Commutative S-algebra Research Articles

Adams filtration and generalized Hurewicz maps for infinite loopspaces

We study the Hurewicz map $$\begin{aligned} h_*: \pi _*(X) \rightarrow R_*(\Omega ^{\infty }X) \end{aligned}$$ where $$\Omega ^{\infty }X$$ is the 0th space of a spectrum X, and $$R_*$$ is the generalized homology theory associated to a connective commutative S-algebra R. We prove that the decreasing filtration of the domain associated to an R-based Adams resolution is compatible with a filtration of the range associated to the augmentation ideal filtration of the augmented commutative S-algebra $$\Sigma ^{\infty }(\Omega ^{\infty }X)_+$$ . The proof of our main theorem makes much use of composition properties of this filtration and its interaction with Topological Andre–Quillen homology. An application is a Connectivity Theorem: localize away from $$(p-1)!$$ and suppose X is $$(c-1)$$ -connected with $$c>0$$ . If $$\alpha \in \pi _*(X)$$ has Adams filtration s and $$|\alpha | < cp^s$$ , then $$h_*(\alpha )=0 \in R_*(\Omega ^{\infty }X)$$ . When specialized to mod p homology, this implies a Finiteness Theorem: if $$H^*(X;{{\mathbb {Z}}}/p)$$ is finitely presented as a module over the Steenrod algebra, then the image of the Hurewicz map in $$H_*(\Omega ^{\infty }X;{{\mathbb {Z}}}/p)$$ is finite. We illustrate these theorems with calculations of the mod 2 Hurewicz image of BO, its connected covers, and $$\Omega ^{\infty }tmf$$ , and the mod p Hurewicz image of all the spaces in the BP and $$BP\langle n \rangle $$ spectra. En route, we get new proofs of theorems of Milnor and Wilson. In the special case when X is the suspension spectrum of a space Z and $$R = H{{\mathbb {Z}}}/2$$ , we recover results announced by Lannes and Zarati in the 1980s, relating the Adams filtration of $$\pi _*^S(Z)$$ to Dyer-Lashof length in $$H_*(QZ;{{\mathbb {Z}}}/2)$$ , and generalize them to all primes p. For any X, we also get parallel results for the Hurewicz map for Morava E-theory, where $$\pi _*(X)$$ is now given the Adams–Novikov filtration.

Relative Loday constructions and applications to higher [formula omitted]-calculations

We define a relative version of the Loday construction for a sequence of commutative S-algebras A→B→C and a pointed simplicial subset Y⊂X. We use this to construct several spectral sequences for the calculation of higher topological Hochschild homology and apply those for calculations in some examples that could not be treated before.

Operad bimodules and composition products on André–Quillen filtrations of algebras

If O is a reduced operad in symmetric spectra, an O-algebra I can be viewed as analogous to the augmentation ideal of an augmented algebra. Implicit in the literature on Topological Andre-Quillen homology is that such an I admits a canonical (and homotopically meaningful) decreasing O-algebra filtration I > I^2 > I^3 > ... satisfying various nice properties analogous to powers of an ideal in a ring. In this paper, we are explicit about these constructions. With R a commutative S-algebra, we study derived versions of the circle product M o_O I, where M is an O-bimodule, and I is an O-algebra in R-modules. Letting M run through a decreasing O-bimodule filtration of O itself then yields the augmentation ideal filtration as above. The composition structure of the operad induces algebra maps from (I^i)^j to I^{ij}, fitting nicely with previously studied structure. As a formal consequence, an O-algebra map from I to J^d induces compatible maps from I^n to J^{dn}, for all n. This is an essential tool in the first author's study of Hurewicz maps for infinite loop spaces, and its utility is illustrated here with a lifting theorem.

Covering homology

We introduce the notion of covering homology of a commutative S-algebra with respect to certain families of coverings of topological spaces. The construction of covering homology is extracted from Bökstedt, Hsiang and Madsen's topological cyclic homology. In fact covering homology with respect to the family of orientation preserving isogenies of the circle is equal to topological cyclic homology. Our basic tool for the analysis of covering homology is a cofibration sequence involving homotopy orbits and a restriction map similar to the restriction map used in Bökstedt, Hsiang and Madsen's construction of topological cyclic homology. Covering homology with respect to families of isogenies of a torus is constructed from iterated topological Hochschild homology. It receives a trace map from iterated algebraic K-theory and there is a hope that the rich structure, and the calculability of covering homology will make it useful in the exploration of J. Rognes' “red shift conjecture”.

On E∞ -structure of the generalized Chern character

In this note we show that we can lift the generalized Chern character to a morphism of commutative S-algebras. Furthermore, we show that we can take a lifting that is equivariant with respect to the action of a Morava stabilizer group.

Topological Hochschild homology of Thom spectra which are E ∞ -ring spectra

We identify the topological Hochschild homology (THH) of the Thom spectrum associated to an E ∞ classifying map X → BG for G an appropriate group or monoid (e.g. U, O, and F). We deduce the comparison from the observation of McClure, Schwanzl, and Vogt that THH of a cofibrant commutative S-algebra (E ∞ -ring spectrum) R can be described as an indexed colimit together with a verification that the Lewis-May operadic Thom spectrum functor preserves indexed colimits and is in fact a left adjoint. We prove a splitting result THH(M f) ≃ eq Mf ∧ BX + , which yields a convenient description of THH(MU). This splitting holds even when the classifying map f: X → BG is only a homotopy commutative A ∞ map, provided that the induced multiplication on Mf extends to an E ∞ -ring structure; this permits us to recover Bokstedt's calculation of THH(Hℤ).

Topological André-Quillen homology for cellular commutative S-algebras

Topological André-Quillen homology for commutative S-algebras was introduced by Basterra following work of Kriz, and has been intensively studied by several authors. In this paper we discuss it as a homology theory on CW commutative S-algebras and apply it to obtain results on minimal atomic p-local S-algebras which generalise those of Baker and May for p-local spectra and simply connected spaces. We exhibit some new examples of minimal atomic commutative S-algebras.

Galois extensions of Lubin-Tate spectra

Let En be the n-th Lubin-Tate spectrum at a prime p. There is a commutative S-algebra E nr n whose coecients are built from the coecients of En and contain all roots of unity whose order is not divisible by p. For odd primes p we show that E nr n does not have any non-trivial connected finite Galois extensions and is thus separably closed in the sense of Rognes. At the prime 2 we prove that there are no non-trivial connected Galois extensions of E nr with Galois group a finite group G with cyclic quotient. Our results carry over to the K(n)-local context.

Uniqueness of $E_\infty$ structures for connective covers

We refine our earlier work on the existence and uniqueness of E∞ structures on K- theoretic spectra to show that at each prime p, the connective Adams summand l has a unique structure as a commutative S-algebra. For the p-completion lp we show that the McClure- Staffeldt model for lp is equivalent as an E∞ ring spectrum to the connective cover of the periodic Adams summand Lp. We establish a Bousfield equivalence between the connective cover of the Lubin-Tate spectrum En and BPh ni .

Realizability of algebraic Galois extensions by strictly commutative ring spectra

We discuss some of the basic ideas of Galois theory for commutative S-algebras originally formulated by John Rognes. We restrict our attention to the case of finite Galois groups and to global Galois extensions. We describe parts of the general framework developed by Rognes. Central roles are played by the notion of strong duality and a trace mapping constructed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Galois extensions to topological ones by applying obstruction theories of Robinson and Goerss-Hopkins to produce topological models for algebraic Galois extensions and the necessary morphisms of commutative S-algebras. Examples such as the complex $ K-theory spectrum as a KO-algebra indicate that more exotic phenomena occur in the topological setting. We show how in certain cases topological abelian Galois extensions are classified by the same Harrison groups as algebraic ones, and this leads to computable Harrison groups for such spectra. We end by proving an analogue of Hilbert's theorem 90 for the units associated with a Galois extension.

Hopf algebra structure on topological Hochschild homology

The topological Hochschild homology THH(R) of a commutative S-algebra (E_infty ring spectrum) R naturally has the structure of a commutative R-algebra in the strict sense, and of a Hopf algebra over R in the homotopy category. We show, under a flatness assumption, that this makes the Boekstedt spectral sequence converging to the mod p homology of THH(R) into a Hopf algebra spectral sequence. We then apply this additional structure to the study of some interesting examples, including the commutative S-algebras ku, ko, tmf, ju and j, and to calculate the homotopy groups of THH(ku) and THH(ko) after smashing with suitable finite complexes. This is part of a program to make systematic computations of the algebraic K-theory of S-algebras, by means of the cyclotomic trace map to topological cyclic homology.

Differentials in the homological homotopy fixed point spectral sequence

We analyze in homological terms the homotopy fixed point spectrum of a T-equivariant commutative S-algebra R. There is a homological homotopy fixed point spectral sequence with E^2_{s,t} = H^{-s}_{gp}(T; H_t(R; F_p)), converging conditionally to the continuous homology H^c_{s+t}(R^{hT}; F_p) of the homotopy fixed point spectrum. We show that there are Dyer-Lashof operations beta^epsilon Q^i acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the $^{2r}-term of the spectral sequence there are 2r other classes in the E^{2r}-term (obtained mostly by Dyer-Lashof operations on x) that are infinite cycles, i.e., survive to the E^infty-term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra R = THH(B) of many S-algebras, including B = MU, BP, ku, ko and tmf. Similar results apply for all finite subgroups C of T, and for the Tate- and homotopy orbit spectral sequences. This work is part of a homological approach to calculating topological cyclic homology and algebraic K-theory of commutative S-algebras.

Localization of André–Quillen–Goodwillie towers, and the periodic homology of infinite loopspaces

Let K ( n ) be the n th Morava K -theory at a prime p, and let T ( n ) be the telescope of a v n -self map of a finite complex of type n. In this paper we study the K ( n ) * -homology of Ω ∞ X , the 0th space of a spectrum X, and many related matters. We give a sampling of our results. Let P X be the free commutative S-algebra generated by X: it is weakly equivalent to the wedge of all the extended powers of X. We construct a natural map s n ( X ) : L T ( n ) P ( X ) → L T ( n ) Σ ∞ ( Ω ∞ X ) + of commutative algebras over the localized sphere spectrum L T ( n ) S . The induced map of commutative, cocommutative K ( n ) * -Hopf algebras s n ( X ) * : K ( n ) * ( P X ) → K ( n ) * ( Ω ∞ X ) , satisfies the following properties. It is always monic. It is an isomorphism if X is n-connected, π n + 1 ( X ) is torsion, and T ( i ) * ( X ) = 0 for 1 ⩽ i ⩽ n - 1 . It is an isomorphism only if K ( i ) * ( X ) = 0 for 1 ⩽ i ⩽ n - 1 . It is universal. The domain of s n ( X ) * preserves K ( n ) * -isomorphisms, and if F is any functor preserving K ( n ) * -isomorphisms, then any natural transformation F ( X ) → K ( n ) * ( Ω ∞ X ) factors uniquely through s n ( X ) * . The construction of our natural transformation uses the telescopic functors constructed and studied previously by Bousfield and the author, and thus depends heavily on the Nilpotence Theorem of Devanitz, Hopkins, and Smith. Our proof that s n ( X ) * is always monic uses Topological André-Quillen Homology and Goodwillie Calculus in nonconnective settings.

Brave new Hopf algebroids and extensions of $MU$-algebras

We apply recent work of A. Lazarev which develops an obstruction theory for the existence of R-algebra structures on Rmodules, where R is a commutative S-algebra. We show that certain MU-modules have such A1 structures. Our results are often simpler to state for the related BP-modules under the currently unproved assumption that BP is a commutative Salgebra. Part of our motivation is to clarify the algebra involved in Lazarev’s work and to generalize it to other important cases. We also make explicit the fact that BP admits an MU-algebra structure as do E(n) and [ E(n), in the latter case rederiving and

Algebraic K-theory of topological K-theory

We are interested in the arithmetic of ring spectra. To make sense of this we must work with structured ring spectra, such as S-algebras [EKMM], symmetric ring spectra [HSS] or Γ-rings [Ly]. We will refer to these as Salgebras. The commutative objects are then commutative S-algebras. The category of rings is embedded in the category of S-algebras by the Eilenberg– MacLane functor R →HR. We may therefore view an S-algebra as a generalization of a ring in the algebraic sense. The added flexibility of S-algebras provides room for new examples and constructions, which may eventually also shed light upon the category of rings itself. In algebraic number theory the arithmetic of the ring of integers in a number field is largely captured by its Picard group, its unit group and its Brauer group. These are

On the Adams spectral sequence forR–modules

We discuss the Adams Spectral Sequence for R-modules based on commutative localized regular quotient ring spectra over 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 the calculation of its E_2-term involves the cohomology of certain `brave new Hopf algebroids' E^R_*E. In working 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 S_R based on a commutative localized regular quotient R ring spectrum E=R/I[X^{-1}] converges to the homotopy of the E-nilpotent completion pi_*hat{L}^R_ES_R=R_*[X^{-1}]^hat_{I_*}. We also show that when the generating regular sequence of I_* is finite, hatL^R_ES_R is equivalent to L^R_ES_R, the Bousfield localization of S_R with respect to E-theory. The spectral sequence here collapses at its E_2-term but it 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 the construction of an I-adic tower R/I <-- R/I^2 <-- ... <-- R/I^s <-- R/I^{s+1} <-- ... whose homotopy limit is hatL^R_ES_R. We describe some examples for the motivating case R=MU.

Idempotents and Landweber exactness in brave new algebra

We explain how idempotents in homotopy groups give rise to splittings of homotopy categories of modules over commutative S-algebras, and we observe that there are naturally occurring equivariant examples involving idempotents in Burnside rings. We then give a version of the Landweber exact functor theorem that applies to MU-modules. In 1997, not long after [6] was written, I gave an April Fool’s talk on how to prove that BP is an E1 ring spectrum or equivalently, in the language of [6], a commutative S-algebra. Unfortunately, the problem of whether or not BP is an E1 ring spectrum remains open. However, two interesting remarks emerged and will be presented here. One concerns splittings along idempotents and the other concerns the Landweber exact functor theorem. One of the nicest things in [6] is its one line proof that KO and KU are commutative S-algebras. This is an application of the following theorem [6, VIII.2.2], or rather the special case that follows. Theorem 1. Let R be a cell commutative S-algebra, A be a cell commutative Ralgebra, and M be a cell R-module. Then the BousÞeld localization i : A A! AM of A at M can be constructed as the inclusion of a subcomplex in a cell commutative R-algebra. In particular, the commutative R-algebra AM is a commutative S-algebra by neglect of structure.

André–Quillen cohomology of commutative S-algebras

We define topological André–Quillen cohomology of commutative S-algebras and construct a spectral sequence that calculates the André–Quillen cohomology of commutative S-algebras over H F p .

Higher differential algebras of discrete valuation rings

Let S be a commutative ring with identity. Let T be a commutative S-algebra. A8(T) denotes the higher differential algebra over S, in the sense of Berger [1] or Kawahara-Yokoyama [5], with an index set consisting of all nonnegative integers. {dr/#}n=o,i,2,... denotes the canonical higher ^-derivation of T into AS(T). In case S is the ring of all integers, we use simplified notations A(T) and d? (n=0, 1, 2, • • •)• Let R denote a complete discrete valuation ring, of a valuation v of unequal characteristic with maximal ideal m = (n). Assume that the characteristic of k=Rjm is p. Let P be a coefficient ring of R. Let {cj ter be a ^-independent base of k and let ct be a representative of ct chosen from P for every i e I The symbol A means the/?-adic completion of P-algebra. By arguments developed by Berger or Kawahara-Yokoyama in the cited papers, and formal smoothness and flatness of P over the prime local ring, we can deduce the following theorem.

Higher differential algebras of discrete valuation rings

Let S be a commutative ring with identity. Let T be a commutative S-algebra. A8(T) denotes the higher differential algebra over S, in the sense of Berger [1] or Kawahara-Yokoyama [5], with an index set consisting of all nonnegative integers. {dr/#}n=o,i,2,... denotes the canonical higher ^-derivation of T into AS(T). In case S is the ring of all integers, we use simplified notations A(T) and d? (n=0, 1, 2, • • •)• Let R denote a complete discrete valuation ring, of a valuation v of unequal characteristic with maximal ideal m = (n). Assume that the characteristic of k=Rjm is p. Let P be a coefficient ring of R. Let {cj ter be a ^-independent base of k and let ct be a representative of ct chosen from P for every i e I The symbol A means the/?-adic completion of P-algebra. By arguments developed by Berger or Kawahara-Yokoyama in the cited papers, and formal smoothness and flatness of P over the prime local ring, we can deduce the following theorem.