aa r X i v : . [ m a t h . A T ] J un THE REAL JOHNSON-WILSON COHOMOLOGY OF CP ∞ VITALY LORMANA
BSTRACT . We completely compute the Real Johnson-Wilson cohomology of CP ∞ . Applying techniquesfrom equivariant stable homotopy theory to the Bockstein spectral sequence, we produce permanent cyclesand solve extension problems to give an explicit description of the ring ER ( n ) ∗ ( CP ∞ ) .
1. I
NTRODUCTION
The complex cobordism spectrum,
M U , carries an action of C , the group of order two,coming from complex conjugation and may be constructed as a genuine (indexed on acomplete universe) C -equivariant spectrum, MR [Lan68, Fuj76, AM78]. At the prime 2,the Johnson-Wilson spectrum, E ( n ) , is an M U -algebra with coefficients E ( n ) ∗ = Z (2) [ v , . . . , v n − , v ± n ] where v k is in cohomological degree − k − . E ( n ) may also be constructed as a genuine C -equivariant spectrum, Real Johnson-Wilson Theory, ER ( n ) [HK01]. On the category of C -equivariant spaces, ER ( n ) naturally gives rise to a multiplicative cohomology theoryvalued in commutative MR -algebras [KLW16b]. Its underlying nonequivariant spectrumis E ( n ) . Let ER ( n ) denote the fixed point spectrum ER ( n ) C .Fixed point spectra associated to MR have proved to be powerful tools for algebraic topol-ogists, for example in Hill, Hopkins, and Ravenel’s resolution of the Kervaire invariantone problem [HHR09] (in all dimensions besides 126). At heights n = 1 and , the ER ( n ) are familiar cohomology theories. ER (1) is KR (2) , Atiyah’s Real K-theory with underly-ing nonequivariant spectrum KU (2) , and ER (1) is KO (2) , real K-theory. After a suitablecompletion, the spectrum ER (2) is (additively) equivalent to the spectrum T M F (3) oftopological modular forms with level structure [MR09, HM15], and so the results of thispaper apply there. At present, this identification provides the most powerful approach tocomputing T M F (3) -cohomology (for example, see [LO16]).Just as there is a fibration Σ KO η / / KO / / KU , η ∈ KO − there is for each n a fibration Σ λ ( n ) ER ( n ) x ( n ) / / ER ( n ) / / E ( n ) Date : September 13, 2018. here x ( n ) ∈ ER ( n ) − λ ( n ) is (2 n +1 − -nilpotent and λ ( n ) = 2(2 n − − n +1 − n +2 +1 .This was constructed in [KW07] and leads to a Bockstein spectral sequence of the form E i,j = E ( n ) iλ ( n )+ j − i ( X ) ⇒ ER ( n ) j − i ( X ) . Our main result is a computation of ER ( n ) ∗ ( CP ∞ ) for all n . ER ( n ) is not a complexoriented theory, and so the computation is nontrivial. The Atiyah-Hirzebruch spectralsequence is unwieldy, so we use the above Bockstein spectral sequence instead. Eventhis has nontrivial higher differentials, but what makes ER ( n ) ∗ ( CP ∞ ) computable is thefact that, after some rearranging, the only interesting differential is d and the higherdifferentials all play out in the coefficients. We develop methods of producing permanentcycles and obtain control over the extension problems in the Bockstein spectral sequenceto produce a complete description of the ring ER ( n ) ∗ ( CP ∞ ) .In the case of ER (1) = KO (2) , this problem has a long history. KO ∗ ( CP ∞ ) was first com-puted in degree zero by Sanderson [San64], then in all degrees with ring structure on KO even ( CP ∞ ) by Fujii [Fuj67]. Yamaguchi [Yam07] gave the first complete descriptionof KO ∗ ( CP ∞ ) as a ring. All three computations use the Atiyah-Hirzebruch spectral se-quence. Bruner and Greenlees [BG10] computed the connective real K -theory of CP ∞ using the Bockstein spectral sequence. Our result gives the ER ( n ) -cohomology of CP ∞ for all n in the same level of detail as Yamaguchi’s description and reproduces n = 1 as anearly degenerate case (see Remarks 1.4 and 10.9). The answer for n > is richer in thesense that it yields a great deal more 2-torsion.This paper forms part of a program to compute the ER ( n ) -cohomology of basic spaces.The spaces whose ER ( n ) -cohomology is known at present may be divided into two fami-lies. Building on the computations in [KW08a, KW08b, KW14], the results of [KLW16a]identify a class of spaces whose ER ( n ) -cohomology is directly computed from E ( n ) -cohomology (which is known) by base change. These include BO ( q ) for q ≤ ∞ , theconnective covers BSO , BSpin and
BString (the last for n ≤ only), and half of allEilenberg MacLane spaces—those of the form K ( Z , k + 1) and K ( Z / q , k ) . The space ofinterest in this paper, CP ∞ = K ( Z , , does not belong to this family and is the first spaceof ‘exotic type’ whose ER ( n ) -cohomology is nevertheless computable. Our computationshere open the door to the results of [KLW16c], which deal with the ER (2) -cohomology of B Z / q and truncated complex projective spaces CP k . The computation of ER ( n ) ∗ ( CP ∞ ) also points the way toward the ER ( n ) -cohomology of the spaces Q ji =1 CP ∞ , BU ( q ) andits connective covers, and the other half of the Eilenberg MacLane spaces.One motivation for developing ER ( n ) (and ER (2) in particular) as a computable theoryis the applications to proving nonimmersions of projective spaces. Kitchloo and Wilson[KW08a, KW08b] and Banerjee [Ban13] established new nonimmersion results for realprojective spaces by constructing obstructions in ER (2) -cohomology. These obstructionsare given by powers of a generating class surviving beyond the skeletal truncation of theprojective space—they are 2-torsion and undetectable by any complex oriented theory.The computations in [KLW16c] reveal the same sorts of extra powers of a generating class(the class b p described in the next paragraph) in the ER (2) -cohomology of CP k , and one oal of this computational program is to attack the nonimmersion problem and relatedquestions for complex projective spaces.Before stating the main result, we fix some notation. For any z ∈ E ( n ) k ( CP ∞ ) , let b z de-note zv k (2 n − n . In particular, we have b v i := v i v − (2 i − n − n and, letting u ∈ E ( n ) ( CP ∞ ) denote the complex orientation, b u := v n − n u . Let c denote the involution on E ( n ) ∗ ( CP ∞ ) coming from the action of C on ER ( n ) . We show that c ( b u ) = b u ∗ is given by the powerseries v n − n [ − F ( u ) , where F is the formal group law over E ( n ) ∗ with [2] F ( u ) = v u + F v u + F · · · + F v n u n . Under the identification CP ∞ = BSO (2) , the product b u b u ∗ ∈ E ( n ) ∗ ( CP ∞ ) is v n − n times the first Pontryagin class. In Proposition 5.1, we describe a class in ER ( n ) ∗ ( CP ∞ ) which lifts b u b u ∗ ∈ E ( n ) ∗ ( CP ∞ ) . We denote the lift by b p . In E ( n ) ∗ ( CP ∞ ) ,we also have the sum b u + b u ∗ . We show that this lifts to ER ( n ) ∗ ( CP ∞ ) as a power series in b p , denoted ξ ( b p ) . We have the main theorem of this paper. Theorem 1.1.
There is a short exact sequence of modules over ER ( n ) ∗ −→ im( N res ∗ ) −→ ER ( n ) ∗ ( CP ∞ ) −→ ER ( n ) ∗ [[ b p ]]( ξ ( b p )) −→ where im( N res ∗ ) is the image of the restricted norm Z (2) [ b v , . . . , b v n − , v ± n ][[ b u b u ∗ ]] { b u, v n b u } ⊂ E ( n ) ∗ ( CP ∞ ) N ∗ / / ER ( n ) ∗ ( CP ∞ ) such that for all z , N ∗ ( z ) maps to z + c ( z ) under the map ER ( n ) ∗ ( CP ∞ ) −→ E ( n ) ∗ ( CP ∞ ) . Remark 1.2.
Although the middle and right terms of the short exact sequence of Theo-rem 1.1 are rings, the ER ( n ) ∗ -module im( N res ∗ ) is not an ideal of ER ( n ) ∗ ( CP ∞ ) and theright hand map is not a ring homomorphism. However, we give a complete answer for ER ( n ) ∗ ( CP ∞ ) as an algebra in terms of generators and relations in Theorem 10.8. A sim-pler answer than either Theorem 1.1 or 10.8 is given by restricting to degrees multiples of n +2 (note that ER ( n ) is n +2 (2 n − -periodic), though this portion of it contains none ofthe 2-torsion: ER ( n ) n +2 ∗ ( CP ∞ ) = ER ( n ) n +2 ∗ ( pt )[[ b p ]] Remark 1.3.
Note that neither the complex orientation u nor b u lift to ER ( n ) ∗ ( CP ∞ ) , but b u b u ∗ does. The characteristic class ξ ( b p ) by which we quotient in the right hand term aboveis not zero in ER ( n ) ∗ ( CP ∞ ) but is in the image of the norm, ξ ( b p ) = N ∗ ( b u ) . It has geometricsignificance as we discuss further in Remark 10.2. Remark 1.4.
In Theorem 1.1 for n = 1 , it turns out that ξ ( b p ) = − b p , and so the right handterm reduces to the coefficients ER (1) ∗ . This is not true for n > (see Remark 10.9).In sections 2 and 3 we review the Bockstein spectral sequence and describe the computa-tion for the coefficients. In sections 4 and 5 we begin the computation for CP ∞ , identifyingthe key permanent cycle, b u b u ∗ , and giving a convenient reformulation of E ∗ , ∗ . In section 6we compute E ∗ , ∗ . From there, sections 7-9 break up the Bockstein spectral sequence intoa short exact sequence of spectral sequences and show that the remainder of the compu-tation happens in the coefficients via a Landweber flatness argument. In section 10, weprove Theorem 1.1, describe the multiplicative structure, and state the most explicit form f the answer as an algebra over ER ( n ) ∗ . Finally, section 11 describes the very clean formof the answer that occurs after a certain completion. The appendix at the end containssome key equivariant lemmas necessary for our computations.This paper forms part of the author’s thesis [Lor16]. Acknowledgements:
This work would not be possible without the patience, support, andenthusiasm of my advisor Nitu Kitchloo. I am also deeply grateful to Steve Wilson for hisinterest in this project and insightful comments. Finally, I am grateful for the commentsand suggestions of an anonymous referee.2. ER ( n ) AND THE B OCKSTEIN SPECTRAL SEQUENCE
We begin by reviewing some facts about ER ( n ) we need to set up our computational ma-chinery. Let α denote the sign representation of C . By a genuine C -equivariant spectrum E , we mean a collection of spaces E V ranging over finite-dimensional C -representations V = s + tα together with a transitive system of based C -equivariant homeomorphisms E V −→ Ω W − V E W , for V ⊆ W. Such spectra represent bigraded cohomology theories E ⋆ ( − ) given by E s + tα ( − ) = [ − , Σ s + tα E ] C . The C -action means there is an involution on E . Letting ι ∗ E denote the underlyingnonequivariant spectrum, there is an induced involution on the (nonequivariant) coho-mology groups ( ι ∗ E ) ∗ ( − ) . This is the same c described in the introduction.The ER ( n ) are genuine C -equivariant spectra. We draw attention to two classes in thecoefficients, ER ( n ) ⋆ ( pt ) = ER ( n ) ⋆ . As shown in [KW07], there is an invertible class y ( n ) ∈ ER ( n ) − λ ( n ) − α . It restricts to the (nonequivariant) class v n − n ∈ E ( n ) − λ ( n ) − . Additionally,there is a class x ( n ) ∈ ER ( n ) − λ ( n ) = ER ( n ) − λ ( n ) with x ( n ) n +1 − = 0 . Henceforth, we dropthe ‘ n ’ and simply write x, y, and λ .In [KW07], Kitchloo and Wilson construct the fibration Σ λ ER ( n ) x −−−→ ER ( n ) −−−→ E ( n ) . Applying [ X, − ] yields an exact couple ER ( n ) ∗ ( X ) x / / ER ( n ) ∗ ( X ) w w ♦♦♦♦♦♦♦♦♦♦♦ E ( n ) ∗ ( X ) g g ❖❖❖❖❖❖❖❖❖❖❖ and produces the Bockstein spectral sequence (BSS). Remark 2.1.
Depending on whether one truncates the multiplication-by- x tower, there aretwo spectral sequences that can arise from the above. One converges to ER ( n ) ∗ ( X ) (as in[KW14]), the other to (as in [KW08a], [KW08b], and [KLW16c]). In the latter case, one ust go back to reconstruct the answer from the differentials. Both have their advantagesand ultimately contain equivalent information, but it is the truncated BSS converging to ER ( n ) ∗ ( X ) that we use in this paper.The BSS has the following properties. Theorem 2.2. [KW14](i) There is a first and fourth quadrant spectral sequence of ER ( n ) ∗ -modules, E i,jr ⇒ ER ( n ) j − i ( X ) .The differential d r has bidegree ( r, r + 1) for r ≥ .(ii) The E -term is given by E i,j = E ( n ) iλ + j − i ( X ) with d ( z ) = v − n n (1 − c )( z ) where c ( v i ) = − v i . The differential d r increases cohomological degree by rλ between theappropriate subquotients of E ( n ) ∗ ( X ) .(iii) E n +1 ( X ) = E ∞ ( X ) , which is described as follows. Filter M = ER ( n ) ∗ ( X ) by M r = x r M so that M = M ⊃ M ⊃ M ⊃ · · · ⊃ M n +1 − = { } . Then E r, ∗∞ ( X ) is canonically isomorphic to M r /M r +1 .(iv) d r ( ab ) = d r ( a ) b + c ( a ) d r ( b ) . In particular, if c ( z ) = z ∈ E r ( X ) then d r ( z ) = 0 , r > . Remark 2.3.
As in [KW14], we note that when X is a space, there is a canonical classin E , − λ that corresponds to ∈ E ( n ) λ +1 − λ − ( X ) = E ( n ) ( X ) and is a permanent cyclerepresenting x ∈ ER ( n ) − λ . We abuse notation and give its representative in E , − λ thename x as well. Note that though x is a permanent cycle, x n +1 − does not survive thespectral sequence and is equal to zero in ER ( n ) ∗ ( X ) . We may rewrite the E -page toindex the vertical lines by powers of x : E ∗ , ∗ = E , ∗ [ x ] = E ( n ) ∗ ( X )[ x ] d ( z ) = v − n n (1 − c )( z ) x, v n ∈ E , − n )1 Remark 2.4.
To make things even more confusing, the representative of x = x ( n ) in E , − λ was previously called y in [KW14] and is not the same as y ( n ) ∈ ER ( n ) − λ ( n ) − α as describedabove. Since our x ∈ E , − λ represents x = x ( n ) ∈ ER ( n ) − λ , we choose the lesser of twoevils and henceforth use our notation instead.3. T HE SPECTRAL SEQUENCE FOR X = pt When X = pt , we have E ∗ , ∗ = E ( n ) ∗ = Z (2) [ v , . . . , v n − , v ± n ][ x ] , | v k | = − k − . None of the generators v k are permanent cycles as c ( v k ) = − v k . However, there is a trickwe can do to replace v k for k < n by permanent cycles b v k . As in [HK01], each class v k ∈ M U − k − has an equivariant lift in MR − (2 k − α ) . For ≤ k ≤ n , the MR -algebra tructure on ER ( n ) produces these classes in ER ( n ) ⋆ . We may use y ∈ ER ( n ) − α − λ to shiftthe “diagonal” v k classes to integer grading. For ≤ k < n , let b v k ∈ ER ( n ) (2 k − λ − = ER ( n ) (2 k − λ − denote v k y − (2 k − . Then by construction, this class restricts to v k v − (2 k − n − n ∈ E ( n ) (2 k − λ − and represents a permanent cycle in E , (2 k − λ − .We have now shifted all of the differentials onto powers of v n . Let R n = Z (2) [ b v , . . . , b v n − ] , I j = (2 , b v , . . . , b v j − ) , and I = (0) . The Bockstein spectral sequence computing ER ( n ) ∗ goes as follows. Theorem 3.1. [KW14] In the spectral sequence E r ( pt ) ⇒ ER ( n ) ∗ ,(i) E ∗ , ∗ ∼ = Z (2) [ b v , b v , . . . , b v n − , v ± n ][ x ] That is, E m, ∗ = Z (2) [ b v , . . . , b v n − , v ± n ] on x m . (ii) The only non-zero differentials are generated by d k +1 − ( v − k n ) = b v k v − n + k n x k +1 − for ≤ k ≤ n. (iii) E ∗ , ∗ k = E ∗ , ∗ k +1 = · · · = E ∗ , ∗ k +1 − , for ≤ k ≤ n , and E ∗ , ∗ n +1 = E ∗ , ∗∞ .(iv) For ≤ j < k ≤ n + 1 , E m, ∗ k = R n [ v ± k n ] /I j M j
Note that v n does not survive the spectral sequence, but v n +1 n does. Since itis invertible, this makes ER ( n ) periodic with period | v − n +1 n | = 2 n +2 (2 n − .4. T HE SPECTRAL SEQUENCE FOR X = CP ∞ The BSS for CP ∞ starts with E ∗ , ∗ = E ( n ) ∗ ( CP ∞ )[ x ] = E ( n ) ∗ [[ u ]][ x ] . Again, we hat off v i , ≤ i < n so that E ( n ) ∗ = Z (2) [ b v , . . . , b v n − , v ± n ] . As described in theintroduction, in general, for a class z ∈ E ( n ) j ( X ) , we set b z := v j (2 n − n z ∈ E ( n ) j (1 − λ ) ( X ) . However, note that for arbitrary z , b z need not be a permanent cycle. In fact, b u = v n − n u ∈ E ( n ) − λ ( CP ∞ ) is not. In any case, we replace u by b u as the power series generator of E ( n ) ∗ ( CP ∞ ) , which is valid since v n is a unit. We may similarly hat off the coefficients ofthe formal group law so that b F ( b u , b u ) is a homogenous expression of degree − λ andsatisfies v n − n F ( u , u ) = b F ( b u , b u ) . e now have E ∗ , ∗ = Z (2) [ b v , . . . , b v n − , v ± n ][[ b u ]][ x ] with the following bidegrees: | b v k | = (cid:18) , − λ | v k | (cid:19) = (0 , ( λ − k − , | v n | = (0 , − n − | b u | = (0 , − λ ) , | x | = (1 , − λ ) To compute d , by Theorem 2.2(ii), we need the action of c on E ∗ , ∗ . The classes b v k , ≤ k < n , as well as x are permanent cycles and in particular have trivial c -action. We have c ( v n ) = − v n . It remains to identify c ( b u ) . Lemma 4.1. c ( b u ) = [ − b F ( b u ) Proof.
We view u as an equivariant map u : CP ∞ −→ Σ α E ( n ) . The diagram CP ∞ inv (cid:15) (cid:15) u / / S α ∧ E ( n ) ( − ∧ c (cid:15) (cid:15) CP ∞ u / / S α ∧ E ( n ) commutes, where inv denotes the involution on CP ∞ classifying the conjugate line bundlewith inv ∗ ( u ) = [ − F ( u ) . The above diagram shows that c ( u ) = − [ − F ( u ) . Then on b u = v n − n u , we have c ( b u ) = c ( v n − n u ) = − v n − n c ( u ) = v n − n [ − F ( u ) = [ − b F ( b u ) . (cid:3) From now on, we let b u ∗ denote c ( b u ) = [ − b F ( b u ) . For future reference, it will be helpful tohave some terms of this power series, so we pause to derive some formulas. Lemma 4.2.
We have the following congruences in E ( n ) ∗ ( CP ∞ ) : b u ∗ ≡ − b u mod ( b u ) b u ∗ ≡ b u + b v k b u k mod ( b v , . . . , b v k − , b u k +1 ) for < k < n Proof.
Both follow from the formula for the -series [2] b F ( b u ) = n X i =0 b F b v i b u i and the equation b u ∗ + b F [2] b F ( b u ) = b u. (cid:3) . A TOPOLOGICAL BASIS FOR E ∗ , ∗ The next step in computing d is finding a convenient topological basis for E ∗ , ∗ . To thatend, we identify a large collection of permanent cycles in our spectral sequence. Note that E ∗ , ∗ is a power series ring over E ( n ) ∗ [ x ] , and throughout, by a basis for E ∗ , ∗ we mean atopological basis. Proposition 5.1. b u b u ∗ is a permanent cycleProof. Our starting point is ER ( n ) ⋆ ( BU (2)) . It may be computed using the Real Atiyah-Hirzebruch spectral sequence completely analogously to the complex-oriented case (see[HK01]). We have ER ( n ) ⋆ ( BU (2)) = ER ( n ) ⋆ [[ c , c ]] with | c i | = i (1 + α ) . We hat c to produce b c = c y in degree − λ ) + 0 α which restricts to c v n − n ∈ E ( n ) ∗ ( BU (2)) . When we take fixed points to land in ER ( n ) ∗ ( BO (2)) and mapover to ER ( n ) ∗ ( BSO (2)) = ER ( n ) ∗ ( CP ∞ ) , we claim this will produce a permanent cyclewhich lifts b u b u ∗ ∈ E ( n ) ∗ ( CP ∞ ) . That is, consider the following commutative diagram: [ BU (2) , ER ( n )] C (cid:15) (cid:15) / / [ BSO (2) , ER ( n )] C [ BU (2) , ER ( n )] [ BSO (2) , ER ( n )] (cid:15) (cid:15) [ BU (2) , E ( n )] / / [ BSO (2) , E ( n )] Here the two horizontal maps are induced by the inclusion
BSO (2) −→ BU (2) . Fromthe diagram, we conclude that the image of b c ∈ ER ( n ) ⋆ ( BU (2)) in ER ( n ) ∗ ( BSO (2)) isa permanent cycle, whose representative on E is given by mapping to the bottom rightcorner. Since b c restricts to c v n − n in E ( n ) ∗ ( BU (2)) , it remains to show that the imageof this class in E ( n ) ∗ ( BSO (2)) is b u b u ∗ . This follows from the homotopy commutativity ofthe following diagram: BU (1) ∆ / / ≃ (cid:15) (cid:15) BU (1) × BU (1) × c / / BU (1) × BU (1) m (cid:15) (cid:15) BSO (2) / / BU (2) (cid:3) Remark 5.2.
The above argument shows that, as an element of E ( n ) ∗ ( CP ∞ ) = E ( n ) ∗ ( BSO (2)) ,the class b u b u ∗ is in fact v n − n times the first Pontryagin class in E ( n ) -cohomology and liftsto ER ( n ) ∗ ( CP ∞ ) . We denote its lift by b p as in Theorem 1.1.Now that we have permanent cycles ( b u b u ∗ ) l for l ≥ , we can use these to form half of ourbasis for E ∗ , ∗ . The other half of the basis will consist of classes b u ( b u b u ∗ ) l , l ≥ . Since ( b u b u ∗ ) l ≡ ( − l b u l mod ( b u l +1 ) nd b u ( b u b u ∗ ) l ≡ ( − l b u l +1 mod ( b u l +2 ) it follows that { b u ǫ ( b u b u ∗ ) l : ǫ = 0 or , l ≥ } clearly forms a topological basis for E ∗ , ∗ over E ( n ) ∗ [ x ] . 6. C OMPUTING E ∗ , ∗ To determine d on this basis, it is necessary to distinguish between odd and even expo-nents of v n , since c ( v ln ) = ( − l v ln . In what follows, recall that b v i are permanent cycles andnote that d ( v n ) = 0 . We have d ( v pn ( b u b u ∗ ) l ) = 0 d ( v p +1 n ( b u b u ∗ ) l ) = 2 v p − n n ( b u b u ∗ ) l xd ( v pn ( b u ( b u b u ∗ ) l )) = v p − (2 n − n ( b u − b u ∗ )( b u b u ∗ ) l xd ( v p +1 n ( b u ( b u b u ∗ ) l )) = v p − n n ( b u + b u ∗ )( b u b u ∗ ) l x. Set R = Z (2) [ b v , . . . , b v n − , v ± n ][ x ] so that E ∗ , ∗ = ( R ⊕ v n R )[[ b u ]] . To analyze the image andkernel of d , we begin with a technical lemma concerning b u ∗ . Lemma 6.1. b u ∗ is in R [[ b u ]] .Proof. Consider R as a submodule of Z (2) [ b v , . . . , b v n − , v ± n ][ x ] over the ring Z (2) [ b v , . . . , b v n − ] .Notice b v n = v − (2 n − +1 n is in R . Thus, the coefficients of b F , formed by hatting the coeffi-cients of F , are also in R , as is [2] b F ( b u ) . We have b u = [2] b F ( b u ) + b F b u ∗ . Reducing modulo the submodule R [[ b u ]] , we have ≡ b F b u ∗ mod R [[ b u ]] . Thus, b u ∗ ∈ R [[ b u ]] . (cid:3) It follows from the formulas for d above together with Lemma 6.1 that d interchangesclasses in R [[ b u ]] with classes in v n R [[ b u ]] . We now describe a convenient (topological) basisfor the kernel of d . Proposition 6.2.
A basis for the kernel of d over R = Z (2) [ b v , . . . , b v n − , v ± n ][ x ] is given by { ( b u b u ∗ ) l , v n ( b u − b u ∗ )( b u b u ∗ ) l : l ≥ } . Proof.
Let f be in the kernel of d . We may write f = f e + v n f o with f e ∈ R [[ b u ]] and v n f o ∈ v n R [[ b u ]] . Since d interchanges classes in R [[ b u ]] and v n R [[ b u ]] , we must have both d ( f e ) = 0 and d ( v n f o ) = 0 . We will show that f e ∈ span { ( b u b u ∗ ) l } and v n f o ∈ span { v n ( b u − b u ∗ )( b u b u ∗ ) l } .Let f e = µ b u j mod ( b u j +1 ) with µ ∈ R . By Lemma 4.2, b u ∗ j ≡ ( − j b u j mod ( b u j +1 ) . Since d ( f e ) ≡ µv − n n ( b u j − b u ∗ j ) ≡ µv − n n ( b u j + ( − j +1 b u j ) mod ( b u j +1 ) ust be zero, j must be even. Then f e − ( − j µ ( b u b u ∗ ) j is in R [[ b u ]] , in the kernel of d , andhas b u -adic valuation strictly larger than that of f e . Thus, f e may be approximated to anydegree by polynomials in R [ b u b u ∗ ] , which proves the claim for f e .We need to consider the first two terms in the case of f o . Let v n f o ≡ µv n b u j + νv n b u j +1 mod ( b u j +2 ) . Applying d modulo ( b u j +1 ) shows that j must now be odd. Next we apply d modulo ( b u j +2 ) . The congruences in Lemma 4.2 give b u j + b u ∗ j ≡ b v j b u j +1 mod (2 , b u j +2 ) b u j +1 + b u ∗ j +1 ≡ mod (2 , b u j +2 ) . Thus, d ( v n f o ) ≡ v − n n µ ( b u j + b u ∗ j ) + v − n n ν ( b u j +1 + b u ∗ j +1 ) ≡ b v j v − n n µ b u j +1 mod (2 , b u j +2 ) . It follows that µ = 2 γ for some γ ∈ R . Then γv n ( b u − b u ∗ )( b u b u ∗ ) j − ≡ µ b u j mod ( b u j +1 ) . Thus, v n f o − ( − j − γv n ( b u − b u ∗ )( b u b u ∗ ) j − is in v n R [[ b u ]] , is in ker ( d ) , and has b u -adic valuationstrictly larger than that of v n f o . This shows that f o may be approximated to any degree byelements of span { v n ( b u − b u ∗ )( b u b u ∗ ) l } , which proves the claim for f o .That the above set of elements is linearly independent follows from inspecting their lead-ing terms. (cid:3) The next step is to relate the image of d to its kernel. This consists of analyzing the class b u + b u ∗ in terms of the above basis for the kernel. Lemma 6.3. b u + b u ∗ is in Z (2) [ b v , . . . , b v n − , v ± n ][[ b u b u ∗ ]] . In other words, there is a power series ξ with coefficients in Z (2) [ b v , . . . , b v n − , v ± n ] such that b u + b u ∗ = ξ ( b u b u ∗ ) .Proof. It follows from the proof of Proposition 6.2 that b u + b u ∗ ∈ R [[ b u b u ∗ ]] , as it showsthat any class that is in R [[ b u ]] and in ker ( d ) is also in R [[ b u b u ∗ ]] . Lemma 6.1 shows that b u + b u ∗ ∈ R [[ b u ]] and c ( b u + b u ∗ ) = b u + b u ∗ shows it is in ker ( d ) . Since x does not divide b u + b u ∗ ,the coefficients of ξ lie in Z (2) [ b v , . . . , b v n − , v ± n ] . (cid:3) Remark 6.4.
In fact, something stronger is true. In the proof of Proposition 6.2, if wereplace R by the ring Z (2) [ b v , . . . , b v n ] , the same argument applies to show that b u + b u ∗ is in Z (2) [ b v , . . . , b v n ][[ b u b u ∗ ]] .We will say more about the power series expansion of b u + b u ∗ in b u b u ∗ in section 11. For now,we describe the E -page. We will present the result as a module over E ∗ , ∗ ( pt ) . Recall that E , ∗ ( pt ) = Z (2) [ b v , . . . , b v n − , v ± n ] x E s, ∗ ( pt ) = Z / b v , . . . , b v n − , v ± n ] x s for s > . We then have heorem 6.5. The E -page is given by E , ∗ = E , ∗ ( pt )[[ b u b u ∗ ]] { , v n ( b u − b u ∗ ) } E s, ∗ = E s, ∗ ( pt )[[ b u b u ∗ ]] / ( b u + b u ∗ ) for s > . Proof.
In the basis for the kernel given in Proposition 6.2, for s > , the classes v n ( b u − b u ∗ )( b u b u ∗ ) l x s are targets of differentials as are the classes b u b u ∗ ) l x s . Thus, these classes onlysurvive on the zero line. Away from the zero line, we just have R [[ b u b u ∗ ]] x s modulo theimage of d . Lemma 6.3 shows that the ideal generated by ( b u + b u ∗ ) x in E ∗ , ∗ is containedin R [[ b u b u ∗ ]][ x ] . This proves the theorem. (cid:3)
7. T
HE IMAGE OF THE NORM
We now find ourselves in a very nice place. We know how the differentials act on thecoefficients, and we have a large collection of permanent cycles. The remaining perma-nent cycles live on the zero line and the next step is to find representatives for them in ER ( n ) ∗ ( CP ∞ ) . We do this using the norm described by Proposition 12.1 in the appendix.We let N ∗ denote the map N ∗ : E ( n ) ∗ ( CP ∞ ) −→ ER ( n ) ∗ ( CP ∞ ) and N ∗ denote the map resulting from postcomposing N ∗ with the inclusion of fixedpoints map ER ( n ) ∗ ( CP ∞ ) → E ( n ) ∗ ( CP ∞ ) , N ∗ : E ( n ) ∗ ( CP ∞ ) −→ ER ( n ) ∗ ( CP ∞ ) −→ E ( n ) ∗ ( CP ∞ ) . Thus, for any z ∈ E ( n ) ∗ ( CP ∞ ) , N ∗ ( z ) is a permanent cycle on the zero line represented in ER ( n ) ∗ ( CP ∞ ) by N ∗ ( z ) . From the appendix, we have N ∗ ( z ) = z + c ( z ) . For any w suchthat c ( w ) = w , we have N ∗ ( wz ) = w N ∗ ( z ) . Let S = Z (2) [ b v , . . . , b v n − , v ± n ] (so that R aboveis S [ x ] ). As a module over S [[ b u b u ∗ ]] = Z (2) [ b v , . . . , b v n − , v ± n ][[ b u b u ∗ ]] , we may write E ( n ) ∗ ( CP ∞ ) = S [[ b u b u ∗ ]] { , v n , b u, v n b u } . Since c fixes S [[ b u b u ∗ ]] , N ∗ is a map of modules over S [[ b u b u ∗ ]] . We restrict N ∗ to the submod-ule S [[ b u b u ∗ ]] { b u, v n b u } ⊂ E ( n ) ∗ ( CP ∞ ) and let im( N res ∗ ) denote the image. (We restrict to thesubmodule because we do not want the coefficients, in particular 2, to be in im( N res ∗ ) . Thisis because we will mod out by im( N ∗ ) later, and we will want multiplication by 2 to beinjective on the quotient.) We have N ∗ ( b u ) = b u + b u ∗ N ∗ ( v n b u ) = v n ( b u − b u ∗ ) so im( N res ∗ ) = S [[ b u b u ∗ ]] { b u + b u ∗ , v n ( b u − b u ∗ ) } . This is a submodule of E , ∗ , and furthermore, since elements of im( N res ∗ ) are permanentcycles, it is contained in ker ( d ) . Since no differentials have their targets in the zero line, t follows that im( N res ∗ ) is a submodule of E , ∗ . We have a short exact sequence −→ im( N res ∗ ) −→ E ∗ , ∗ −→ e E ∗ , ∗ −→ where e E ∗ , ∗ is by definition the quotient. Since all differentials on im( N res ∗ ) are zero, wemay further view it as a sub-spectral sequence. Furthermore, since im( N res ∗ ) injects into E ∗ , ∗ r at each stage, it follows that the above short exact sequence is in fact a short exactsequence of spectral sequences. From Theorem 6.5 we conclude e E ∗ , ∗ = E ∗ , ∗ ( pt )[[ b u b u ∗ ]]( b u + b u ∗ ) . In the short exact sequence above im( N res ∗ ) collapses immediately, so it remains to com-pute the spectral sequence e E ∗ , ∗ .8. L ANDWEBER FLATNESS
Let b E ( n ) ∗ = Z (2) [ b v , . . . , b v n − , b v ± n ] . There is an isomorphism of rings (but not graded rings)between E ( n ) ∗ and b E ( n ) ∗ sending v k to b v k . b E ( n ) ∗ consists entirely of permanent cycles.Thus b E ( n ) ∗ ⊂ ER ( n ) ∗ and ER ( n ) ∗ is a module over b E ( n ) ∗ . We may view E ∗ , ∗ r ( pt ) asa spectral sequence of b E ( n ) ∗ -modules. The E -page of the spectral sequence of interest, e E ∗ , ∗ , may be written as e E ∗ , ∗ = E ∗ , ∗ ( pt )[[ b u b u ∗ ]] / ( b u + b u ∗ ) = E ∗ , ∗ ( pt ) ⊗ b E ( n ) ∗ b E ( n ) ∗ [[ b u b u ∗ ]] / ( b u + b u ∗ ) . The right hand coordinate of the tensor product consists entirely of permanent cycles, sowe will be done if we can show that we can commute taking homology past the tensorproduct at each stage. That is, we need to know that tensoring with b E ( n ) ∗ [[ b u b u ∗ ]] / ( b u + b u ∗ ) over b E ( n ) ∗ is exact. This would be true if b E ( n ) ∗ [[ b u b u ∗ ]] / ( b u + b u ∗ ) were flat over b E ( n ) ∗ , butwe can in fact show that a weaker condition holds.If we identify E ( n ) ∗ with b E ( n ) ∗ as above, then we may view E ∗ , ∗ r ( pt ) as a spectral sequenceof E ( n ) ∗ -modules. Starting with this observation, it is shown in [KW14] that E ∗ , ∗ r ( pt ) infact lives in the category of E ( n ) ∗ E ( n ) -comodules that are finitely presented as E ( n ) ∗ -modules.Thus, to solve our problem we only need to show that b E ( n ) ∗ [[ b u b u ∗ ]] / ( b u + b u ∗ ) is flat on thecategory of finitely presented E ( n ) ∗ E ( n ) -comodules, i.e. that it is Landweber flat .In [HS05], Hovey and Strickland prove an E ( n ) -version of the Landweber filtration the-orem. We state and prove an E ( n ) -exact functor theorem, which follows as a corollaryof Hovey and Strickland’s work. To be consistent with the literature, we prove the resultfor E ( n ) ∗ -modules, keeping in mind that E ( n ) ∗ is formally isomorphic (as rings but notgraded rings) to E ( n ) ∗ so everything below holds for E ( n ) ∗ -modules as well. Proposition 8.1.
Let M be an E ( n ) ∗ -module. The functor ( − ) ⊗ E ( n ) ∗ M is exact on the categoryof E ( n ) ∗ E ( n ) -comodules that are finitely presented as E ( n ) ∗ -modules if and only if for each k ≥ ultiplication by v k is monic on M/ ( v , . . . , v k − ) M , i.e. ( v , v , v , . . . ) is a regular sequence on M . Remark 8.2.
Since E ( n ) ∗ is height n in the sense of Hovey-Strickland, there is only some-thing to check for ≤ k ≤ n . For k > n , M/ ( v , . . . , v k ) = 0 so multiplication by v k istrivially monic. Proof.
We follow Landweber’s original proof over
M U ∗ in [Lan76]. Applying ( − ) ⊗ E ( n ) ∗ M to the sequence −−−→ E ( n ) ∗ p −−−→ E ( n ) ∗ −−−→ E ( n ) ∗ / ( p ) −−−→ shows that p : M −→ M is monic if and only if Tor E ( n ) ∗ ( E ( n ) ∗ / ( p ) , M ) = 0 . For k > ,applying ( − ) ⊗ E ( n ) ∗ M to the sequence −−−−→ E ( n ) ∗ / ( v , . . . , v k − ) v k −−−−→ E ( n ) ∗ / ( v , . . . , v k − ) −−−−→ E ( n ) ∗ / ( v , . . . , v n ) −−−−→ shows that multiplication by v k is monic if and only ifTor E ( n ) ∗ ( E ( n ) ∗ / ( v , . . . , v k − ) , M ) −→ Tor E ( n ) ∗ ( E ( n ) ∗ / ( v , . . . , v k ) , M ) is surjective. It follows that multiplication by v k is monic on M/ ( v , . . . , v k − ) for all k ifand only if Tor E ( n ) ∗ ( E ( n ) ∗ / ( v , . . . , v k ) , M ) is zero for all k . In [HS05] it is shown that every E ( n ) ∗ E ( n ) -comodule N that is finitely presented over E ( n ) ∗ admits a finite filtration bysubcomodules N ⊆ N ⊆ · · · ⊆ N s = N for some s with N r /N r − ≡ Σ t E ( n ) ∗ / ( v , . . . , v j ) for some j ≤ n and some t , both de-pending on r . In view of this, Tor E ( n ) ∗ ( E ( n ) ∗ / ( v , . . . , v k ) , M ) = 0 for all k is equivalentto Tor E ( n ) ∗ ( N, M ) = 0 for all finitely presented E ( n ) ∗ E ( n ) -comodules, N . Finally, this isequivalent to ( − ) ⊗ E ( n ) ∗ M being an exact functor on the category of E ( n ) ∗ E ( n ) -comodulesfinitely presented over E ( n ) ∗ . (cid:3) We now show that M = b E ( n ) ∗ [[ b u b u ∗ ]] / ( b u + b u ∗ ) satisfies the algebraic criterion given above. Lemma 8.3. ( b v , . . . , b v n − , b v n ) is a regular sequence in b E ( n ) ∗ [[ b u b u ∗ ]] / ( b u + b u ∗ ) .Proof. Recall our notation I k = ( b v , b v , . . . , b v k − ) and I = (0) . Suppose f ( b u b u ∗ ) ∈ b E ( n ) ∗ [[ b u b u ∗ ]] / ( I k , b u + b u ∗ ) is such that b v k f ( b u b u ∗ ) = 0 . Then b v k f ( b u b u ∗ ) = g ( b u b u ∗ )( b u + b u ∗ ) mod I k . Further modding out by b v k , we have g ( b u b u ∗ )( b u + b u ∗ ) mod I k +1 . By Lemma 4.2, we have that b u + b u ∗ = b v k +1 ( b u b u ∗ ) k +1 mod ( I k +1 , ( b u b u ∗ ) k +1 +1 ) which means b u + b u ∗ = 0 mod I k +1 . Since I k +1 is prime in b E ( n ) ∗ , it follows that g ( b u b u ∗ ) = 0 mod I k +1 . Then g ( b u b u ∗ ) = b v k h ( b u b u ∗ ) mod I k for some h ( b u b u ∗ ) . Hence, f ( b u b u ∗ ) = h ( b u b u ∗ )( b u + b u ∗ ) mod I k o that f ( b u b u ∗ ) = 0 in b E ( n ) ∗ [[ b u b u ∗ ]] / ( I k , b u + b u ∗ ) . Thus, multiplication by b v k is injective. (cid:3) Thus, M is Landweber flat. That is, tensoring with M over b E ( n ) ∗ is an exact functor on thecategory of finitely presented E ( n ) ∗ E ( n ) -comodules. Since E ∗ , ∗ r ( pt ) lives in this category,we may commute homology past the tensor product at each stage of e E ∗ , ∗ r, = E ∗ , ∗ r ( pt ) ⊗ b E ( n ) ∗ M . Furthermore, since M consists entirely of permanent cycles by Proposition 5.1, theentire spectral sequence can be evaluated on the coefficients. In other words, Theorem 4.3in [KW14] applies to show that e E ∗ , ∗ r is isomorphic to ( E ∗ , ∗ r ( pt ) ⊗ b E ( n ) ∗ M, d r ⊗ b E ( n ) ∗ id M ) asspectral sequences of ER ( n ) ∗ -modules and converges to ER ( n ) ∗ ⊗ b E ( n ) ∗ M . We conclude Proposition 8.4. e E ∗ , ∗∞ = E ∗ , ∗∞ ( pt ) ⊗ b E ( n ) ∗ M = E ∗ , ∗∞ ( pt )[[ b u b u ∗ ]] / ( b u + b u ∗ )
9. T HE E ∞ - PAGE
We will now put all of the pieces together. Let us return to the short exact sequence of E -terms we had in Section 7. −→ im( N res ∗ ) −→ E ∗ , ∗ −→ e E ∗ , ∗ −→ Lemma 9.1.
Upon taking homology the induced long exact sequence collapses into short exactsequences at each stage. In particular, we have a short exact sequence −→ im( N res ∗ ) −→ E ∗ , ∗∞ −→ E ∗ , ∗∞ ( pt )[[ b u b u ∗ ]] / ( b u + b u ∗ ) −→ . Proof.
Since the connecting homomorphism of the long exact sequence must increase fil-tration degree (because the differentials do), yet im( N res ∗ ) is concentrated in filtration de-gree zero, the connecting homomorphism is zero at each stage. Thus we have a shortexact sequence at E ∞ . Recall that the left hand spectral sequence collapses immediately.The right hand spectral sequence was computed in the previous section. (cid:3) We analyze this short exact sequence, starting away from the zero line. In strictly positivefiltration degree, the left hand term is zero and so we have E s, ∗∞ = E s, ∗∞ ( pt )[[ b u b u ∗ ]] / ( b u + b u ∗ ) for s > . The zero line is more involved. We begin by giving a “polite” answer.
Proposition 9.2.
The zero-line E , ∗∞ injects into its rationalization, and the rationalization maybe computed as the algebraic invariants of the rationalization of E , ∗ : E , ∗∞ (cid:31) (cid:127) / / E , ∗∞ ⊗ Q ∼ = / / ( E , ∗ ⊗ Q ) C Proof.
Since no classes on the zero line are targets of differentials, E , ∗∞ contains no torsionand so injects into its rationalization. E , ∗ is exactly the invariants in E , ∗ . Away from thezero line, everything is 2-torsion from E onward. Thus, for any class in E , ∗ r , twice it is n the kernel of d r . Since d n +1 − is the last possible differential, we have that n +1 − timesany class on E , ∗ survives to E ∞ . After we rationalize, the isomorphism E , ∗∞ ⊗ Q ∼ = ( E , ∗ ⊗ Q ) C follows. (cid:3) We now describe the zero line explicitly. Recall that E , ∗ ( pt ) = Z (2) [ b v , . . . , b v n − , v ± n ] E , ∗∞ ( pt ) = Z (2) [ b v , . . . , b v n − , v ± n +1 n ] M
Theorem 9.3.
As a module over E , ∗∞ ( pt ) , we have E , ∗∞ = E , ∗∞ ( pt )[[ b u b u ∗ ]] { , v pn ( b u + b u ∗ ) , v p +1 n ( b u − b u ∗ ) } K , ≤ p < n where K encodes the relations generating J in im( N res ∗ ) over E , ∗∞ above together with the relation v n ( b u + b u ∗ ) = ξ ( b u b u ∗ ) · .
0. E
XTENSION PROBLEMS AND MULTIPLICATIVE STRUCTURE
Most of the hard work in solving extension problems is already done by Propositions5.1 and 12.1 as they provide canonical lifts to ER ( n ) ∗ ( CP ∞ ) of our generators of E ∗ , ∗∞ .Proposition 5.1 produces a class b p ∈ ER ( n ) ∗ ( CP ∞ ) whose image in E ( n ) ∗ ( CP ∞ ) is b u b u ∗ .On the zero line, we also have classes v kn ( b u +( − k b u ∗ ) which are the images under the normof classes v kn b u ∈ E ( n ) ∗ ( CP ∞ ) . Since the norm factors through the map ER ( n ) ∗ ( CP ∞ ) −→ E ( n ) ∗ ( CP ∞ ) , these classes have canonical lifts in ER ( n ) ∗ ( CP ∞ ) as well.In Lemma 6.3 we showed that in E ( n ) ∗ ( CP ∞ ) , b u + b u ∗ may be written as a power series ξ ( b u b u ∗ ) , with the coefficients of ξ in b E ( n ) ∗ . We have independently constructed lifts N ∗ ( b u ) of b u + b u ∗ and b p of b u b u ∗ , so we must verify this equality lifts to ER ( n ) ∗ ( CP ∞ ) . This turnsout to be true for degree reasons: Lemma 10.1. N ∗ ( b u ) = ξ ( b p ) in ER ( n ) ∗ ( CP ∞ ) .Proof. The two classes have the same image in E ( n ) ∗ ( CP ∞ ) , so their difference is a mul-tiple of x . If N ∗ ( b u ) − ξ ( b p ) = 0 in ER ( n ) ∗ ( CP ∞ ) , let r be the maximal power of x thatdivides N ∗ ( b u ) − ξ ( b p ) . Suppose r ≥ . Then N ∗ ( b u ) − ξ ( b p ) is represented by a nonzeroclass z ∈ E r, − λ + r ∞ . Since r ≥ , we have j − < r ≤ j +1 − . Since E r, ∗∞ = 0 for r ≥ n +1 ,we must have j < n + 1 . By inspection of degrees, we have that E r,l ∞ = 0 unless l = 0 mod j +1 . Then − λ + r = 0 mod j +1 . Since − λ = − n +2 (2 n − − , it follows that r = 0 mod j +1 which is impossible. Thus, N ∗ ( b u ) − ξ ( b p ) = 0 in ER ( n ) ∗ ( CP ∞ ) . (cid:3) Before we go on to solve the remaining extension problems, we pause to remark on thesignificance of the class N ∗ ( b u ) = ξ ( b p ) above which appears in the denominator of theright hand term of the short exact sequence of Theorem 1.1. Remark 10.2. N ∗ ( b u ) is a canonical class in ER ( n ) ∗ ( CP ∞ ) in the following sense. The “hat-ted” orientation b u induces a map B ( U (1) ⋊ C ) + = CP ∞ + ∧ C EC + b u ∧ / / Σ − λ ER ( n ) ∧ C EC + . Postcomposing with the Adams isomorphism ER ( n ) ∧ C EC + ≃ ER ( n ) hC = ER ( n ) and precomposing with BU (1) + → B ( U (1) ⋊ C ) + yields the class N ∗ ( b u ) in the ER ( n ) -cohomology of BU (1) = CP ∞ whose image in E ( n ) ∗ ( CP ∞ ) is the b u + b u ∗ described above.This description also gives an alternate argument that N ∗ ( b u ) is a power series on b p as follows. Identifying B ( U (1) ⋊ C ) above with BO (2) , it is shown in [KW14] that ER ( n ) ∗ ( BO (2)) is the quotient of a power series ring over ER ( n ) ∗ on two classes, b c and b c . The above description of N ∗ ( b u ) ∈ ER ( n ) ∗ ( CP ∞ ) shows that it is the image of a classin ER ( n ) ∗ ( BO (2)) , i.e. some power series in b c and b c . Identifying BU (1) with BSO (2) , itcan be shown that the map ER ( n ) ∗ ( BO (2)) −→ ER ( n ) ∗ ( BSO (2)) above sends b c to zeroand b c to b p (see Proposition 5.1). It follows that N ∗ ( b u ) is a power series on b p over ER ( n ) ∗ . ote that this argument does not identify the power series explicitly—to do that, we stillneed Lemmas 6.3 and 10.1.With canonically determined lifts in hand, we now solve all extension problems. Therelations of Theorem 9.3 need to be lifted to ER ( n ) ∗ ( CP ∞ ) . Additionally, we describehow classes in im( N res ∗ ) multiply together. In both cases, we use Proposition 12.1 in theappendix. In a ) and ( b ) of the following lemma, recall that classes in E ( n ) ∗ of the form b v j v i +1 m +2 i n with ≤ j < i are permanent cycles and lift to classes of the same name in ER ( n ) ∗ . Lemma 10.3.
The following relations hold in ER ( n ) ∗ ( CP ∞ ) :1. Write p = 2 i +1 m +2 i and suppose ≤ j < i . Then, as a module over ER ( n ) ∗ , ER ( n ) ∗ ( CP ∞ ) satisfies(a) b v j · N ∗ ( v pn b u ) = ( b v j v pn ) · N ∗ ( b u ) (b) b v j · N ∗ ( v p +1 n b u ) = ( b v j v pn ) · N ∗ ( v n b u ) (c) x · N ∗ ( v pn b u ) = 0 (d) x · N ∗ ( v p +1 n b u ) = 0 .2. Let ≤ p, l < n +1 and write p + 2 l = 2 n +1 q + 2 r with ≤ r < n +1 and q = 0 , . Asan algebra over ER ( n ) ∗ , ER ( n ) ∗ ( CP ∞ ) satisfies(a) N ∗ ( v pn b u ) N ∗ ( v ln b u ) = v n +1 qn N ∗ ( v rn b u ) N ∗ ( b u ) (b) N ∗ ( v p +1 n b u ) N ∗ ( v ln b u ) = v n +1 qn N ∗ ( v r +1 n b u ) N ∗ ( b u ) (c) N ∗ ( v p +1 n b u ) N ∗ ( v l +1 n b u ) = v n +1 qn ( N ∗ ( v r +2 n b u ) N ∗ ( b u ) − v r +2 n b p ) .Proof. The key facts are that N ∗ is a map of modules over ER ( n ) ∗ ( CP ∞ ) and that the ER ( n ) ∗ ( CP ∞ ) -module (really, algebra) structure on E ( n ) ∗ ( CP ∞ ) comes from the quotient-by- x map ER ( n ) ∗ ( CP ∞ ) −→ E ( n ) ∗ ( CP ∞ ) . The module structure is described by the fol-lowing diagram: ER ( n ) ∗ ( CP ∞ ) ⊗ E ( n ) ∗ ( CP ∞ ) (cid:15) (cid:15) ⊗ N ∗ / / ER ( n ) ∗ ( CP ∞ ) ⊗ ER ( n ) ∗ ( CP ∞ ) (cid:15) (cid:15) E ( n ) ∗ ( CP ∞ ) N ∗ / / ER ( n ) ∗ ( CP ∞ ) We prove a ) , c ) and a ) ; the other relations are proved similarly. For a ) , note that inthe upper left corner, the classes b v j ⊗ v pn b u and b v j v pn ⊗ b u have as their images in the bottomright corner the classes b v j · N ∗ ( v pn b u ) and ( b v j v pn ) · N ∗ ( b u ) , respectively. But both b v j ⊗ v pn b u and b v j v pn ⊗ b u map to the same class, b v j v pn b u , in the bottom left corner, which proves a ) . In c ) , x ⊗ v pn b u maps to zero in the bottom left corner (since x maps to zero in E ( n ) ∗ ); thus, x · N ∗ ( v pn b u ) = 0 . Finally, for a ) , note that the classes v pn b u ⊗ N ∗ ( v ln b u ) and v p + l ) n b u ⊗ N ∗ ( b u ) map to the same class under the left vertical map. Thus, their images in ER ( n ) ∗ ( CP ∞ ) are equal. But these are exactly N ∗ ( v pn b u ) N ∗ ( v ln b u ) and N ∗ ( v p + l ) n b u ) N ∗ ( b u ) , respectively. Forany α ∈ E ( n ) ∗ ( CP ∞ ) which admits a lift to ER ( n ) ∗ ( CP ∞ ) and any z ∈ E ( n ) ∗ ( CP ∞ ) , theabove diagram shows that N ∗ ( αz ) = αN ∗ ( z ) . Applying this to α = v n +1 qn and z = v rn b u completes the proof of a ) . (cid:3) e now prove Theorem 1.1 as stated in the introduction. Proof of Theorem 1.1. ER ( n ) ∗ ( CP ∞ ) is (topologically) generated over ER ( n ) ∗ by im( N res ∗ ) and the classes b p l . To see the isomorphism (of ER ( n ) ∗ -modules) ER ( n ) ∗ ( CP ∞ ) / im( N res ∗ ) ∼ = ER ( n ) ∗ [[ b p ]] / ( ξ ( b p )) we must show that the intersection of im( N res ∗ ) and the submodule of ER ( n ) ∗ ( CP ∞ ) gen-erated over ER ( n ) ∗ by { b p l | l ≥ } is precisely the ideal of ER ( n ) ∗ ( CP ∞ ) generated by ξ ( b p ) . Clearly, ( ξ ( b p )) is in this intersection. We now prove the reverse inclusion. Firstrecall that the domain of N res ∗ is by definition the submodule Z (2) [ b v , . . . , b v n − , v ± n ][[ b u b u ∗ ]] { b u, v n b u } of E ( n ) ∗ ( CP ∞ ) . Choose some z = X i ( a i b u + b i v n b u )( b u b u ∗ ) i with a i , b i ∈ Z (2) [ b v , . . . , b v n − , v ± n ] so that (since N ∗ is a map of modules over ER ( n ) ∗ ( CP ∞ ) ) N ∗ ( z ) = X i ( N ∗ ( a i b u ) + N ∗ ( b i v n b u )) b p i . (10.4)Suppose that N ∗ ( z ) is also in span { b p l | l ≥ } over ER ( n ) ∗ , i.e. that N ∗ ( z ) = X i λ i b p i with λ i ∈ ER ( n ) ∗ . (10.5)We claim that ξ ( b p ) = N ∗ ( b u ) divides N ∗ ( z ) in ER ( n ) ∗ ( CP ∞ ) . This will follow from twoclaims: for all i , (i) b i = 0 and (ii) a i is in the subalgebra E , ∗∞ ( pt ) of Z (2) [ b v , . . . , b v n − , v ± n ] .Together, they imply that N ∗ ( z ) = P i N ∗ ( a i b u ) b p i with each a i having a representative in ER ( n ) ∗ . Since N ∗ is a map or ER ∗ ( CP ∞ ) -modules, it follows that N ∗ ( a i b u ) = a i N ∗ ( b u ) andso N ∗ ( b u ) divides N ∗ ( z ) in ER ( n ) ∗ ( CP ∞ ) .To prove both claims, we map into E ( n ) ∗ ( CP ∞ ) . Then (10.4) becomes N ∗ ( z ) = X i ( a i ( b u + b u ∗ ) + b i v n ( b u − b u ∗ ))( b u b u ∗ ) i (10.6)and (10.5) becomes N ∗ ( z ) = X i λ i ( b u b u ∗ ) i (10.7)with λ i now in E , ∗∞ ( pt ) (by abuse of notation, we denote the image of λ i in E , ∗∞ ( pt ) bythe same name). Recall that b u − b u ∗ = 2 b u + . . . and b u + b u ∗ is the span of { ( b u b u ∗ ) l | l ≥ } over E , ∗∞ ( pt ) (since b v i ∈ E , ∗∞ ( pt ) for all i ). The collection { ( b u b u ∗ ) l , b u ( b u b u ∗ ) l } is clearly linearlyindependent over E ( n ) ∗ . Hence, from inspecting the right hand sides of (10.6) and (10.7),it follows that b i = 0 for all i .To prove the second claim, suppose that i is the first index such that a i / ∈ E , ∗∞ ( pt ) . Let l bethe maximal such that a i = f ( v l n ) with the coefficients of f in Z (2) [ b v , . . . , b v n − ] . Note that f cannot be a constant polynomial since a i is not in E , ∗∞ by assumption. Then inspection f E , ∗∞ ( pt ) shows that b v k a i ∈ E , ∗∞ ( pt ) for ≤ k < l and b v k a i / ∈ E , ∗∞ ( pt ) for k ≥ l . Recallthat b u + b u ∗ ≡ b v l ( b u b u ∗ ) l + . . . mod (2 , b v , . . . , b v l − ) . Since we know that X i a i ( b u + b u ∗ )( b u b u ∗ ) i = X i λ i ( b u b u ∗ ) i with λ i ∈ E , ∗∞ it follows from looking at lowest degree terms mod (2 , b v , . . . , b v l − ) that b v l a i is in E , ∗∞ , acontradiction. Thus, a i ∈ E , ∗∞ for all i . This proves Theorem 1.1. (cid:3) Theorem 1.1 is the nice form of the answer, but we have in fact shown something stronger.The following theorem presents ER ( n ) ∗ ( CP ∞ ) explicitly as an algebra over ER ( n ) ∗ . Theorem 10.8. ER ( n ) ∗ ( CP ∞ ) = ER ( n ) ∗ ( pt )[[ b p , N ∗ ( v jn b u )]] /K where ≤ j < n +1 and K is the ideal generated by the relation N ∗ ( b u ) = ξ ( b p ) of Lemma 10.1together with the relations of Lemma 10.3Proof. This is a consequence of Lemmas 10.1 and 10.3 together with the description of E ∗ , ∗∞ in section 9. (cid:3) Remark 10.9.
We conclude by returning to the case n = 1 . It is somewhat degenerate,as N ∗ ( b u ) = b u + b u ∗ = − b u b u ∗ in KU ∗ ( CP ∞ ) , so N ∗ ( b u ) = − b p . It follows that x · b p = 0 andpowers of b p do not generate any 2-torsion. This is not true for n > . In general, b p supports higher powers of x up to and including x n +1 − . In light of this, when n = 1 , b p is redundant and it suffices to take { , N ∗ ( b u ) , N ∗ ( v b u ) , N ∗ ( v b u ) , N ∗ ( v b u ) } as a set of algebragenerators with x · N ∗ ( v k b u ) = 0 for all k . The relations come from Lemma 10.3 and ouranswer matches exactly the answer in Corollary 2.13 of [Yam07].11. C OMPLETING AT I We obtain an especially nice form of the answer if we complete at I := I n = (2 , b v , . . . , b v n − ) .It turns out that the right hand side of the short exact sequence of Theorem 1.1 is free aftercompletion. To see this, we need a technical lemma concerning b u + b u ∗ . Lemma 11.1.
The following congruence holds in b E ( n )[[ b u b u ∗ ]] : b u + b u ∗ ≡ b v n ( b u b u ∗ ) n − mod (( b u b u ∗ ) n − +1 , I ) Since b E ( n ) ⊂ ER ( n ) ∗ , this lifts to ξ ( b p ) ≡ b v n b p n − mod ( b p n − +11 , I ) in ER ( n ) ∗ [[ b p ]] . roof. By Lemma 6.3, b u + b u ∗ ∈ b E ( n )[[ b u b u ∗ ]] . By Lemma 4.2, b u + b u ∗ ≡ b v n b u n mod ( b u n +1 , I ) . It follows that b u + b u ∗ ≡ b v n ( b u b u ∗ ) n − mod (( b u b u ∗ ) n − +1 , I ) . (cid:3) We will use the following version of the Weierstrass Preparation Theorem in [HKR00].
Lemma 11.2. [HKR00] Let A be a graded commutative ring, complete in the topology defined bypowers of an ideal I . Suppose α ( x ) ∈ A [[ x ]] satisfies α ( x ) ≡ ωx d mod ( x d +1 , I ) , with ω ∈ A aunit. Then the ring A [[ x ]] / ( α ( x )) is a free A -module with basis { , x, x , . . . , x d − } . In what follows, we apologize for the poor notation: the hat of completion and the hat in b p are unrelated. Recall the short exact sequence of Theorem 1.1: −→ im( N res ∗ ) −→ ER ( n ) ∗ ( CP ∞ ) −→ ER ( n ) ∗ [[ b p ]]( ξ ( b p )) −→ After completing at I , if we set A = ER ( n ) ∗∧ I we find that the Weierstrass PreparationTheorem applies to the right hand term. Proposition 11.3. ER ( n ) ∗∧ I [[ b p ]] / ( ξ ( b p )) is a free module over ER ( n ) ∗∧ I with basis { , b p , b p , . . . , b p n − − } .
12. A
PPENDIX
The purpose of this section is to establish some technical lemmas that enable us to solveextension problems in the Bockstein spectral sequence. We construct a norm map N ∗ : E ( n ) ∗ ( X ) −→ ER ( n ) ∗ ( X ) and prove that (i) it is a map of ER ( n ) ∗ ( X ) -modules and (ii) the image of this map is rep-resented on the E -page of the Bockstein spectral sequence by the group-theoretic norm.For more details and more generality, see [LMSM86], [RV14] (for equivariant orthogonalspectra), and the author’s thesis [Lor16].We begin with some general setup. Let G be a finite group. Let p : G −→ G/G = { e } denote projection onto the quotient. Let ι : { e } −→ G denote inclusion of the trivialsubgroup. Let U be a complete G -universe. Let i : U G −→ U denote the inclusion ofuniverses. Let GSU denote the category of (genuine) G -equivariant spectra indexed over U . Let E ∈ GSU be a ring spectrum. Let D = i ∗ E ∈ GSU G .We construct the norm N : ι ∗ D −→ D G as follows. First, we construct e N : EG + ∧ D −→ p ∗ ( D G ) . We get N by applying ι ∗ ( − ) and noting that the map EG + ∧ D −→ D is anequivalence on underlying nonequivariant spectra. o construct e N : EG + ∧ D −→ p ∗ ( D G ) , we start with the unit of the ( − /G, p ∗ ) adjunction, EG + ∧ D −→ p ∗ (( EG + ∧ D ) /G ) . As EG + ∧ D is G -free, we may then compose with theAdams isomorphism (see [LMSM86]), A : p ∗ (( EG + ∧ D ) /G ) −→ p ∗ (( EG + ∧ D ) G ) . Finally, we compose with p ∗ (( EG + ∧ D ) G ) −→ p ∗ ( D G ) . We call this composite e N . EG + ∧ D is a module over p ∗ ( D G ) with action given by EG + ∧ D ∧ p ∗ ( D G ) / / EG + ∧ D ∧ D / / EG + ∧ D .p ∗ ( D G ) is a module over p ∗ ( D G ) with action given by p ∗ ( D G ) ∧ p ∗ ( D G ) / / p ∗ (( D ∧ D ) G ) / / p ∗ ( D G ) . Thus, both the source and target of e N are modules over p ∗ ( D G ) and a diagram chaseshows that e N is a map of p ∗ ( D G ) -modules. Thus, N : ι ∗ D −→ D G is a map of D G -modules.We now postcompose N with the inclusion of fixed points D G −→ ι ∗ D . We thus have aself-map of ι ∗ D which we denote by N . N has a very nice expression on the underlyingnonequivariant homotopy groups of D , π ∗ ( ι ∗ D ) = π u ∗ ( D ) . It is given by composing thediagonal ι ∗ D / / G + ∧ ι ∗ D W g ∈ G ι ∗ D with the action map G + ∧ ι ∗ D −→ ι ∗ D . For z ∈ π u ∗ ( D ) this gives N ∗ ( z ) = X g ∈ G g · z. We now specialize to the case of interest. Let G = C . Let X be a space with trivial C -action (in our present applications, X = CP ∞ with C acting trivially and not by con-jugation). Let E = F ( X, ER ( n )) . Then E is a ring spectrum via the multiplication on ER ( n ) . Note that the underlying nonequivariant spectrum is ι ∗ E = F ( X, E ( n )) . This isthe source of the norm, with π u ∗ E = E ( n ) ∗ ( X ) . Since C acts trivially on X , the target is E C = F ( X, ER ( n )) C = F ( X, ER ( n ) C ) = F ( X, ER ( n )) with π ∗ E C = ER ( n ) ∗ ( X ) . Thus, we have shown the following.
Proposition 12.1. (a) There is a norm map N : F ( X, E ( n )) = ι ∗ E −→ E C = F ( X, ER ( n )) . b) Give F ( X, ER ( n )) and F ( X, E ( n )) module structures over F ( X, ER ( n )) via the multipli-cation on ER ( n ) and the map of ring spectra ER ( n ) −→ E ( n ) , respectively. Then N is amap of modules over E C = F ( X, ER ( n )) . On homotopy, it is given as a map of modulesover ER ( n ) ∗ ( X ) by N ∗ : E ( n ) ∗ ( X ) −→ ER ( n ) ∗ ( X ) . (c) After composing with the inclusion of fixed points ER ( n ) ∗ ( X ) −→ E ( n ) ∗ ( X ) (which is oneof the maps in our exact couple) we denote the composite by N ∗ , N ∗ : E ( n ) ∗ ( X ) −→ ER ( n ) ∗ ( X ) −→ E ( n ) ∗ ( X ) . It is given in homotopy by N ∗ ( z ) = z + c ( z ) .(d) In the Bockstein spectral sequence, every class N ∗ ( z ) in E ( n ) ∗ ( CP ∞ ) = E , ∗ is a permanentcycle represented by N ∗ ( z ) ∈ ER ( n ) ∗ ( X ) . R EFERENCES [AM78] Sh ˆor ˆo Araki and Mitutaka Murayama. τ -cohomology theories. Japan. J. Math. (N.S.) , 4(2):363–416, 1978.[Ban13] Romie Banerjee. On the ER (2) -cohomology of some odd-dimensional projective spaces. Topol-ogy Appl. , 160(12):1395–1405, 2013.[BG10] Robert R. Bruner and J. P. C. Greenlees.
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EPARTMENT OF M ATHEMATICS , J
OHNS H OPKINS U NIVERSITY , B
ALTIMORE , USA
E-mail address : [email protected]@math.jhu.edu