the pair LLI7 V7 Vt comes with structure maps 77L the obvious map V A Vt7 773 the map that classi es the p typical FGL over Vt given by fFf 1zf 1y where F is the group law given by UL and f is the series de ned by f 1z sztZzpl7 a conjugation 5 corresponding to FfG goes to G7 f 1F and a diagonal map col 1 quot to A T of llUJJJUJJJULlei m Also like for the pair LLI there are formulas for these maps in terms of the tis and the generators of V7 but they are more complicated because the tils are only the formal coef cients of f 1 rather than the actual coef cients of There is an easy corollary of the LandweberNovikov theorem that applies here LLI BPX7 BEEP For the next section we7ll want a certain choice of polynomial generators of V De ne M to be the image of mp1 in V Q A0 l Araki de nes generators vi inductively by lt6 pi ZAivi Caution there is a similar formula due to Hazewinkel that also de nes a set of generators for V those are also called oils but are slightly different Proposition 57 The vi as de ned above are polynomial generators for V Proof We sum 6 over all n E pAkzpk Z Aivglzpl k0 ij This gives 00 1710ng 210ng vjzpl j0 Applying eXpFP7 we get p Z vjzpl 120 Since the p series of Fp is de ned over V7 and not just V Q7 this shows the vjs are de ned over V To see they generated V mod decomposables7 note that on E pAn 7 ppnAn mod decomposables by 6 FINITE HEIGHT FORMAL GROUP LAWS IN CHARACTERISTIC p Let F be a FGL over a eld 16 characteristic p Then since k is a Z Zalgebra7 F is strictly isomorphic to a p typical formal group law7 F F is given by 9in where 9 z A Is So Mpz minim 2 MM F It follows that plpz and hence has leading term uzph for some unit u E k unless p 0 De nition 61 We de ne the height of a formal group law F over a eld of characteristic p to be h the leading term of is a unit times mph If 0 we say the height is Note that iff F A F is an isomorphism of formal group laws7 then from which it follows that height is an isomorphism invariant Example 61 If F is the additive formal group law7 p pm So over a eld of characteristic p7 the p series is 07 and the height is 00 If Fz7 y zyzy the multiplicative formal group law so that 11 1y l Fzy then 1 I l E l 1p modulo p So over a eld of characteristic p7 p 1p and the height is 1 Consider the example Fum of section 1 We proved that this formal group law has height 00 in characteristic 2 and height 1 at odd characteristics De nition 62 Let R be an Fpalgebra De ne a map 9 V A R by 9vi 0 except 9vn 1 Let Fn be the corresponding formal group law 38 Theoreom 63 Lazard fF is a formal group law of height it over a separably closed eld K F is isomorphic to Fn Proof VVLOG7 F is p typicali Let 9 V A K classify Fl 9vi 0 for i lt n7 9vn 0 We want an iso f Fn A Fl f 1z will then equal 2 tizpl Pk where we abuse notation by writing ti for ti where 9 Vt A K is the map classifying the triple F77 f7 Fi Note that vi 0 except vn l7 nRvZ 9vii In order to proceed we need a formula for 773 Lemma 64 In V Z vbtgbzpb Z tbnRvcpbzpbni beon 1on Proof Recall Fp is the universal p typical FGL over Vi Write F3 for the p typical FGL over Vt induced by 773 V A Vti Note that logFP EAlzpl and logFRI EURO DIPI Write f 1z EFPtizpz so that f is a strict isomorphism Fp A F Then IOgFRI longf71I Zlogp pl ZZWM gt271 mean ZAztili Therefore From 6 we recall pAn Z M11571 0937 Applying 773 we get n n i 7 7 7 E PAitiii E E A i vnei i0 i0 j0 Now we apply 6 on the left to get n n i I 7 1 2mm Z Ajvijtii i0 i0 j0 Re indexing7 we get a ab a Z Aavg t1 Z Aatg nRvcabi abcn abcn Summing over it and adding powers of z to produce a homogeneous power series of degree 2 ab4 Z Aavbt bpazpabn Z Aalttmltvcgtpbgtfzp abc aJnc Now we apply eprP 7 log1 7 to get Z my 2 WWW 17 F 17 F D Now to nish the proof of Proposition 63 we apply 9 to the formula from Lemma Gilli We write ti for 92 This gives 2 tifzpw Z tbt9vcpbzpbai 0 FW IncZ nFW Examining the term of lowest degree we get tgn t0t9vni The equation tgnil 9vn is a purely separable equation no factor of tga 7 6 so we can solve it over Ki Suppose we have found to i i tj1 as elements of Ki Then if we examine the n7 117 term of the equation above we get tquot a tjt9vnpn a which is also purely separable again no factor of ti 7 6 so we can solve for tj E Ki B Our next goal is to understand something about EndF where F is height n over a characteristic p e Exercise 61 Show that if F is p typical over a Zpalgebra and p is odd then 1l1 IA Proposition 65 Let F be a formal group low E EndF 1 E is a ring 2 E is a domain if R is 3 E is a Zpalgebra ifR is a eld of characteristic p torsion free if the height is nite and an Fpvector space if the height is in nite Exercise 62 Prove the rst two facts For the rst one if f E E E RHIH and the multiplication is composition of power series the addition is formal addi tion Proof For the last property of E note that we7ve already shown that E is a Z07 algebra if R is We have the nseries 1 1 1 1 1 1 F z n times and if n is a unit in R this series begins nz hence is invertible by associativity and F n m So the map r gt gt r gives a ring homomorphism Z07 A El To extend to Zp note that p E azph mod zph1 if F has height h 0 if mil height is in nite Therefore prz 10 E aplf ljhilzpfh modulo zp urli So if 71177127 7137 is a p adically convergent sequence of integers then lnll17n2l17lnsl1 is a convergent sequence of power series So there is a power series corresponding to the limit of the nil 40 Lemma 66 Let F be ptypz39cal ff 6 EndF then EZOFalzpl Proof The argument is a simple adaptation of the proof of Lemma 55 Suppose WLOG EDIFCZzli Then 4 q q 00 0fZ 4 Z fltizZ Z cjcw i1F i1F i1Fj1F As in the proof of Lemma 55 since F is assumed to be p typical when 4 does not divide j Eg1FCjltijIj 0 When 4 does divide j we get Eg1FCjltijIj q 011 This gives since q is invertible So Cj 0 unless j is not divisible by any prime except p D Finally weld like to specialize to Fh as we can determine EndFh preciselyi Of course if F is some other isomorphic formal group law when K is separably closed this is any other height h formal group law then we will also know EndFi Let Fh is as de ned above Then we have p zphi This follows because Fh is de ned using a map 9 V A K that takes all ms to 0 except Uh to l and the p series for Fp is EZOF vizpli First note that the power series 11 is in EndFhi This is because Fh is de ned over Fp so FhIvyp FMIpvypl Let be any endomorphism of Fhi We observe that the p series is central in fact the n series is central for any n in Zpi To see this take n an ordinary integeri Then fFhFhi iFhzzziHz FhFhi iFhfz i i Now write EZOFhaZzpzi We have fl0l1 Z aiIPITh i0Fh On the other hand fawn pfz Z Z i0Fh i0Fh So 2 ainITh Z agthITh i0Fh i0Fh and hence ai aft This tells us that ai E thi So the power series that actually occur in EndFh are of a very special form and EndFh over th is the same as EndFh over any larger eld Welll assume then that K 2 thi 41 We next look at an intermediate formal group law between F17 and Fhi De ne the graded ring Mnh Fplvhl Vp7v17 i 7vh717vh17vh27 The map 9 V A K factors through the quotient V A Denote the quotient map by 1 is a graded map7 so dinQ y zyE 11213141 y is a homogeneous power series of degree 27 where z and y have degree 2 and Uh has degree 2 E 2 i Therefore Pk I y is a homogeneous polynomial of degree Mink E l 1 in z and y So if a E th amiss my ax ay Z viaWMHIPm y 11 ay a Zv P z y adiFALy In other words7 the power series am is an endomorphism of din and hence of Fhi e now have a precise description of the noncommutative ring EndFhi EndFh aizpl i0Fh ai E th Note that if K doesn7t contain all of th we have the same description with the restriction ai E th Ki Although this description is explicit7 it hides the structure of the ring Proposition 67 EquotdFhP th lt5gt5hA Here p is the ideal generated by the element of the 7ing p which is just the pseries and S is a noncommutative indeterminate that commutes by the rule Sa apS for a E thi Proof Note rst that EZhFhaZzpz E p if the ai E thi This is just because 2 WWW WM ihFh ihFh ihFh So in EndFhp7 00 hil Z aizpl E Z aizph i0Fh i0Fh To get a map thlt5gtSh A EndFhP we send 5 to Iquot We send the element a E th to the series azi We ve already ver i ed that this gives an endomorphismi The map from th is clearly multiplicative To see it is additive7 note that Fhazbz E az bx mod zphi So in EndFnp7 az Fn bz a 101 Now we extend to thltSgtSh multiplicatively7 so ST gt gt 11725 gt gt mph E 0 This is clearly a vector space isomorphism if it is well de ned7 we only need to check that the relation Sa apS is respected But Sa gt gt a1 apzp 42 which is just the image of apSi D Before giving our description of EndFn we need a little bit of algebra De nition 68 We will denote the ring ZpQ where Q is aprimitive phil root of unity by prhi This ring is called the Witt vectors of the eld th It is a special case of a general construction WR the Witt vectors of a commutative ring R but we won t deal with the general construction The ring prh should be thought of as the ring of integers of the unrami ed extension of Zp corresponding to thFpi It is a complete local ring with residue eld thi It is a Zpalgebra of rank h as a free Zpmodulei The generator of the Galois group GalthFp is z gt gt Iquot This lifts to an automorphism a of prh which generates GalWFPhZpi By analogy with Zp it is possible to choose a multiplicative map X X th a wFP taking the roots of zph l 7 1 to the roots of the same equationi Then we can write any element w E prh as 00 w E wipl i0 17 where each wi satis es wi 7 win So each wi is either 0 or in the image of thi Then we get 00 0w Za o pli i0 We de ne a ring Eh by Eh WFPh lt5gt5h 107510 0w5A Note that Eh is a noncommutative prh algebra of rank h hence of rank h2 as a free Zpmodulei It is also the maximal order in the division algebra En Q which has center Q17 and Hasse invariant lhi Theoreom 69 En EndFn Proof We construct a map from Eni As before 5 gt gt Iquot r E Zp goes to This gives immediately that the relation Sh p is respected by the map we are constructingi Next ifQ is a ph 71 root of unity in prh send Q to the series Qz here I abuse notation by thinking of Q as an element of th Notice that then Qz is a ph 7 1 root of unity in End 7 i As before we observe that Sa aaSi Finally to see our map is an isomorphism we note that it is surjective since it is surjective mod p and it is iso since both domain and range have the same dimension as free Zpmodules and the map is surjectivei D 7i LUBIN TATE THEORY OF STAR ISOMORPHISMS l7m mostly not going to include proofs in this chapter since I can7t improve much on 11 lld like to discuss the point of that paper though We now understand formal group laws over a eld of characteristic p rather well If the eld is big enough they are characterized by their height and for a particular formal group law of each height we know the endomorphism ring and hence the automorphism ringi lf we7d like to understand the action of the group of invertible power series 5K over a eld K on formal group laws over K FCLK we can separate the space of FGLs into components by height FCLMK and then in the simplest case 5K acts transitively on FCLMK with isotropy group isomorphic to Weld like to extend our understanding to the simplest characteristic 0 case De nition 71 If D is a complete Noetherian local ring with maximal ideal m and Dm 16 let 39i39 be some FGL over 16 F is a lift of ifF is a FGL over D such that mF 39i39 If f F A G is an isomorphism between two lifts of Q f is a isomorphism if z ie f lifts the identity Now let R be some ring with maximal ideal I and residue eld RI h 39i39 a formal group law of height h We paraphrase the results of 11 for D in the category of complete Noetherian local Ralgebras with maximal ideals m containing DIi If one is concerned with a particular complete Noetherian local ring D it makes sense to take R D and it makes sense to think of R Zp or prhi I paraphrase the results from 11 belowi o If two lifts are isomorphic the isomorphism is unique The set of isomorphism classes of lifts of 39i39 is represented by ring maps from Rt1 i i i th71ll so is given naturally in complete Noetherian local R algebras by D gt gt mh li There is a natural choice of representative for each isomorphism class of lifts of Q If E thHX then f gives a wellde ned correspondence between isomorphism classes of lifts of 39i39 and isomorphism classes of lifts of the target off f filzf 1yi So if we wish to study say the action of X on FGLD we can rst divide FGLD into components FGLhD corresponding to the height of the reduction to the residue eldi Then we study 1 The free action of kerDz X A 2 The action of End on the functor D gt gt mh l or equivalently on the representing object Rt1i i i th1 here End acts by R algebra maps 3 The free action of gtltEnd 39 on FGLh In terms of topology we are interested in the cohomology of these group ac tions Because the rst and the last are essentially free actions we really care about the middle actioni One of the fundamental problems in stable homotopy is understanding HEndFhWFPht1Hi th1 where the c in the subscript stands for continuous iiei Galois cohomologyi Now some comments on the details of 11 We can prove Proposition 1 1 a little more easily Since we7ve studied the structure of the Lazard ring we can see that if 44 we map L A Rt1i i i th71H by sending 11 to ti ifi lt h zj to 0 ifj lt ph and not a power of p and the other IT to lifts of the image of the IT in the map classifying 39i39 then we get a lquot as desired Moving on to Proposition 24 there are two points to that proposition The more transparent one is that if two FGLs are isomorphic over a satisfying the hypothesis then there is a unique isomorphismi The other point is that one can understand the obstructions as living in some kind of cohomology groupi Proposition 26 is used in proving 31 which is the main theorem that identi es isomorphism classes of lifts and canonical representatives of isomorphism classesi inally from the homotopy theoretic viewpoint I wish to emphasize 34 which gives an important application to homotopy theory The point here is that if R is a complete Noetherian local ring with a formal group law 39i39 over RI then the functor D gt gt HomRialg Rt1 i i i th1 D is naturally equivalent to D gt gt isomorphism classes of lifts of if Aut 39 acts on this second functor lift E Aut 39 to some E RHIHX and then given a isomorphism class take a representative formal group law F over Di That goes to the class of fFf 1z f lyi So Aut 39 acts on the isomorphism classes of lifts Since that functor is represented by Ptch i i i th71ll Aut acts on Ptch i i i th1 by ring automorphismsi 71 A hint about the relation of LubinTate to homotopy theory The connection to homotopy theory is approximately as follows There is a cohomology theory Eh with E WFPh ul i i i uh1uu 1 the ui are degree 0 and the u is degree 72 This is a complex oriented theoryi If you pretend u 1 then the orientation gives a LubinTate formal group law lquotz y for the formal group law Fh over thi This tells us that E AutFn acts on the coefficients to straighten out the business of u see In fact it turns out to act on the cohomology theory Ehi Finally it turns out there is sort of an Eh based Adams spectral sequence with HgtAutFnEX gt 7rLKhXi Because ofvarious nice properties known about AutFn the E2 term of the spectral sequence is relatively tractable compared for example to the usual Adams spectral sequencei Furthermore the spectral sequence frequently collapsesi The previous paragraph contains one lie and several completely unde ned termsi When X is not a nite complex something related to but possibly different re places E LKhX is Bous eld localization of X with respect to the homology theory Kh the hth Morava K theoryi77 You should think of mLKOL X as an approximation to mX when X is a nite complex whose accuracy increases as h increases The group Aut Fh is called the hth Morava stabilizer group and is written Sh sometimes The subscript c on the indicates Our group Sh is a subgroup of the topological group of all power series over thi So it has a topology E also has a topology since it is a power series ring and Sh u 1 45 acts continuously on Continuous cohornology means look only at continuous cochainsi See 16 for a real de nition and discussion REFERENCES 1 J F Adams Stable Homotopy and Generalised Homology University of Chicago Press Chicago 1974 M 2 Boardman Stable homotopy theory Johns Hopkins University 3 E H Brown Cohomology theories Annals of Mathematics 754677484 1962 4 E H Brown and F P Peterson A spectrum whose Zp cohomology is the algebra of reduced pith powers Topology 51497154 1966 5 E Cartan and S Eilenberg Homological Algebra Princeton University Press 1956 6 E Devinatz and M Hopkins The action of the Morava stabilizer groups on the LubineTate moduli space of lifts American Journal of Mathematics 1995 7 M Hazewinkel Formal Groups and Applications Academic Press New York 1978 8 D Husemoller Fibre Bundles SpringeriVerlag 1994 9 M Laz mmutatiye Formal Groups SpringeriVerlag 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