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We provide a very general approach to placing model structures and semi-model structures on algebras over symmetric colored operads. Our results require minimal hypotheses on the underlying model category $\mathcal{M}$, and these hypotheses vary depending on what is known about the colored operads in question. We obtain results for the classes of colored operad which are cofibrant as a symmetric collection, entrywise cofibrant, or arbitrary. As the hypothesis on the operad is weakened, the hypotheses on $\mathcal{M}$ must be strengthened. Via a careful development of the categorical algebra of colored operads we provide a unified framework which allows us to build (semi-)model structures for all three of these classes of colored operads. We then apply these results to provide conditions on $\mathcal{M}$, on the colored operad $\mathcal{O}$, and on a class $\mathcal{C}$ of morphisms in $\mathcal{M}$ so that the left Bousfield localization of $\mathcal{M}$ with respect to $\mathcal{C}$ preserves $\mathcal{O}$-algebras., Comment: 56 pages, comments welcome, v2 contains a new section on applications along with minor changes in exposition in section 5. Version 2 matches the submitted version

Suppose that R is a ring and that A is a chain complex over R. Inside the derived category of differential graded A-modules there are naturally defined subcategories of A-torsion objects and of A-complete objects. Under a finiteness condition on A, we develop a Morita theory for these subcat egories, find conceptual interpretations for some associated algebraic functors, and, in appropriate commutative situations, identify the associated functors as local homology or local cohomology. Some of the results are suprising even in the case R = Z and A = Z/p. 1. Introduction. Let R be a ring and R-mod the derived category of chain complexes of left /?-modules (see Section 1.2). We choose a fixed complex A which is perfect, in other words, isomorphic in /?-mod to a complex of finite length in which the entries are finitely generated projective /?-modules. We de clare another complex N to be A-trivial if Hom/?(A,A0 = 0, where Hom^(-,) denotes the chain complex of morphisms in R-mod. Going further, we say that X is A-torsion if Hom/?(X, AT) = 0 for all A-trivial N9 and that X is A-complete if HomR(N9X) = 0 for all A-trivial N. We then study the category Ators of A-torsion complexes and the category Acomp of A-complete complexes (both of these are triangulated full subcategories of R-mod). It turns out that these categories are equivalent to one another, and also equivalent to the derived category of dif ferential graded modules over the endomorphism complex of A. We construct approximation functors CelU: R-mod ?> At0rs and (-)A: R-mod ?> Acomp. For an object M of R-mod9 the complex CelU(M) is a kind of A-cellular approxi mation to M, in the sense that it is the best approximation to M which can be cobbled together from A and its suspensions; the complex MA is the Bousfield localization of M with respect to a homology theory on R-mod derived from A. We provide algebraic formulas for the functors, and find that the functors are related in interesting ways, one of which involves an arithmetic square. We also show that if CcIIa(R) has certain finite-dimensionality properties, then an object M of R-mod is A-torsion or A-complete if and only if the homology groups 77/M individually satisfy appropriate torsion or completeness conditions. Suppose now that R is a commutative ring, that 7 C R is a finitely generated ideal, that A = R/I9 and that K is the associated Koszul complex. The complex A

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, Comment: Published 7 April 2001 by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-9.abs.html . Erratum added 9 May 2001

Recent results of Voevodsky and others have effectively led to the proof of the Lichtenbaum-Quillen conjectures at the prime 2, and consequently made it possible to determine the 2-homotopy type of the $K$-theory spectra for various number rings. The basic case is that of $ BGL({\Bbb Z})$; in this note we use these results to determine the 2-local (topological) $K$-theory of the space $BGL({\Bbb Z})$, which can be described as a completed tensor product of two quite simple components; one corresponds to a real `image of $J$' space, the other to $BBSO$.

We construct the Bousfield-Kan (unstable Adams) spectral sequence based on certain nonconnective periodic homology theories E such as complex periodic /^-theory, and define an E completion of a space X. For X = S2n+X and E = K we calculate the E2-term and show that the spectral sequence converges to the homotopy groups of the tf-completion of the sphere. This also determines all of the homotopy groups of the (unstable) tf-theory localization of S2n+l including three divisible groups in negative stems. 1. Introduction. In (6) the first author with E. Curtis and H. Miller gen eralized the Bousfield-Kan construction of an unstable Adams spectral sequence based on homology with coefficients in a ring to generalized homology based on certain connective ring spectra E. In this paper we extend that generalization to the case of certain nonconnective periodic ring spectra such as the ^-theory spectrum. In the case of K-theovy, localized at an odd prime p, we calculate the E2 term of the resulting spectral sequence for a sphere and show that it converges to a A'-theory completion, whose simply connected cover is the unstable ZC-theory Bousfield localization. Of particular interest is the way in which the divisible groups in the negative stems arise in connection with higher derived functors of the primitive element functor, and also in connection with the failure of Bousfield localization to commute with loops and fiber sequences. In (19) the second author with Mark Mahowald computed the mod p homotopy groups of the Bousfield localization of an unstable sphere with re spect to K, and also the integral homotopy groups in dimensions greater than or equal to the dimension of the sphere. As a corollary of the results of the present paper, we obtain the remaining homotopy groups in dimensions less than the dimension of the sphere.

Examples are constructed to illustrate: (i) The LS category of a 1-connected, finite type CW-complex X which is the homotopy colimit of a sequence X1→X2→X3→⋯ of 1-connected, finite CW-complexes may exceed the LS category of each Xi; and (ii) LS category is not an invariant of the localization genus of a 1-connected, finite type CW-complex.

Let V(0) denote the mod p Moore spectrum and L2 denote the Bousfield localization functor with respect to v2−1BP. The E2-term E2∗(L2V(0)) of the Adams–Novikov spectral sequence for computing homotopy groups π∗(L2V(0)) is E2∗(L2V(0))=ExtΓ∗(A,A) over the Hopf algebroid (A,Γ)=(E(2)∗,E(2)∗E(2)) for the Johnson–Wilson spectrum E(2), and obtained immediately from the chromatic E1-term H∗M11=ExtΓ∗(A,A/(v1∞)) for the generator v1 of A=Z/p[v1,v2±1]. In this paper, we determine the module structure of the E2-term E2∗(L2V(0)) at the prime 2 after determining H∗M11.

Suppose C is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the smash product in C, and the Picard group Pic(C) is the collection of isomorphism classes of such invertible objects. The Picard group need not be a set in general, but if it is then it is an abelian group canonically associated with C. There are many examples of symmetric monoidal categories in stable homotopy theory. In particular, one could take the whole stable homotopy category S. In this case, it was proved by Hopkins that the Picard group is just Z, where a representative for n can be taken to be simply the n-sphere S [HMS94, Str92]. It is more interesting to consider Picard groups of the E-local category, for various spectra E (all of which will be p-local for some fixed prime p in this paper). Here the smash product of two E-local spectra need not be E-local, so one must relocalize the result by applying the Bousfield localization functor LE . The most well-known case is E = K(n), the nth Morava K-theory, considered in [HMS94]. In this paper we study the case E = E(n), where E(n) is the Johnson-Wilson spectrum. In this case the E-localization functor is universally denoted Ln, and we denote the category of E-local spectra by L. Our main theorem is the following result.

Let G G be a finitely generated nilpotent group. The object of this paper is to identify the Bousfield localization L h B G L_hBG of the classifying space B G BG with respect to a multiplicative complex oriented homology theory h ∗ h_{*} . We show that L h B G L_hBG is the same as the localization of B G BG with respect to the ordinary homology theory determined by the ring h 0 h_0 .

The category of rational O(2)-equivariant cohomology theories has an algebraic model A(O(2)), as established by work of Greenlees. That is, there is an equivalence of categories between the homotopy category of rational O(2)-equivariant spectra and the derived category of the abelian model DA(O(2)). In this paper we lift this equivalence of homotopy categories to the level of Quillen equivalences of model categories. This Quillen equivalence is also compatible with the Adams short exact sequence of the algebraic model., 30 pages. Final version, to appear in Homology, Homotopy and Applications. Section 5 on dihedral O(2) spectra has been simplified

This paper is the third paper of a series devoted to higher dimensional transition systems. The preceding paper proved the existence of a left determined model structure on the category of cubical transition systems. In this sequel, it is proved that there exists a model category of labelled symmetric precubical sets which is Quillen equivalent to the Bousfield localization of this left determined model category by the cubification functor. The realization functor from labelled symmetric precubical sets to cubical transition systems which was introduced in the first paper of this series is used to establish this Quillen equivalence. However, it is not a left Quillen functor. It is only a left adjoint. It is proved that the two model categories are related to each other by a zig-zag of Quillen equivalences of length two. The middle model category is still the model category of cubical transition systems, but with an additional family of generating cofibrations. The weak equivalences are closely related to bisimulation. Similar results are obtained by restricting the constructions to the labelled symmetric precubical sets satisfying the HDA paradigm., Comment: 31 pages ; final version

We study left and right Bousfield localisations of stable model categories which preserve stability. This follows the lead of the two key examples: localisations of spectra with respect to a homology theory and A-torsion modules over a ring R with A a perfect R-algebra. We exploit stability to see that the resulting model structures are technically far better behaved than the general case. We can give explicit sets of generating cofibrations, show that these localisations preserve properness and give a complete characterisation of when they preserve monoidal structures. We apply these results to obtain convenient assumptions under which a stable model category is spectral. We then use Morita theory to gain an insight into the nature of right localisation and its homotopy category. We finish with a correspondence between left and right localisation., Comment: 30 pages

For a tensor triangulated category which is well generated in the sense of Neeman, it is shown that the collection of Bousfield classes forms a set. This set has a natural structure of a complete lattice which is then studied, using the notions of stratification and support., 25 pages; minor changes from earlier version; this will appear in the Math. Z

We give a general method of constructing positive stable model structures for symmetric spectra over an abstract simplicial symmetric monoidal model category. The method is based on systematic localization, in Hirschhorn's sense, of a ceratin positive projective model structure on spectra, where positivity basically means the truncation of the zero slice. The localization above is by the set of stabilizing morphisms, or their truncated version., Comment: 20 pages

In this paper, we discuss the theory of quasi-fibrations in proper Bousfield localizations of model categories of simplicial sheaves. We provide a construction of fibrewise localization and use this construction to generalize a criterion for locality of fibre sequences due to Berrick and Dror Farjoun. The result allows a better understanding of unstable A^1-homotopy theory., 23 pages, completely revised version

Given a monoidal model category [math] and an object [math] in [math] , Hovey constructed the monoidal model category [math] of [math] –symmetric spectra over [math] . In this paper we describe how to lift a model structure on the category of [math] –enriched categories to the category of [math] –enriched categories. This allow us to construct a (four step) zig-zag of Quillen equivalences comparing dg categories to [math] –categories. As an application we obtain: (1) the invariance under weak equivalences of the topological Hochschild homology (THH) and topological cyclic homology (TC) of dg categories; (2) non-trivial natural transformations from algebraic [math] –theory to THH.

In this paper, we examine the `derived completion' of the representation ring of a pro-p group G_p^ with respect to an augmentation ideal. This completion is no longer a ring: it is a spectrum with the structure of a module spectrum over the Eilenberg-MacLane spectrum HZ, and can have higher homotopy information. In order to explain the origin of some of these higher homotopy classes, we define a deformation representation ring functor R[-] from groups to ring spectra, and show that the map R[G_p^] --> R[G] becomes an equivalence after completion when G is finitely generated nilpotent. As an application, we compute the derived completion of the representation ring of the simplest nontrivial case, the p-adic Heisenberg group., This is the version published by Algebraic & Geometric Topology on 26 February 2006

We study locally constant coefficients. We first study the theory of homotopy Kan extensions with locally constant coefficients in model categories, and explain how it characterizes the homotopy theory of small categories. We explain how to interpret this in terms of left Bousfield localization of categories of diagrams with values in a combinatorial model category. Finally, we explain how the theory of homotopy Kan extensions in derivators can be used to understand locally constant functors.

We examine the proof of a classical localization theorem of Bousfield and Friedlander and we remove the assumption that the underlying model category be right proper. The key to the argument is a lemma about factoring in morphisms in the arrow category of a model category., A small mistake in the published version is fixed

We continue to study the notions of Y-category and strong Vcategory which we introduced in [2]. We give a characterization of them in terms of homotopy colimits and then use it to prove some product theorems in this context. In [2] we introduced new homotopy invariants by generalizing the notions of Lusternik-Schnirelmann category and strong category as follows: For any given class of spaces v we define the X'-category X-cat(X) of a space X to be the smallest integer k for which there exists a numerable covering (X1, ..., Xk) of X such that each inclusion X. c X factors through some space Ai E v up to homotopy. If no such covering exists we set X-cat(X) oo. X-cat(X) is an invariant of the homotopy type of X [2, 1.4]. If we replace the condition that each X c X factors through some space in v up to homotopy by asking each X to have the homotopy type of some space in X, then the number V-gcat(X) thus obtained is not a homotopy invariant [3]. So we define the strong X-category sl-Cat(X) of X to be the minimum of Ygcat(X') for all spaces X' having the homotopy type of X. If v consists only of the one-point-space then X-cat (X) is just LusternikSchnirelmann's category cat (X) of X and V-Cat (X) is the strong category Cat (X) introduced by Ganea in [5]. Other interesting examples are the qconnective (strong) category catq(X) (Catq(X)) of X obtained by taking -v to be the class of q-connected CW-complexes, q > 0, and the q-dimensional (strong) category catq(X) (Catq (X)) of X obtained by taking for -v the class of all CW-complexes of dimension < q [2, 1.2]. In this paper we shall prove the following product theorems, which generalize the well known ones for the classical case [7]. Unlike [7] however, we do not restrict ourselves to polyhedra but rather exploit the linear structure given by the numeration of the coverings. Received by the editors November 11, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 55M30; Secondary 55P50.

A large variety of cohomology theories is derived from complex cobordism MU^*(-) by localizing with respect to certain elements or by killing regular sequences in MU_*. We study the relationship between certain pairs of such theories which differ by a regular sequence, by constructing topological analogues of algebraic I-adic towers. These give rise to Higher Bockstein spectral sequences, which turn out to be Adams spectral sequences in an appropriate sense. Particular attention is paid to the case of completed Johnson--Wilson theory E(n)-hat and Morava K-theory K(n) for a given prime p., Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-65.abs.html

We introduce a notion of a filtered model structure and use this notion to produce various model structures on pro-categories. This framework generalizes several known examples. We give several examples, including a homotopy theory for $G$-spaces, where $G$ is a pro-finite group. The class of weak equivalences is an approximation to the class of underlying weak equivalences., Comment: 29 pages

We introduce a construction adding low-dimensional cells to a space that satisfies certain low-dimensional conditions; it preserves high-dimensional homology with appropriate coefficients. This includes as special cases Quillen’s plus construction, Bousfield’s integral homology localization, the existence of Moore spaces M(G, 1) and Bousfield and Kan’s partial k-completion of spaces. We also use it to generalize counterexamples to the zero-in-the-spectrum conjecture found by Farber and Weinberger, and by Higson, Roe and Schick.

We construct a ``logarithmic'' cohomology operation on Morava E-theory, which is a homomorphism defined on the multiplicative group of invertible elements in the ring E^0(K) of a space K. We obtain a formula for this map in terms of the action of Hecke operators on Morava E-theory. Our formula is closely related to that for an Euler factor of the Hecke L-function of an automorphic form., 44 pages; revised version, including an expanded introduction and a complete reworking of sections 4 and 8 (of the original version), which now correspond to sections 4, 5, and 9 of the new version

We show that if ς is a proper model category, then the pro-category pro-ς has a proper model structure in which the weak equivalences are the levelwise weak equivalences. The structure is simplicial when ς is simplicial. This is related to a major result of [10] and is the starting point for many homotopy theories of pro-objects such as those described in [5], [17], and [19].

We introduce a fibre homotopy relation for maps in a category of cofibrant objects equipped with a choice of cylinder objects. Weak fibrations are defined to be those morphisms having the weak right lifting property with respect to weak equivalences. We prove a version of Dold's fibre homotopy equivalence theorem and give a number of examples of weak fibrations. If the category of cofibrant objects comes from a model category, we compare fibrations and weak fibrations, and we compare our fibre homotopy relation, which is defined in terms of left homotopies and cylinders, with the fibre homotopy relation defined in terms of right homotopies and path objects. We also dualize our notion of weak fibration in a category of cofibrant objects to a notion of weak cofibration in a category of fibrant objects, and give examples of these weak cofibrations. A section is devoted to the case of chain complexes in an abelian category.

2-Theories are a canonical way of describing categories with extra structure. 2-theory-morphisms are used when discussing how one structure can be replaced with another structure. This is central to categorical coherence theory. We place a Quillen model category structure on the category of 2-theories and 2-theory-morphisms where the weak equivalences are biequivalences of 2-theories. A biequivalence of 2-theories (Morita equivalence) induces and is induced by a biequivalence of 2-categories of algebras. This model category structure allows one to talk of the homotopy of 2-theories and discuss the universal properties of coherence., Comment: 32 pages; XY-Pic

Bousfield localization is a technique that has been used extensively in algebraic topology, specifically in stable homotopy theory, over the last quarter century. Its name derives from the fundamental work of Bousfield [3], although this is based on earlier work of Brown [4] and Adams [1]. In abstract terms, Bousfield localization deals with the inclusion of a thick subcategory (i.e., a triangulated subcategory closed under direct summands) into a triangulated category and the existence of adjoint functors to such an inclusion. In the original topological setting, the triangulated category was typically the stable homotopy category, and the thick subcategory was defined by the vanishing of some homology theory, but the techniques involved work much more generally. In particular, the triangulated category can be one of interest to representation theorists, such as a stable module category or the derived category of a module category: since the techniques rely heavily on limiting procedures, however, one is forced to work with infinite dimensional modules.

We show that the notion ofHZ-local groups due to A. Bousfield which is based on considerations from algebraic topology can be defined and understood in terms of solubility of certain systems of equations overG.

IN [8] WE studied localizations of spaces with respect to homology, and we now develop the analogous stable theory. Let Ho” denote the stable homotopy category of CW-spectra. We show that each spectrum E E Ho” gives rise to a natural E*localization functor ( )E: Ho” -+HoS and n : 1 +( )E. For A E Ho”, v:A-+AE is the terminal example of an E*-equivalence going out of A in Ho”. After proving the existence of ES-localizations, we develop their basic properties and discuss in detail the cases where E is connective and E = K. Using EJocalization theory, we obtain results on the convergence of the E*-Adams spectral sequence, and on the class A(Ho”) of acyclicity types of spectra, see [9]. The paper is sectioned as follows: 91. E*-localizations and E*-acyclizations of spectra; 42. Localizations with respect to Moore spectra; 93. General examples of E*-localizations; 34. K-theoretic localizations of spectra; 05. The E*-Adams spectral sequence and the E-nilpotent completion; 06. Convergence theorems for the E*Adams spectral sequence; P7. Duality in &Ho”). E*-localizations of spectra were previously studied by Adams, who sketched a useful outline of the subject in 141. However, the first complete existence proof for such localizations was obtained by the author as an immediate corrollary of work on EJocalizations of spaces, see [8]. The relevant proofs in [81 may be stabilized by using Kan’s semi-simplicial spectra in place of simplicial sets. That approach also shows the existence of “ES-factorizations” for maps of spectra and leads to a Quillen model category framework for “stable homotopy theory with respect to Es.” To make the present exposition more understandable we omit these topics and work in the homotopy category of CW-spectra. This paper grew out of an attempt to understand Doug Ravenel’s work on localizations of spectra with respect to certain periodic homology theories [17], and we are indebted to him for explaining his ideas. We are also indebted to Mark Mahowald and Haynes Miller for supplying the homotopy theoretic theorem underlying our approach to K*-localizations, and we have been significantly influenced by Guido Mislin’s work on K*-localizations of spaces [ 161. We understand that various results in this paper, particularly in 54, were obtained by Frank Adams and David Baird in earlier unpublished work, and that some have been obtained independently by W. Dwyer (unpublished). We essentially use the notation and terminology of [4]. However, we let Ho” be the category of CW-spectra and homotopy classes of maps of degree 0, see [4], p. 144. Thus HO” is an additive category equipped with an equivalence C : Ho” --f Ho” induced by the “shift” suspension for CW-spectra. We let [X, Y] be the group of morphisms

We introduce and study in detail generalizations of the notion of Lusternik-Schnirelmann category which give information about the topology of the critical set of a differentiable function. We also improve a result of T. Ganea about the equality of the strong category and the category (even in the classical case). The category cat(X) of a space X in the sense of Lusternik and Schnirelmann [14] is the smallest number k such that there exists a covering { X1,..., Xk } of X (of a certain kind, cf. 1.2(1)) for whicheach inclusion X c X is nullhomotopic. The motivation for introducing this concept was that it gives a lower bound for the number of critical points of a function. More precisely, if M is a closed differentiable manifold and f is a differentiable real function on M then the number of critical points of f is at least cat(M). We propose the following generalization: If _V is any class of spaces we replace the condition that Xj c X is nullhomotopic by requiring that it factors through some A E _ up to homotopy and we obtain the notion of -V-category, -V-cat(X). If _/ consists only of the one-point space, this is the classical cat(X). If _V is the class of q-connected spaces s/-cat is the "q-dimensional homotopy category" introduced by Fox in [7]. Another interesting example is the class _V of q-dimensional spaces. If f: M -+ R is as above then -V-cat(M) does not give any new information about the number of critical points, because it is less than or equal to cat(M). It does give, however, under certain conditions, some new information on the topological structure of the critical set. Roughly, one can say that either there are at least -V-cat(M) critical values of f or there is one critical value -y of f such that the corresponding set of critical points K n f 1(y) is not of the homotopy type of any space in _V (cf. ?2 for more details). Quite a number of papers have appeared on Lusternik-Schnirelmann category and related notions (cf. [13] for a survey). In particular there has been a revival of interest in recent years. We shall present a systematic theory for our more general Received by the editors November 25, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 55M30; Secondary 58E05, 55P50.

Let (X, O_X) be a noetherian formal scheme and consider D_qct(X) its derived category of sheaves with quasi-coherent torsion homology. We show that there is a bijection between the set of rigid (i.e. \tensor-ideals) localizing subcategories of D_qct(X) and subsets in X, generalizing previous work by Neeman. If moreover X is separated, the associated localization and acyclization functors are described in certain cases. When Z is a stable for specialization subset of X, its associated acyclization is \Gamma_Z. When X is an scheme, the corresponding localizing subcategories are generated by perfect complexes and we recover Thomason's classification of thick subcategories. On the other hand, if Y is a generically stable subset of X, we give an expression for the associated localization functor., Comment: Relaxed the separation hypothesis. Added connection with Thomason's classification of thick subcategories

The category of modules over an S -algebra ( A ∞ or E ∞ ring spectrum) has many of the good properties of the category of spectra. When the homotopy groups of the S -algebra in question form a sufficiently nice ring, it is possible to see the deviation of the category of modules over an S -algebra from the corresponding algebraic module category. In particular, many algebraic modules are realized as homotopy groups of topological modules over S -algebras. Examples studied include real and complex K -theory, both connective and periodic. Further, Bousfield localization by a smashing spectrum is shown to yield a category of modules over the localized sphere. For periodic K -theory, these methods yield an algebraic criterion to determine when a local spectrum is a module over the K -theory S -algebra, real or complex.

K-theory localization of an unstable odd sphere is constructed by Mahowald and Thompson in [8]. Earlier work on unstable K-theory localization includes Bousfield’s construction of the K-theory localization of infinite loop spaces in [4] and Mislin’s construction of the K-theory localization of Eilenberg-MacLane spaces in [13]. The con- struction of Mislin demonstrates that the unstable K-theory localization functor does not commute with taking loops on a space, nor does it preserve the connectivity of a space. This being the case, we note that the K-theory localization of loops on an odd sphere cannot be directly given by knowing the K-theory localization of an odd sphere. In recent work Bousfield avoids this difficulty by defining a new vi localization functor which behaves well with respect to fibrations and then shows that it is closely related to the unstable K-theory localization functor. In this paper a more direct approach will be taken to construct the K-theory localiza- tion of loops on an odd sphere and double loops on an odd sphere. The construction and proof is a generalization of the results of Mahowald and Thompson in [IS]. As an application, these results will be used to construct the K-theory localization of the fiber of the Hopf map. Construction and Results: We will begin by summarizing the construction of the K-theory localization of loops on an odd sphere done in [S]. By [14] the localization with respect to real and complex K-theory give the same result. Throughout this paper K-theory will refer to complex K-theory. For simplicity, we will take the localization with respect to KCpJ at each prime separately, where K,,, denotes p local complex K-theory. These localiza- tions may then be pieced together to give the K-theory localization. Let F, denote the homotopy fiber of the Snaith map where QX = C2= C”X, q = 2p - 2, and B, denotes a stunted BXP, localized at p with bottom cell in dimension m. The inclusion map ii :S2”+’ -+ QS2”+’ lifts to a map ji:S”‘+i -+ F,. Mahowald and Thompson show in [S] that the map jr induces an isomor- phism in p local K-theory. This is the first step in showing that their construction is the K-theory localization of S2”+ ‘. Now let G, denote the homotopy fiber of the localized Snaith map

We define a plus-construction on connective augmented algebras over operads in symmetric spectra using Quillen homology. For associative and commutative algebras, we show that this plus-construction is related to both Bousfield localization and Carlsson’s derived completion.

The homotopy groups π ∗ (L 2 S 0 ) of the L2-localized sphere are determined by studying the Bockstein spectral sequence. The results also indicate the homotopy groups π ∗ (L K(2) S 0 ) and we observe that the fiber of the localization map L2S30→LK(2)S0 is homotopic to Σ−2L1S30. Here S30 denotes the 3-completed sphere.

When G is a profinite group and H and K are closed subgroups, with H normal in K, it is not known, in general, how to form the iterated homotopy fixed point spectrum ( Z h H ) h K / H , where Z is a continuous G-spectrum and all group actions are to be continuous. However, we show that, if G = G n , the extended Morava stabilizer group, and Z = L ˆ ( E n ∧ X ) , where L ˆ is Bousfield localization with respect to Morava K-theory, E n is the Lubin–Tate spectrum, and X is any spectrum with trivial G n -action, then the iterated homotopy fixed point spectrum can always be constructed. Also, we show that ( E n h H ) h K / H is just E n h K , extending a result of Devinatz and Hopkins.

We proved in a previous work that Cattani-Sassone's higher dimensional transition systems can be interpreted as a small-orthogonality class of a topological locally finitely presentable category of weak higher dimensional transition systems. In this paper, we turn our attention to the full subcategory of weak higher dimensional transition systems which are unions of cubes. It is proved that there exists a left proper combinatorial model structure such that two objects are weakly equivalent if and only if they have the same cubes after simplification of the labelling. This model structure is obtained by Bousfield localizing a model structure which is left determined with respect to a class of maps which is not the class of monomorphisms. We prove that the higher dimensional transition systems corresponding to two process algebras are weakly equivalent if and only if they are isomorphic. We also construct a second Bousfield localization in which two bisimilar cubical transition systems are weakly equivalent. The appendix contains a technical lemma about smallness of weak factorization systems in coreflective subcategories which can be of independent interest. This paper is a first step towards a homotopical interpretation of bisimulation for higher dimensional transition systems., Comment: 39 pages ; erratum added

Existence of proper Gorenstein projective resolutions and Tate cohomology is proved over rings with a dualizing complex. The proofs are based on Bousfield Localization which is originally a method from algebraic topology.