On the Adams Spectral Sequence for R-modules

Transcription

1 ISSN (on-line) (pinted) 173 Algebaic & Geometic Topology Volume 1 (2001) Published: 7 Apil 2001 ATG On the Adams Spectal Sequence fo -modules Andew Bake Andey Lazaev Abstact We discuss the Adams Spectal Sequence fo -modules based on commutative localized egula quotient ing specta ove a commutative S -algeba in the sense of Elmendof, Kiz, Mandell, May and Stickland. The fomulation of this spectal sequence is simila to the classical case and the calculation of its E 2 -tem involves the cohomology of cetain bave new Hopf algeboids E E. In woking out the details we esuect Adams oiginal appoach to Univesal Coefficient Spectal Sequences fo modules ove an ing spectum. We show that the Adams Spectal Sequence fo S based on a commutative localized egula quotient ing spectum E = /I[X 1 ] conveges to the homotopy of the E -nilpotent completion π L E S = [X 1 ] Î. We also show that when the geneating egula sequence of I is finite, L ES is equivalent to L E S, the Bousfield localization of S with espect to E -theoy. The spectal sequence hee collapses at its E 2 -tem but it does not have a vanishing line because of the pesence of polynomial geneatos of positive cohomological degee. Thus only one of Bousfield s two standad convegence citeia applies hee even though we have this equivalence. The details involve the constuction of an I -adic towe /I /I 2 /I s /I s+1 whose homotopy limit is L ES. We descibe some examples fo the motivating case = MU. AMS Classification 55P42, 55P43, 55T15; 55N20 Keywods S -algeba, -module, ing spectum, Adams Spectal Sequence, egula quotient c Geomety & Topology Publications

2 174 Andew Bake and Andey Lazaev Eatum While this pape was in e-pess, the authos discoveed that the oiginal vesions of Theoems 6.3 and 6.4 wee incoect since they did not assume that the egula sequence u j was finite. With the ageement of the Editos, we have evised this vesion to include the appopiate finiteness assumptions. We have also modified the Abstact and Intoduction to eflect this and in Section 7 have eplaced Bousfield localizations L E X by E -nilpotent completions L E X. As fa as we ae awae, thee ae no futhe poblems aising fom this mistake. Andew Bake and Andey Lazaev 9 May 2001 Intoduction We conside the Adams Spectal Sequence fo -modules based on localized egula quotient ing specta ove a commutative S -algeba in the sense of [11, 16], making systematic use of ideas and notation fom those two souces. This wok gew out of a pepint [4] and the wok of [6]; it is also elated to ongoing collaboation with Alain Jeanneet on Bockstein opeations in cohomology theoies defined on -modules [7]. One slightly supising phenomenon we uncove concens the convegence of the Adams Spectal Sequence based on E = /I[X 1 ], a commutative localized egula quotient of a commutative S -algeba. We show that the spectal sequence fo π S collapses at E 2, howeve fo 2, E has no vanishing line because of the pesence of polynomial geneatos of positive cohomological degee which ae infinite cycles. Thus only one of Bousfield s two convegence citeia [10] (see Theoems 2.3 and 2.4 below) apply hee. Despite this, when the geneating egula sequence of I is finite, the spectal sequence conveges to π L E S,wheeL E is the Bousfield localization functo with espect to E -theoy on the categoy of -modules and π L E S = [X 1 ] Î, the I -adic completion of [X 1 ]; we also show that in this case L E S L E S,theE-nilpotent completion of S. In the final section we descibe some examples fo the impotant case of = MU, leaving moe delicate calculations fo futue wok. To date thee seems to have been vey little attention paid to the detailed homotopy theoy associated with the categoy of -modules, apat fom geneal

3 On the Adams Spectal Sequence fo -modules 175 esults on Bousfield localizations and Wolbet s wok on K -theoetic localizations in [11, 19]. We hope this pape leads to futhe wok in this aea. Acknowledgements The fist autho wishes to thank the Univesity and City of Ben fo poviding such a hospitable envionment duing many visits, also Alain Jeanneet, Us Wügle and othe paticipants in the Topology woking semina duing sping and summe Finally, the authos wish to thank the membes of Tanspennine Topology Tiangle (funded by the London Mathematical Society) fo poviding a timely boost to this poject. Backgound assumptions, teminology and technology We wok in a setting based on a good categoy of specta S such as the categoy of L-specta of [11]. Associated to this is the subcategoy of S -modules M S and its deived homotopy categoy D S. Thoughout, will denote a commutative S -algeba in the sense of [11]. Thee is an associated subcategoy M of M S consisting of the -modules, and its deived homotopy categoy D and ou homotopy theoetic wok is located in the latte. Because we ae woking in D, we fequently make constuctions using cell -modules in place of non-cell modules (such as itself). Fo -modules M and N,weset whee D (M,N) n = D (M,Σ n N). M N=π M N, N M = D (M,N), We will use the following teminology of Stickland [16]. If the homotopy ing = π is concentated in even degees, a localized quotient of will be an ing spectum of the fom /I[X 1 ]. A localized quotient is commutative if it is a commutative ing spectum. A localized quotient /I[X 1 ]isegula if the ideal I is geneated by a egula sequence u 1,u 2,... say. The ideal I extends to an ideal of [X 1 ] which we will again denote by I ;then as -modules, /I[X 1 ] [X 1 ]/I. We will make use of the language and ideas of algebaic deived categoies of modules ove a commutative ing, mildly extended to deal with evenly gaded

4 176 Andew Bake and Andey Lazaev ings and thei modules. In paticula, this means that chain complexes ae often bigaded (o even multigaded) objects with thei fist gading being homological and the second and highe ones being intenal. 1 Bave new Hopf algeboids and thei cohomology If E is a commutative -ing spectum, the smash poduct E E is also a commutative -ing spectum. Moe pecisely, it is natually an E -algeba spectum in two ways induced fom the left and ight units E = E E E E = E. Theoem 1.1 Let E E be flat as a left o equivalently ight E -module. Then the following ae tue. i) (E,E E) is a Hopf algeboid ove. ii) fo any -module M, E M is a left E E -comodule. Poof This is poved using essentially the same agument as in [1, 15]. The natual map E M = E M E E M induces the coaction ψ : E M π E E M = E E E M, E which uses an isomophism π E E M = E E E M. E that follows fom the flatness condition. Fo late use we ecod a geneal esult on the Hopf algeboids associated with commutative egula quotients. A numbe of examples fo the case = MU ae discussed in Section 7. Poposition 1.2 Let E = /I be a commutative egula quotient whee I is geneated by the egula sequence u 1,u 2,...ThenasanE -algeba, E E =Λ E (τ i :i 1),

5 On the Adams Spectal Sequence fo -modules 177 whee deg τ i = degu i +1. Moeove, the geneatos τ i ae pimitive with espect to the coaction, and E E is a pimitively geneated Hopf algeba ove E. Dually, as an E -algeba, E E = Λ E (Q i : i 1), whee Q i is the Bockstein opeation dual to τ i with deg Q i = degu i +1 and Λ E ()indicates the completed exteio algeba geneated by the anticommuting Q i elements. The poof equies the Künneth Spectal Sequence fo -modules of [11], E 2 p,q =To p,q (E,E )= Ep+q E. This spectal sequence is multiplicative, howeve thee seems to be no published poof in the liteatue. At the suggestion of the efeee, we indicate a poof of this due to M. Mandell and which oiginally appeaed in a pepint vesion of [12]. Lemma 1.3 If A and B ae ing specta then the Künneth Spectal Sequence To (A,B )= A B=π A B is a spectal sequence of diffeential gaded -algebas. Sketch poof To deal with the multiplicative stuctue we need to modify the oiginal constuction given in Pat IV section 5 of [11]. We emind the eade that we ae woking in the deived homotopy categoy D. Let f p f 1 f 0 F p, F p 1, F0, A 0 be an fee -esolution of A. Using feeness, we can choose a map of complexes µ: F, F, F, which lifts the multiplication on A. Fo each p 0letF p be a wedge of sphee -modules satisfying π F p = F p,. Set A 0 = F 0 and choose a map ϕ 0 : A 0 A inducing f 0 in homotopy. If Q 0 is the homotopy fibe of ϕ 0 then π Q 0 =kef 0

6 178 Andew Bake and Andey Lazaev and we can choose a map F 1 Q 0 fo which the composition ϕ 1 : F 1 Q 0 F 0 induces f 1 in homotopy. Next take A 1 to be the cofibe of ϕ 1. The map ϕ 0 has a canonical extension to a map ϕ 1 : A 1 A. If Q 1is the homotopy fibe of ϕ 1 then π Σ 1 Q 1 =kef 1, and we can find a map F 2 ΣQ 1 fo which the composite map ϕ 2 : F 2 Q 1 F 1 induces f 2 in homotopy. We take A 2 to be the cofibe of ϕ 2 and find that thee is a canonical extension of ϕ 1 to a map ϕ 2 : A 2 A. Continuing in this way we constuct a diected system A 0 A 1 A p (1.1) whose telescope A is equivalent to A. Since we can assume that all consecutive maps ae inclusions of cell subcomplexes, thee is an associated filtation on A. Smashing this with B we get a filtation on A B and an associated spectal sequence conveging to A B. The identification of the E 2 -tem is outine. ecall that A and theefoe A ae ing specta. Smashing the diected system of (1.1) with itself we obtain a filtation on A A, A 0 A 0 i+j =k A i A j i+j =k+1 A i A j, (1.2) whee the filtations tems ae unions of the subspecta A i A j. Poceeding by induction, we can ealize the multiplication map A A A as a map of filteed -modules so that on the cofibes of the filtation tems of (1.2) it agees with the paiing µ. We have constucted a collection of maps A i A j A i+j. Using these maps and the multiplication on B we can now constuct maps A i B A j B A i+j B which induce the equied paiing of spectal sequences. Poof of Poposition 1.2 As in the discussion peceding Poposition 5.1, making use of a Koszul esolution we obtain E 2, =Λ E (e i :i 1). The geneatos have bidegee bideg e i =(1, u i ), so the diffeentials d :E p,q E p,q+ 1

7 On the Adams Spectal Sequence fo -modules 179 ae tivial on the geneatos e i fo dimensional easons. Togethe with multiplicativity, this shows that spectal sequence collapses, giving E E =Λ E (τ i :i 1), whee the geneato τ i has degee deg τ i =degu i + 1 and is epesented by e i. Fo each i, (/u i ) (/u i)=λ /(ui )(τ i ) with deg τ i = u i + 1. Unde the copoduct, τ i is pimitive fo degee easons. By compaing the two Künneth Spectal Sequences we find that τ i E E can be chosen to be the image of τ i unde the evident ing homomophism (/u i ) (/u i) E E, which is actually a mophism of Hopf algeboids ove. Hence τ i is coaction pimitive in E E. Fo E E, we constuct the Bockstein opeation Qi using the composition /u i Σ u i +1 Σ u i +1 /u i to induce a map E Σ ui +1 E, then use the Koszul esolution to detemine the Univesal Coefficient Spectal sequence E p,q 2 =Ext p,q (E,E )= E p+q E which collapses at its E 2 -tem. Futhe details on the constuction of these opeations appea in [16, 7]. Coollay 1.4 i) The natual map E = E E E induced by the unit /I is a split monomophism of E -modules. ii) E E is a fee E -module. Poof An explicit splitting as in (i) is obtained using the multiplication map E E E which induces a homomophism of E -modules E E E. We will use Coext to denote the cohomology of such Hopf algeboids athe than Ext since we will also make heavy use of Ext goups fo modules ove ings; moe details of the definition and calculations can be found in [1, 15]. ecall that fo E E -comodules L and M whee L is E -pojective, Coext s,t E E(L,M ) can be calculated as follows. Conside a esolution 0 M J 0, J 1, J s, in which each J s, is a summand of an extended comodule E E N s,, E

8 o < o o 180 Andew Bake and Andey Lazaev fo some E -module N s,. Then the complex 0 Hom E E(L,J 0, ) Hom E E(L,J 1, ) Hom E E(L,J s, ) has cohomology H s (Hom E E(L,J, )) = Coext s, E E (L,M ). The functos Coext s, E E(L, ) ae the ight deived functos of the left exact functo M Hom E E(L,M ) on the categoy of left E E -comodules. By analogy with [15], when L = E we have Coext s, E E (E,M )=Coto s, E E (E,M ). 2 The Adams Spectal Sequence fo -modules We will descibe the E -theoy Adams Spectal Sequence in the homotopy categoy of -module specta. As in the classical case of sphee spectum = S, it tuns out that the E 2 -tem is can be descibed in tems of the functo Coext E E. Let L, M be -modules and E a commutative -ing spectum with E E flat as a left (o ight) E -module. Theoem 2.1 If E L is pojective as an E -module, thee is an Adams Spectal Sequence with E s,t 2 (L, M) =Coexts,t E E(E L, E M). Poof Woking thoughout in the deived categoy D, the poof follows that of Adams [1], with S eplacing the sphee spectum S. The canonical Adams esolution of M is built up in the usual way by splicing togethe the cofibe tiangles in the following @ E M y y y y y y E M I I I I I I $ E E M 9 E E M B B B B B B B B The algebaic identification of the E 2 -tem poceeds as in [1].

9 On the Adams Spectal Sequence fo -modules 181 In the est of this pape we will have L = S, andset E s,t 2 (M) =Coexts,t E E(E,E M). We will efe to this spectal sequence as the Adams Spectal Sequence based on E fo the -module M. To undestand convegence of such a spectal sequence we use a citeion of Bousfield [10, 14]. Fo an -module M,letD s M (s 0) be the -modules defined by D 0 M = M and taking D s M to be the fibe of the natual map D s 1 M = D s 1 M E D s 1 M. Also fo each s 0letK s Mbe the cofibe of the natual map D s M M. Then the E -nilpotent completion of M is the homotopy limit L EM = holim K s M. s emak 2.2 It is easy to see that if M N is a map of -modules which is an E -equivalence, then fo each s, thee is an equivalence K s M K s N, hence L EM L EN. Theoem 2.3 If fo each pai (s, t) thee is an 0 fo which E s,t (M) =E s,t (M) wheneve 0, then the Adams Spectal Sequence fo M based on E conveges to π L E M. Although thee is a natual map L E M L E M, it is not in geneal a weak equivalence; this equivalence is guaanteed by anothe esult of Bousfield [10]. Theoem 2.4 Suppose that thee is an 1 such that fo evey -module N thee is an s 1 fo which E s,t (N) =0wheneve 1 and s s 1. Then fo evey -module M the Adams Spectal Sequence fo M based on E conveges to π L E M and L EM L EM. 3 The Univesal Coefficient Spectal Sequence fo egula quotients Let be a commutative S -algeba and E = /I a commutative egula quotient of, wheeu 1,u 2,... is a egula sequence geneating I.

10 182 Andew Bake and Andey Lazaev We will discuss the existence of the Univesal Coefficient Spectal Sequence E 2,s =Ext,s E (E M,N )= NM, (3.1) whee M and N ae -modules and N is also an E -module spectum in M. The classical pototype of this was descibed by Adams [1] (who genealized a constuction of Atiyah [2] fo the Künneth Theoem in K -theoy) and used in setting up the E -theoy Adams Spectal Sequence. It is outine to veify that Adams appoach can be followed in D.WeemakthatifEwee a commutative -algeba then the Univesal Coefficient Spectal Sequence of [11] would be applicable but that condition does not hold in the geneality we equie. The existence of such a spectal sequence depends on the following conditions being satisfied. Conditions 3.1 E is a homotopy colimit of finite cell -modules E α whose -Spanie Whitehead duals D E α = F (E α,) satisfy the two conditions (A) E D E α is E -pojective; (B) the natual map N M Hom E (E M,N ) is an isomophism. Theoem 3.2 Fo a commutative egula quotient E = /I of, E can be expessed as a homotopy colimit of finite cell -modules satisfying the conditions of Condition 3.1. In fact we can take E D E α to be E -fee. The poof will use the following Lemma. Lemma 3.3 Let u 2d be non-zeo diviso in. Suppose that P is an -module fo which E P is E -pojective and fo an E -module -spectum N, N P = Hom E (E P, N ). Then E P /u is E -pojective and N P /u = Hom E (E P /u, N ). Poof Smashing E P with the cofibe sequence (3.2) and taking homotopy, we obtain an exact tiangle E P u / E P dj JJJ J JJJ J E P /u zt ttt t ttt t

11 On the Adams Spectal Sequence fo -modules 183 As multiplication by u induces the tivial map in E -homology, this is actually a shot exact sequence of E -modules, 0 E P E P /u E P 0 which clealy splits, so E P /u is E -pojective. In the evident diagam of exact tiangles N P / N P kv VVVVVVVVVVVVVV N P /u sh hhhhhhhhhhhhhh Hom E (E P, N ) / Hom E (E P, N ) jv VVVVVVVV Hom E (E P /u, N ) th hhhhhhhh the map N P Hom E (E P, N ) is an isomophism, so N P /u Hom E (E P /u, N ) is also an isomophism by the Five Lemma. Poof of Theoem 3.2 Let u 1,u 2,... be a egula sequence geneating I. Using the notation /u = /(u), we ecall fom [16] that E = hocolim /u 1 /u 2 /u k. k Fo u 2d a non-zeo diviso, the -fee esolution 0 u /(u) 0 coesponds to an -cell stuctue on /u with one cell in each of the dimensions 0 and 2d + 1. Thee is an associated cofibe sequence Σ 2d u /u Σ 2d+1, (3.2) fo which the induced long exact sequence in E -homology shows that E /u is E -fee. The dual D /u is equivalent to Σ (2d+1) /u, hence /u is essentially self dual.

12 184 Andew Bake and Andey Lazaev Fo an E -module spectum N in D, thee ae two exact tiangles and mophisms between them, The identifications N / N jv VVVVVVVVVVVVV N /u th hhhhhhhhhhhhh Hom E (E,N ) / Hom E (E,N ) jv VVVVVVVV and the Five Lemma imply that Hom E (E /u, N ) th hhhhhhhh N = N = Hom E (E,N ), N /u = Hom E (E /u, N ). Lemma 3.3 now implies that each of the specta /u 1 /u 2 /u k satisfies conditions (A) and (B). 4 The Adams Spectal Sequence based on a egula quotient Fo an -module M,letM (s) denote the s-fold -smash powe of M, If M is an [X 1 ]-module, then M (s) = M M M. M (s) = M M M. [X 1 ] [X 1 ] [X 1 ] Let E = /I[X 1 ] be a localized egula quotient and u 1,u 2,... aegula sequence geneating I. We will discuss the Adams Spectal Sequence based on E. By emak 2.2, we can wok in the categoy of [X 1 ]-modules and eplace the Adams Spectal Sequence of S by that of S [X 1 ]. To simplify notation, fom now on we will eplace by [X 1 ] and theefoe assume that E = /I is a egula quotient of. Fist we identify the canonical Adams esolution giving ise to the Adams Spectal Sequence based on the egula quotient E = /I. We will elate this to a

13 On the Adams Spectal Sequence fo -modules 185 towe descibed by the second autho [12], but the eade should bewae that his notation fo I (s) is I s which we will use fo a diffeent spectum. Thee is a fibe sequence I /I and a towe of maps of -modules I I (2) I (s) I (s+1) in which I (s+1) I (s) is the evident composite I (s+1) I (s) = I (s). Setting /I (s) = cofibe(i (s) ), we obtain a towe /I /I (2) /I (s) /I (s+1) which we will efe to as the extenal I -adic towe. The next esult is immediate fom the definitions. Poposition 4.1 and We have D 0 S =, D s S = I (s), (s 1), K s S = /I (s+1) (s 0). It is not immediately clea how to detemine the limit L ES = holim /I (s). s Instead of doing this diectly, we will adopt an appoach suggested by Bousfield [10], making use of anothe E -nilpotent esolution, associated with the intenal I -adic towe to be descibed below. In ode to cay this out, we fist need to undestand convegence. We will see that the condition of Theoem 2.3 is satisfied fo a commutative egula quotient E = /I. Poposition 4.2 The E 2 -tem of the E -theoy Adams Spectal Sequence fo π S is E s,t 2 (S )=Coext s,t E E(E,E )=E [U i :i 1], whee bideg U i =(1, u i +1). Hence this spectal sequence collapses at its E 2 -tem E, 2 (S )=E, (S ) and conveges to π L E S.

14 186 Andew Bake and Andey Lazaev Poof By Poposition 1.2, E E =Λ (τ i :i 1), with geneatos τ i which ae pimitive with espect to the copoduct of this Hopf algeboid. The detemination of Coext, E E (E,E ) is now standad and the diffeentials ae tivial fo degee easons. Induction on the numbe of cells now gives Coollay 4.3 Fo a finite cell -module M,theE-theoy Adams Spectal Sequence fo π M conveges to π L EM. 5 The intenal I -adic towe Suppose that I is geneated by a egula sequence u 1,u 2,... We will often indicate a monomial in the u i by witing u (i1,...,i k ) = u i1 u ik. We will wite E = /I and make use of algebaic esults fom [5] which we now ecall in detail. Fo s 0, we define the -module I s /I s+1 to be the wedge of copies of E indexed on the distinct monomials of degee s in the geneatos u i. Fo an explanation of this, see Coollay 5.4. We will show that thee is an (intenal) I -adic towe of -modules /I /I 2 /I s /I s+1 so that fo each s 0 the fibe sequence coesponds to a cetain element of /I s /I s+1 I s /I s+1 Ext 1 ( /I,I s /I s s+1 ) in E 2 -tem of the Univesal Coefficient Spectal Sequence of [11] conveging to D (/I s,i s /I s+1 ). On setting I s = fibe( /I s ) we obtain anothe towe I I 2 I s I s+1

15 On the Adams Spectal Sequence fo -modules 187 which is analogous to the extenal vesion of [12]. A elated constuction appeaed in [3, 8] fo the case of = Ê(n) (which was shown to admit a not necessaily commutative S -algeba stuctue) and I = I n. Undelying ou wok is the classical Koszul esolution whee K, /I 0, K, =Λ (e i :i 1), which has gading given by deg e i = u i + 1 and diffeential d e i = u i, d(xy) =(dx)y+( 1) x d y (x K,,y K s, ). Hence (K,, d) is an -fee esolution of /I which is a diffeential gaded -algeba. Tensoing with /I and taking homology leads to a well known esult. Poposition 5.1 As an /I -algeba, To, ( /I, /I )=Λ /I (e i : i 1). Coollay 5.2 To, ( /I, /I ) is a fee /I -module. This is of couse closely elated to the topological esult Poposition 1.2. Now etuning to ou algebaic discussion, we ecall the following standad esult. Lemma 5.3 ([13], Theoem 16.2) Fo s 0, I s/is+1 is a fee /I -module with a basis consisting of esidue classes of the distinct monomials u (i1,...,i s) of degee s. Coollay 5.4 Fo s 0, thee is an isomophism of -modules π I s /I s+1 = I s /I s+1. Hence π I s /I s+1 is a fee /I -module with a basis indexed on the distinct monomials u (i1,...,i s) of degee s.

16 188 Andew Bake and Andey Lazaev Let U (s) be the fee -module on a basis indexed on the distinct monomials of degee s in the u i.fos 0, set Q (s), = K, U (s), d (s) Q =d 1, and also fo x K, wite Thee is an obvious augmentation Lemma 5.5 Fo s 1, xũ (i1,...,i s) = x u (i1,...,i s). Q (s), is a esolution by fee -modules. Q (s) 0, I s /Is+1. ε(s) I s /I s+1 0 Given a complex (C,, d C ), the k-shifted complex (C[k],, d C[k] ) is defined by C[k] n, = C n+k,, d C[k] =( 1) k d C. Thee is a mophism of chain complexes (s+1) : Q (s), Q (s+1) [ 1], ; (s+1) e i1 e i ũ (j1,...,j s) = ( 1) k e i1 ê ik e i ũ (j1,...,j s). k=1 Using the identification Q (s+1) [ 1] n, = Q (s+1) n 1,, we will often view (s+1) as a homomophism (s+1) : Q (s), Q (s+1), of bigaded -modules of degee 1. Thee ae also extenal paiings Q (), Q (s), Q (+s), ; xũ (i1,...,i s) yũ (j1,...,j s) xyũ (i1,...,i s,j 1,...,j s) (x, y K, ). In paticula, each Q (), is a diffeential module ove the diffeential gaded -algeba K (0), and (s+1) is a K (0), -deivation.

17 On the Adams Spectal Sequence fo -modules 189 Theoem 5.6 by fee -modules, whee Fo s 1, thee is a esolution K (s 1), ε (s 1) /I s 0, K (s 1), = Q (0), Q (1), Q (s 1),, and the diffeential is d (s 1) =(d (0) Q, (1) +d (1) Q, (2) +d (2) Q,..., (s 1) +d (s 1) Q ). In fact (K (s 1),, d (s 1) ) is a diffeential gaded -algeba which povides a multiplicative esolution of /I s, with the augmentation given by ε (s 1) (x 0,x 1 ũ i1,...,x s 1 ũ is 1 )=x 0 +x 1 u i1 + +x s 1 u is 1. The algebaic extension of -modules 0 /I s /I s+1 I s /Is+1 0 isclassifiedbyanelementof Ext 1 ( /I,I s /I s s+1 )=Hom D ( /I,I s /I s s+1 [ 1]), whee Hom D denotes mophisms in the deived categoy D of the ing [18]. This element is epesented by the composite (s) : K (s 1), poj Q (s 1), (s) Q (s) [ 1],. (5.1) The analogue of the next esult fo ungaded ings was poved in [5]; the poof is easily adapted to the gaded case. Poposition 5.7 Fo each s 2, the following complex is exact: To, ( /I, /I ) (1) To, ( /I,I /I 2 ) Theoem 5.8 Fo s 2, (2) (s 1) To, ( /I,I s 1 /I ). s To, ( /I, /I s )= /I coke (s 1). This is a fee /I -module and with its natual /I -algeba stuctue, To, ( /I, /I ) s has tivial poducts. Given this algebaic backgound, we can now constuct the I -adic towe.

18 190 Andew Bake and Andey Lazaev Theoem 5.9 Thee is a towe of -modules /I /I 2 /I s /I s+1 whose maps define fibe sequences /I s /I s+1 I s /I s+1 which in homotopy ealise the exact sequences of -modules 0 /I s /I s+1 I s /Is+1 0. Futhemoe, the following conditions ae satisfied fo each s 1. is a fee E -module and the unit induces a splitting E /Is = E (ke : E /Is E ); (ii) the pojection map /I s+1 /I s induces the zeo map (ke : E /I s+1 E ) (ke : E /I s E ); (i) E /I s (iii) the inclusion map j s : I s /I s+1 /I s+1 induces an exact sequence E I s 1 /I s (s) E I s /I s+1 js (ke : E /I s+1 E ) 0. Poof The poof is by induction on s. Assuming that /I s exists with the asseted popeties, we will define a suitable map δ s : /I s ΣI s /I s+1 which induces a fibe sequence of the fom /I s X (s+1) I s /I s+1, (5.2) fo which π X (s+1) = /I s+1 as an -module. If M is an -module which is an E module spectum, Theoem 3.2 povides a Univesal Coefficient Spectal Sequence E, 2 =Ext p,q E (E /Is,M )= D (/I s,m) p+q. Since E /I s is E -fee, this spectal sequence collapses to give In paticula, fo M = I s /I s+1, D (/I s,m) =Hom E (E /Is,M ). D (/I s,i s /I s+1 ) n =Hom n E (E /I s,i /I s s+1 ). By (5.1) and Theoem 5.6, thee is an element (s) Hom 0 E (E /Is,I s /Is+1 [ 1]) = Hom 1 E (E /Is,I s /Is+1 ),

19 On the Adams Spectal Sequence fo -modules 191 coesponding to an element δ s : /I s ΣI s /I s+1 inducing a fibe sequence as in (5.2). It still emains to veify that π X (s+1) = /I s+1 as an -module. Fo this, we will use the esolutions K (s 1), /I s 0 and K, /I 0. These fee esolutions give ise to cell -module stuctues on /I s and E.By[11],the-module E /I s admits a cell stuctue with cells in one-one coespondence with the elements of the obvious tenso poduct basis of K,. Hence thee is a esolution by fee -modules K (s 1), K, K (s 1), E /Is 0. Thee ae mophisms of chain complexes K (s 1), ρ s K, K (s 1), δ s Q (s), [ 1], whee ρ s is the obvious inclusion and δ s is a chain map lifting be chosen so that δ s (e i x) =0. (s) which can The effect of the composite δ s ρ s on the geneato e i ũ (j1,...,j s 1 ) K (s 1) 1, tuns out to be (s) e i ũ (j1,...,j s 1 ) = ũ (i,j1,...,j s 1 ), while the elements of fom e i ũ (j1,...,j k 1 ) with k<saeannihilated. The composite homomophism K (s 1) δ sρ s (s) 1, Q 0, [ 1] ε 1 I s /I s+1 [ 1] is a cocycle. Thee is a mophism of exact sequences 0 /I s K (s 1) 0, K (s 1) 1, K (s 1) 2, α 0 α 1 0 /I s /I s+1 I s /I s+1 0 whee the cohomology class [α 1 ] Ext 1, ( /I,I s /I s s+1 ) epesents the extension of -modules on the bottom ow. It is easy to see that [α 1 ]=[ε 1 δs ρ s ], hence this class also epesents the extension of -modules 0 /I s π X s+1 I s /Is+1 0.

20 o o 192 Andew Bake and Andey Lazaev Thee is a diagam of cofibe tiangles /I s+1 /I I/I s+1 o o I s 1 /I s+1 : t t t t t t t t t I/I 2 > ; x x x x x x x x x x I s 1 /I s 8 I s /I s+1 = I s /I s+1 and applying E ( ) we obtain a spectal sequence conveging to E /I s+1 whose E 2 -tem is the homology of the complex 0 E /I (1) E I/I 2 (2) E I 2 /I 3 (s) E I s /I s+1 0, whee the (k) ae essentially the maps used to compute To ( /I, /I s+1 ) in [5]. By Poposition 5.7 and Theoem 5.8, this complex is exact except at the ends, whee we have ke (1) = E. As a esult, this spectal sequence collapses at E 3 giving the desied fom fo E /Is+1. Coollay 5.10 Fo any E -module spectum N and s 1, N /Is = HomE (E /Is,N ). Poof This follows fom Theoem 5.9(i). We will also use the following esult. Coollay 5.11 Fo s 1, the natual map E /Is+1 E /Is, has image equal to E = E. Poof This follows fom Theoem 5.9(ii). Coollay 5.12 Fo any E -module spectum N and s 1, colim s N /Is = N = N. Poof This is immediate fom Coollaies 5.10 and 5.11 since colim s Hom E (E /I s,n ) =Hom E (E,N ).

21 On the Adams Spectal Sequence fo -modules The I-adic towe and Adams Spectal Sequence Continuing with the notation of Section 5, the fist substantial esult of this section is Theoem 6.1 has homotopy limit The I -adic towe /I /I 2 /I s /I s+1 holim s /I s L ES. Ou appoach follows ideas of Bousfield [10] whee it is shown that the following Lemma implies Theoem 6.1. Lemma 6.2 Let E = /I. Then the following ae tue. E -nilpotent. ii) Fo each E -nilpotent -module M, colim s D (/I s,m) =M. i) Each /I s is Poof (i) is poved by an easy induction on s 1. (ii) is a consequence of Coollay Since the maps /I s+1 /I s ae sujective, fom the standad exact sequence fo π ( ) of a homotopy limit we have π L E S = lim s /I s. (6.1) We can genealize this to the case whee E is a commutative localized egula quotient. Theoem 6.3 of. Then Let E = /I[X 1 ] be a commutative localized egula quotient π L E S = [X 1 ] Î = lim s [X 1 ]/I s. If the egula sequence geneating I is finite, then the natual map S L E S is an E -equivalence, hence L E S L E S, π L E S = [X 1 ] Î.

22 194 Andew Bake and Andey Lazaev Poof The fist statement is easy to veify. By emak 2.2, to simplify notation we may as well eplace by [X 1 ]and so assume that E = /I is a commutative egula quotient of. Using the Koszul complex (Λ (e j : j), d), we see that To, (E, ( ) Î )isthe homology of the complex Λ (e j : j) ( ) Î =Λ ( ) (e j :j) Î with diffeential d =d 1. Since the sequence u j emains egula in ( ) Î, this complex povides a fee esolution of E = /I as an ( ) Î -module (this is false if the sequence u j is not finite). Hence we have To, (E, ( ) Î )=To ( )Î, (E,( ) Î )=E. To calculate E L E S we may use the Künneth Spectal Sequence of [11], E s,t 2 =To s,t (E, L E S )= Es+t L E S. By the fist pat, the E 2 -tem is To, (E, ( ) Î )=E =E. Hence the natual homomophism is an isomophism. E S E L E S If the sequence u j is infinite, the calculation of this poof shows that E L ES =( ) Î /I /I = E S and the Adams Spectal Sequence does not convege to the homotopy of the E -localization. An induction on the numbe of cells of M poves a genealization of Theoem 6.3. Theoem 6.4 Let E be a commutative localized egula quotient of and M a finite cell -module. Then π L E M = M [X 1 ] Î = [X 1 ] Î M. If the egula sequence geneating I is finite, then the natual map M L E M is an E -equivalence, hence L E M L EM, π L E M = M [X 1 ] Î = [X 1 ] Î M.

23 On the Adams Spectal Sequence fo -modules 195 The eade may wonde if the following conjectue is tue, the algebaic issue being that it does not appea to be tue that fo a commutative ing A, the extension A A Ĵ is always flat fo an ideal J A, a Noetheian condition nomally being equied to establish such a esult. Conjectue 6.5 The conclusion of Theoem 6.4 holds when E is any commutative localized quotient of. 7 Some examples associated with MU An obvious souce of commutative localized egula quotients is the commutative S -algeba = MU and we will descibe some impotant examples. It would appea to be algebaically simple to wok with BP at a pime p in place of MU, but at the time of witing, it seems not to be known whethe BP admits a commutative S -algeba stuctue. Example A: MU H F p. Let p be a pime. By consideing the Eilenbeg-Mac Lane spectum HF p as a commutative MU-algeba [11], we can fom HF p HF p. The Künneth MU Spectal Sequence gives E 2 MU s,t =ToMU s,t (F p, F p )= HF p s+t HF p. Using a Koszul complex ove MU, it is staightfowad to see that E 2, =Λ Fp (τ j :j 0), the exteio algeba ove F p with geneatos τ j E 2 1,2j. Taking = MU and E = HF p, we obtain a spectal sequence E s,t 2 (MU)=Coexts,t Λ Fp (τ j :j 0) (F p, F p )= π s+t LMU HF p S MU, whee I MU is geneated by p togethe with all positive degee elements, so MU /I = F p.also, π LMU HF p S MU =(MU ) Î. Moe geneally, fo a finite cell MU-module M, the Adams Spectal Sequence has the fom whee E s,t 2 (M) =Coexts,t Λ Fp (τ j :j 0) (F MU p,hf p π LMU HF p M =(M ) Î. M)= π s+t LMU HF p M,

24 196 Andew Bake and Andey Lazaev Example B: MU E (n). By [11, 16], the Johnson-Wilson spectum E(n) atanodd pime p is a commutative MU-ing spectum. Accoding to poposition 2.10 of [16], at the pime 2 a cetain modification of the usual constuction also yields a commutative MUing spectum which we will still denote by E(n) athe than Stickland s E(n). In all cases we can fom the commutative MU-ing spectum E(n) E(n)and MU thee is a Künneth Spectal Sequence E 2 s,t =To MU s,t (E(n),E(n) )= E(n) MU s+t E(n). By using a Koszul complex fo MU n ove MU and localizing at v n,we find that E 2, =Λ E(n) (τ j :j 1andj p k 1with1 k n), whee Λ denotes an exteio algeba and τ j E 2 1,2j.So E(n) MU E(n)=Λ E(n) (τ j :j 1andj p k 1with1 k n) as an E(n) -algeba. When = MU and E = E(n), we obtain a spectal sequence E s,t 2 (MU)=Coexts,t Λ E(n) (τ j :j n+1) (E(n),E(n) )= π s+t LMU E(n)MU, whee π LMU E(n) MU =(MU ) (p) [vn 1 ]Ĵ n+1 and J n+1 =(ke:(mu ) (p) [vn 1 ] E (n) ) MU [vn 1 ]. In the E 2 -tem we have E s,t 2 (MU)=E(n) [U j :0 j p k 1fo0 k n], with geneato U j E 1,2j+1 2 (MU) coesponding to an exteio geneato in E(n) MU E(n) associated with a polynomial geneato of MU in degee 2j lying in ke MU E (n). Moe geneally, fo a finite cell MU-module M, E s,t 2 (M) =Coexts,t whee Λ E(n) (τ j :j n+1) (E(n),E(n) MU π LMU E(n)M = M Ĵn+1 =(MU ) (p) [vn 1 ] Ĵn+1 M. MU M),= π s+t L MU E(n) M,

25 On the Adams Spectal Sequence fo -modules 197 Example C: MU K (n). We know fom [11, 16] that fo an odd pime p, the spectum K(n) epesenting the n th Moava K -theoy K(n) ( ) is a commutative MU ing spectum. Thee is a Künneth Spectal Sequence E 2 s,t =ToMU s,t (K(n),K(n) )= K(n) MU s+t K(n), and we have E 2, =Λ K(n) (τ j :0 j p n 1). Taking = MU and E = K(n), we obtain a spectal sequence E s,t 2 (MU)=Coexts,t Λ K(n) (τ j :0 j n) (K(n),K(n) )= π s+t LMU K(n)MU, whee π LMU K(n)MU =(MU ) În, with I n, =kemu K (n).inthee 2 -tem we have E s,t 2 (MU)=E(n) [U j :0 j p n 1], with geneato U j E 1,2j+1 2 (MU) coesponding to an exteio geneato in E(n) MU E(n) associated with a polynomial geneato of MU in degee 2k lying in ke MU E (n) (o when j = 0, associated with p). Moe geneally, fo a finite cell MU-module M, E s,t 2 (M) =Coexts,t whee Λ K(n) (τ j :0 j n) (K(n),K(n) MU π LMU K(n)M =(M ) În, =(MU ) În, M. MU M)= π s+t LMU K(n)M, Concluding emaks Thee ae seveal outstanding issues aised by ou wok. Apat fom the question of whethe it is possible to weaken the assumptions fom (commutative) egula quotients to a moe geneal class, it seems easonable to ask whethe the intenal I -adic towe is one of ing specta. Since L E = holim /I s (at least when I is finitely geneated), the localization s theoy of [11, 19] shows that this can be ealized as a commutative -algeba.

26 198 Andew Bake and Andey Lazaev Howeve, showing that each /I s is an ing spectum o even an -algeba seem to involve fa moe inticate calculations. We expect that this will tun out to be tue and even that the towe is one of -algebas. This should involve techniques simila to those of [12, 6]. It is also woth noting that ou poofs make no distinction between the cases whee I is infinitely o finitely geneated. Thee ae a numbe of algebaic simplifications possible in the latte case, howeve we have avoided using them since the most inteesting examples we know ae associated with infinitely geneated egula ideals in MU. The specta E n of Hopkins, Mille et al. have Noetheian homotopy ings and thee ae towes based on powes of thei maximal ideals simila to those in the fist autho s pevious wok [3, 8]. We also hope that ou peliminay exploation of Adams Spectal Sequences fo -modules will lead to futhe wok on this topic, paticulaly in the case = MU and elated examples. A moe ambitious poject would be to investigate the commutative S -algeba MSp fom this point of view, pehaps ewoking the esults of Veshinin, Gobounov and Botvinnik in the context of MSpmodules [9, 17]. efeences [1] J. F. Adams, Stable Homotopy and Genealised Homology, Univesity of Chicago Pess (1974). [2] M. F. Atiyah, Vecto bundles and the K=FCnneth fomula, Topology 1 (1962), [3] A. Bake, A stuctues on some specta elated to Moava K -theoy, Quat. J. Math. Oxf. 42 (1991), [4], Bave new Hopf algeboids and the Adams spectal sequence fo - modules, Glasgow Univesity Mathematics Depatment pepint 00/12; available fom ajb/dvi-ps.html. [5], On the homology of egula quotients, Glasgow Univesity Mathematics Depatment pepint 01/1; available fom ajb/dvi-ps.html. [6] A. Bake & A. Jeanneet, Bave new Hopf algeboids and extensions of MUalgebas, Glasgow Univesity Mathematics Depatment pepint 00/18; available fom ajb/dvi-ps.html. [7], Bave new Bockstein opeations, in pepaation. [8] A. Bake & U. Wügle, Bockstein opeations in Moava K -theoy, Foum Math. 3 (1991),

27 On the Adams Spectal Sequence fo -modules 199 [9] B. Botvinnik, Manifolds with singulaities and the Adams-Novikov spectal sequence, Cambidge Univesity Pess (1992). [10] A. K. Bousfield, The localization of specta with espect to homology, Topology 18 (1979), [11] A.Elmendof,I.Kiz,M.Mandell&J.P.May,ings, modules, and algebas in stable homotopy theoy, Ameican Mathematical Society Mathematical Suveys and Monogaphs 47 (1999). [12] A. Lazaev, Homotopy theoy of A ing specta and applications to MUmodules, toappeaink-theoy. [13] H. Matsumua, Commutative ing Theoy, Cambidge Univesity Pess, (1986). [14] D. C. avenel, Localization with espect to cetain peiodic homology theoies, Ame. J. Math. 106 (1984), [15], Complex Cobodism and the Stable Homotopy Goups of Sphees, Academic Pess (1986). [16] N. P. Stickland, Poducts on MU-modules, Tans. Ame. Math. Soc. 351 (1999), [17] V. V. Veshinin, Cobodisms and spectal sequences, Tanslations of Mathematical Monogaphs 130, Ameican Mathematical Society (1993). [18] C. A. Weibel, An Intoduction to Homological Algeba, Cambidge Univesity Pess (1994). [19] J. J. Wolbet, Classifying modules ove K -theoy specta, J. Pue Appl. Algeba 124 (1998), Mathematics Depatment, Glasgow Univesity, Glasgow G12 8QW, UK. Mathematics Depatment, Bistol Univesity, Bistol BS8 1TW, UK. a.bake@maths.gla.ac.uk and a.lazaev@bis.ac.uk UL: ajb and pue/staff/maxal/maxal eceived: 19 Febuay 2001 evised: 4 Apil 2001