Adams Summand Research Articles

Towards an understanding of ramified extensions of structured ring spectra

AbstractWe propose topological Hochschild homology as a tool for measuring ramification of maps of structured ring spectra. We determine second order topological Hochschild homology of the p-local integers. For the tamely ramified extension of the map from the connective Adams summand to p-local complex topological K-theory we determine the relative topological Hochschild homology and show that it detects the tame ramification of this extension. We show that the complexification map from connective topological real to complex K-theory shows features of a wildly ramified extension. We also determine relative topological Hochschild homology for some quotient maps with commutative quotients.

Logarithmic topological Hochschild homology of topological $K$-theory spectra

In this paper we continue our study of logarithmic topological Hochschild homology. We show that the inclusion of the connective Adams summand \ell into the p -local complex connective K -theory spectrum ku_{(p)} , equipped with suitable log structures, is a formally log THH-étale map, and compute the V(1) -homotopy of their logarithmic topological Hochschild homology spectra. As an application, we recover Ausoni's computation of the V(1) -homotopy of the ordinary THH of ku .

Central Cohomology Operations and K-Theory

AbstractFor stable degree 0 operations, and also for additive unstable operations of bidegree (0, 0), it is known that the centre of the ring of operations for complex cobordism is isomorphic to the corresponding ring of connective complex K-theory operations. Similarly, the centre of the ring of BP operations is the corresponding ring for the Adams summand of p-local connective complex K-theory. Here we show that, in the additive unstable context, this result holds with BP replaced by BP〈n⌰ for any n. Thus, for all chromatic heights, the only central operations are those coming from K-theory.

Logarithmic structures on topological

We study a modified version of Rognes' logarithmic structures on structured ring spectra. In our setup, we obtain canonical logarithmic structures on connective K-theory spectra which approximate the respective periodic spectra. The inclusion of the p-complete Adams summand into the p-complete connective complex K-theory spectrum is compatible with these logarithmic structures. The vanishing of appropriate logarithmic topological Andre-Quillen homology groups confirms that the inclusion of the Adams summand should be viewed as a tamely ramified extension of ring spectra.

Existence and uniqueness of $$E_{\infty }$$ E ∞ structures on motivic $$K$$ K -theory spectra

We show that algebraic \({\textit{K}}\)-theory \(\mathsf {KGL}\), the motivic Adams summand \(\mathsf {ML}\) and their connective covers acquire unique \(E_{\infty }\) structures refining naive multiplicative structures in the motivic stable homotopy category. The proofs combine \(\Gamma \)-homology computations and work due to Robinson giving rise to motivic obstruction theory. As an application we employ a motivic to simplicial delooping argument to show a uniqueness result for \(E_\infty \) structures on the \(K\)-theory Nisnevich presheaf of spectra.

Upper triangular matrices and operations in odd primary connective K-theory

Abstract We prove analogues for odd primes of results of Snaith and Barker–Snaith. Let ℓ denote the p-complete connective Adams summand and consider the group of left ℓ-module automorphisms of ℓ∧ℓ in the stable homotopy category which induce the identity on mod p homology. We prove a group isomorphism between this group and a certain group of infinite invertible upper triangular matrices with entries in the p-adic integers. We determine information about the matrix corresponding to the automorphism 1∧Ψ q of ℓ∧ℓ, where Ψ q is the Adams operation and q is an integer which generates the p-adic units.

Formula omitted]-graded [formula omitted]-groups of [formula omitted], [formula omitted] and [formula omitted

We give an algorithm for calculating the R O ( S 1 ) -graded TR -groups of F p , completing the calculation started by the second author. We also calculate the R O ( S 1 ) -graded TR -groups of Z with mod p coefficients and of the Adams summand ℓ of connective complex K -theory with V ( 1 ) -coefficients. These calculations are used elsewhere to compute the algebraic K -theory of certain Z -algebras.

Topological Hochschild homology of ℓ and ko

We calculate the integral homotopy groups of THH $(\ell)$ at any prime and of THH $(ko)$ at $p=2$, where $\ell$ is the Adams summand of the connective complex $p$-local $K$-theory spectrum and $ko$ is the connective real $K$-theory spectrum.

On the cooperation algebra of the connective Adams summand

The aim of this paper is to gain explicit information about the multiplicative structure of l* l, where l is the connective Adams summand at an odd prime p. Our approach differs from Kane's or Lellmann's because our main technical tool is the MU-based Kunneth spectral sequence. We prove that the algebra structure on l* l is inherited from the multiplication on a Koszul resolution of l* BP.

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 .

On the $\Gamma$-cohomology of rings of numerical polynomials

We investigate \Gamma -cohomology of some commutative cooperation algebras E_*E associated with certain periodic cohomology theories. For KU and E(1) , the Adams summand at a prime p , and for KO we show that \Gamma -cohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique E_\infty structures. As a consequence we obtain an E_\infty structure for the connective Adams summand. For the Johnson–Wilson spectrum E(n) with n\geq1 we establish the existence of a unique E_\infty structure for its I_n -adic completion.

Algebras of operations in [formula omitted]-theory

We describe explicitly the algebras of degree zero operations in connective and periodic p-local complex K-theory. Operations are written uniquely in terms of certain infinite linear combinations of Adams operations, and we give formulas for the product and coproduct structure maps. It is shown that these rings of operations are not Noetherian. Versions of the results are provided for the Adams summand and for real K-theory.

Connective K -Theory of the Im(J )-Spectrum

For l the spectrum of the Adams summand of p-local connective K-theory, and A the spectrum of connective Im(J)-theory at the prime p, an l-module splitting of lΛA is constructed.

The homology of the connective K-theory spectrum as an [formula omitted] (1)-module

From an odd prime p, let l be the Adams Summand of p-local connective K-theory and A (1) the subalgebra of the mod p Steenrod algebra generated by Q 0 and the first Steenrod power P 1. The algebra A (1) is an explicitly understood Hopf algebra over F p of F p -dimension 4 p. The mod p homology H ∗(l) of l is a tensor product of a polynomial algebra on countably many generators with an exterior algebra on countably many generators. We describe H ∗(l) as an A (1)-module.

On the Topological Hochschild Homology of bu, I

The purpose of this paper and its sequel is to determine the homotopy groups of the spectrum THH(`). Here p is an odd prime, ` is the Adams summand of p-local connective K-theory (see for example [25]) and THH is the topological Hochschild homology construction introduced by Bokstedt in [3]. In the present paper we will determine the mod p homotopy groups of THH(`) and also the integral homotopy groups of THH(L) (where L denotes the periodic Adams summand). In the sequel we will investigate the integral homotopy groups of THH(`) using our present results as a starting point. The THH construction appears to be of basic importance in algebraic K-theory because it combines two useful properties: it can be used to construct good approximations to the algebraic K-theory functor, and it is very accessible to calculation. We shall review what is known about the first property in a moment; the second property was demonstrated by Bokstedt’s calculation, in his paper [4], of the homotopy groups of THH(HZ/p) and THH(HZ) (here HZ/p and HZ denote the evident Eilenberg-Mac Lane spectra). It is natural to ask about THH(R) for other popular ring spectra R, and our work is a first step in this direction. We pay special attention to the connective case because this is the case which is likely to be relevant in applications (see Subsection 1.4 below). The calculation which we present in this paper is a homotopy-theoretic one which uses the Adams spectral sequence (hereafter abbreviated ASS). This calculation has several interesting features; in particular it is a pleasing example of an ASS calculation in which, although there are infinitely many differentials, it is still possible to get the complete answer. Here is a summary of the contents of the paper. In Section 2 we review the facts we need to know about ordinary Hochschild homology. In Section 3 we do the same

The Conner-Floyd Map for Formal A-Modules

A generalization of the Conner-Floyd map from complex cobordism to complex /C-theory is constructed for formal A-modules when A is the ring of algebraic integers in a number field or its p-adic completion. This map is employed to study the Adams-Novikov spectral sequence for formal A-modules and to confirm a conjecture of D. Ravenel. 0. Introduction. Let BP be the spectrum representing Brown-Peterson cohomology with respect to a prime p and let E be the Adams summand of complex if-theory with respect to this prime. The BP version of the Conner-Floyd map is a map of spectra BP —+ E which induces a natural equivalence BP, X ®Bp. E» ~ E.X. In particular this induces an isomorphism £* ®Bp. BP»BP®bp.£. ~ E.E and so provides a way of computing the Hopf algebra of stable co-operations for E from those for BP. Using this one can obtain a description of E*E similar to that for K*K contained in [AHS]. The study of BP and the computation of BP* BP are based on a study of formal group law, in particular the p-typical formal group law. In [Rl] Ravenel studied a generalization of this situation where the formal group law is replaced by a formal yl-module where A is the ring of integers in an algebraic number field K or its p-adic completion. The purpose of the present paper is to describe the corresponding generalization of the map (BP», BP* BP) —► (E*,E*E) induced by the Conner-Floyd map, and to compute the generalization of E*E. This is of interest because it provides some information about a conjecture (3.10) made in [Rl]. This conjecture concerned the value of a certain Ext group Exty,t(Va, Va) when K is an extension of the field Qp of p-adic numbers. Here (Va, VaT) is the Hopf algebroid corresponding to the A-typical formal A module. This group was conjectured to be, up to small factor, A/J^,^. Here J^iq_1 is the ideal of A generated by the elements of the form aTM — 1 for units a of A congruent to 1 mod(7r) and (it) is the unique prime ideal in A. We will show, using the generalization of the Conner-Floyd map, that A/J*,^ occurs as E\' in the chromatic spectra sequence for formal ^-modules [Rl, Lemma 2.10] and that the small factor in the conjecture is contributed by the nontriviality of the differential d originating from this group. We will analyze this differential and show that it is nonzero for A Received by the editors June 27, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 55T25; Secondary 55N22, 14L05. ©1987 American Mathematical Society 0002-9947/87 SI.00 + $.25 per page 319 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

A Stong-Hattori spectral sequence

Let G ∗ ( ) {G_ \ast }(\;) be the Adams summand of connective K-theory localized at the prime p. Let B P ∗ ( ) B{P_\ast }(\;) be Brown-Peterson homology for that prime. A spectral sequence is constructed with E 2 {E^2} term determined by G ∗ ( X ) {G_ \ast }(X) and whose E ∞ {E^\infty } terms give the quotients of a filtration of B P ∗ ( X ) B{P_ \ast }(X) where X is a connected spectrum. A torsion property of the differentials implies the Stong-Hattori theorem.