On the Adams Spectral Sequence for R-modules

We consider the Adams Spectral Sequence for R-modules based on commutative localized regular quotient ring spectra of a commutative S-algebra R in the sense of Elmendorf, Kriz, Mandell, May and Strickland. The formulation of this spectral sequence is similar to the classical case, and we reduce to algebra involving the cohomology of certain ‘brave new Hopf algebroids’ E ∗ E. In order to work out the details we resurrect Adams’ original approach to Universal Coefficient Spectral Sequences for modules over an R ring spectrum. We show that the Adams Spectral Sequence for SR based on E = R/I[X−1] converges to the homotopy of the E-nilpotent completion which has homotopy π∗b L ESR = R∗[X−1]bI∗ . We also show that b LRESR is equivalent to L E SR, the Bousfield localization of SR with respect to E-theory. This seems surprising since the spectral sequence collapses at E2, but Er does not have a vanishing line because of the presence of polynomial generators of positive cohomological degree, thus only one of Bousfield’s two standard convergence criteria applies here even though we have this equivalence. The details involve a construction of the internal I-adic tower R/I ←− R/I ←− · · · ←− R/I ←− R/I ←− · · · whose homotopy limit is b LRESR. Finally, we describe some examples for the case R = MU . Introduction In this note we consider the Adams Spectral Sequence for R-modules based on localized regular quotient ring spectra of a commutative S-algebra R in the sense of [9, 14] and we make systematic use of ideas and notation from those two sources. This work grew out of a preprint of the first author [3] and the related work of [5]; it is also related to ongoing collaboration with Alain Jeanneret on Bockstein operations in cohomology theories on R-modules [6]. One slightly surprising phenomenon we uncover concerns the convergence of the Adams Spectral Sequence based on E = R/I[X−1], a commutative localized regular quotient of a commutative S-algebra R. We show that the spectral sequence for π∗SR collapses at E2, but Er does not have a vanishing line because of the presence of polynomial generators of positive cohomological degree. Thus only one of Bousfield’s two convergence criteria [8] (see Theorems 2.3 and 2.4) apply here. However we still find that the spectral sequence converges to π∗ LE SR, where L R E is the Bousfield localization functor with respect to E-theory on the category of R-modules and π∗ LE SR = R∗[X ]̂I∗ , the I∗-adic completion of R∗[X], and that LE SR ' ̂ L R ESR, the E-nilpotent completion. Background assumptions, terminology and technology. We work in a good category of spectra S such as the category of L-spectra of [9]. Associated to this is the subcategory of S-modules MS and its derived category DS . Glasgow University Mathematics Department preprint no. 01/2 [Version 12: 8/1/2001] The first author would like to thank the University and City of Bern for providing such a hospitable environment during many visits, also Alain Jeanneret and Urs Würgler and the other participants in the Topology working seminar during spring and summer 2000. 1 2 ANDREW BAKER & ANDREJ LAZAREV Throughout, R will denote a commutative S-algebra in the sense of [9]. There is an associated full subcategory MR of MS consisting of the R-modules, and its derived homotopy category DR. For R-modules M and N , we set MR ∗ N = π∗M∧RN, N ∗ RM = DR(M, N) ∗, where DR(M,N) = DR(M, ΣnN). After Strickland [14], we will use the following terminology. If the homotopy ring R∗ = π∗R is concentrated in even degrees, a localized quotient of R will be an R ring spectrum of the form R/I[X−1]. A localized quotient is commutative if it is a commutative R ring spectrum. A localized quotient R/I[X−1] is regular if the ideal I∗ / R∗ is generated by a regular sequence. The ideal I∗ / R∗ extends to an ideal of R∗[X] which we will again denote by I∗; then as R-modules, R/I[X−1] ' R[X−1]/I. We will make use of the language and ideas of algebraic derived categories of modules over a commutative ring, mildly extended to deal with evenly graded rings and their modules. In particular, this means that chain complexes are often bigraded (or even multigraded) objects with their first grading being homological and the second and higher ones being internal. 1. Brave new Hopf algebroids and their cohomology If E is a commutative R-ring spectrum, the smash product E∧ R E is a commutative R-ring spectrum. It is also naturally an E-algebra spectrum in two ways induced from the left and right units E ∼= −−→ E∧ R R −→ E∧ R E ←− E∧ R R ∼ = ←−− E. Theorem 1.1. Let ER ∗ E be flat as a left or equivalently right E∗-module. Then i) (E∗, ER ∗ E) is a Hopf algebroid over R∗; ii) for any R-module M , ER ∗ M is a left E R ∗ E-comodule. Proof. This is proved using essentially the same argument as in [1, 13]. The natural map E∧ R M ∼ = −−→ E∧ R R∧ R M −→ E∧ R E∧ R M induces the coaction ψ : ER ∗ M −→ π∗E∧RE∧RM ∼= −−→ ER ∗ E⊗ E∗ ER ∗ M, which uses an isomorphism π∗E∧ R E∧ R M ∼= ER ∗ E⊗ E∗ ER ∗ M. that follows from the flatness condition. ƒ For later use we record the following general result on the Hopf algebroids associated to commutative regular quotients. A number of examples associated with the case R = MU are discussed in Section 7. Proposition 1.2. Let E = R/I be a commutative regular quotient. Then as an E∗-algebra, ER ∗ E = ΛE∗(τi : i > 1). Moreover, the generators τi are primitive with respect to the coaction, so ER ∗ E is a primitively generated Hopf algebra over E∗. Dually, as an E∗-algebra, E∗ RE = ̂ ΛE∗(Q i : i > 1), where Qi is the Bockstein operation dual to τi and ̂ ΛE∗( ) indicates the completed exterior algebra generated by the anti-commuting Qi elements. ON THE ADAMS SPECTRAL SEQUENCE FOR R-MODULES 3 Proof. The algebra structure follows from the Künneth Spectral Sequence for R-modules [9], E2 p,q = Tor R∗ p,q(E∗, E∗) =⇒ ER p+qE. As in Proposition 5.2, with the aid of a Koszul resolution we obtain E2 ∗,∗ = ΛE∗(ei : i > 1), The generators have bidegree bideg ei = (1, |ui|), so the differentials on the generators ei are trivial for dimensional reasons. By [10] Lemma 10.1, this spectral sequence is multiplicative and hence it collapses, showing that ER ∗ E = ΛE∗(τi : i > 1), with each generator τi represented by ei in the spectral sequence and having degree deg τi = |ui|+ 1. For each i, (R/ui)∗ (R/ui) = ΛR∗/(ui)(τ ′ i) with deg τ ′ i = |ui| + 1. Under the coproduct, τ ′ i is primitive for degree reasons. By comparing the two Künneth Spectral Sequences we find that τi ∈ ER ∗ E can be chosen to be the image of τ ′ i under the evident ring homomorphism (R/ui)∗ (R/ui) −→ ER ∗ E, which is actually a morphism of Hopf algebroids over R∗. Hence τi is coaction primitive in ER ∗ E. For E∗ RE, we construct the Bockstein operation Q i using the composition R/ui −→ Σ|ui|+1R −→ ΣiR/ui to induce a map E −→ Σ|ui|+1E, then use the Koszul resolution to determine the Universal Coefficient Spectral sequence E 2 = Ext p,q R∗(E∗, E∗) =⇒ E p+q R E which collapses at E2. For details on the construction of these operations, see [14, 6]. ƒ Corollary 1.3. i) The natural map E∗ = ER ∗ R −→ ER ∗ E induced by the unit R −→ R/I is a split monomorphism of E∗-modules. ii) ER ∗ E is a free E∗-module. Proof. An explicit splitting as in (i) is obtained using the multiplication map E∧ R E −→ E which induces a homomorphism of E∗-modules ER ∗ E −→ E∗. ƒ To denote the cohomology of such Hopf algebroids we will use Coext rather than the more usual Ext since we will also make heavy use of Ext groups for modules over rings; more details of the definition and calculations can be found in [1, 13]. We recall that for ER ∗ E-comodules L∗ and M∗ where L∗ is E∗-projective, we define Coext s,t ER ∗ E (L∗,M∗) as follows. Consider a resolution 0 → M∗ −→ J0,∗ −→ J1,∗ −→ · · · −→ Js,∗ −→ · · · in which each Js,∗ is a summand of an extended comodule of the form ER ∗ E ƒ E∗ Ns,∗, where Ns,∗ is an E∗-module. Then the complex 0 → HomER ∗ E(L∗, J0,∗) −→ Hom ∗ ER ∗ E (L∗, J1,∗) −→ · · · −→ HomER ∗ E(L∗, Js,∗ −→ · · · has cohomology Hs(Hom∗ ER ∗ E(L∗, J∗,∗)) = Coext s,∗ ER ∗ E (L∗,M∗). The functors Coexts,∗ ER ∗ E(L∗, ) are the right derived functors of the left exact functor M∗ Hom∗ ER ∗ E(L∗,M∗) 4 ANDREW BAKER & ANDREJ LAZAREV on the category of left ER ∗ E-comodules. When L∗ = E∗, in analogy with [13], we have Coexts,∗ ER ∗ E(E∗,M∗) = Cotor (E∗, M∗). 2. The Adams Spectral Sequence for R-modules We will describe the E-theory Adams Spectral Sequence in the homotopy category of Rmodule spectra. As in the classical case of R = S, the sphere spectrum, it turns out that the E2-term is built up from the Coext functors. Let L,M be R-modules and E a commutative R-ring spectrum with ER ∗ E flat as a left or right E∗-module. Theorem 2.1. If ER ∗ L is projective as an E∗-module, there is an Adams Spectral Sequence with E 2 (L,M) = Coext s,t ER ∗ E (ER ∗ L,E R ∗ M). Proof. The proof follows that of Adams [1], replacing the sphere spectrum S with SR ' R and working in the derived category DR throughout. The Adams resolution of M is built up in the usual way by splicing together cofibre triangles as in the following diagram. M @ @ @ @ @ @ @ E∧ R M oo $$ I I I I I I E∧ R E∧ R M o o