Local-to-global spectral sequences for the cohomology of diagrams

We construct local-to-global spectral sequences for the cohomology of a diagram, which compute the cohomology of the full diagram in terms of smaller pieces. These are motivated by the obstruction theory of D. Blanc et al. [D. Blanc, M.W. Johnson, J.M. Turner, On realizing diagrams of Π-algebras, Algebraic Geom. Topol. 6 (2006) 763-807] for realizing a diagram of Π-algebras, but are valid in quite general algebraic settings.     

On the Grothendieck construction for ∞-categories

We provide, among other things: (i) a Bousfield–Kan formula for colimits in ∞-categories (generalizing the 1-categorical formula for a colimit as a coequalizer of maps between coproducts); (ii) ∞-categorical generalizations of Barwick–Kan's Theorem B_n and Dwyer–Kan–Smith's Theorem C_n (regarding homotopy pullbacks in the Thomason model structure, which themselves vastly generalize Quillen's Theorem B); and (iii) an articulation of the simultaneous and interwoven functoriality of colimits (or dually, of limits) for natural transformations and for pullback along maps of diagram ∞-categories.     

The realization space of a Π-algebra: a moduli problem in algebraic topology

A Π-algebra A is a graded group with all of the algebraic structure possessed by the homotopy groups of a pointed connected topological space. We study the moduli space R(A) of realizations of A, which is defined to be the disjoint union, indexed by weak equivalence classes of CW-complexes X with , of the classifying space of the monoid of self homotopy equivalences of X. Our approach amounts to a kind of homotopical deformation theory: we obtain a tower whose homotopy limit is R(A), in which the space at the bottom is BAut(A) and the successive fibres are determined by Π-algebra cohomology. (This cohomology is the analog for Π-algebras of the Hochschild cohomology of an associative ring or the André-Quillen cohomology of a commutative ring.) It seems clear that the deformation theory can be applied with little change to study other moduli problems in algebra and topology.     

Localization and duality in topology and modular representation theory

We develop a duality theory for localizations in the context of ring spectra in algebraic topology. We apply this to prove a theorem in the modular representation theory of finite groups.Let G be a finite group and k be an algebraically closed field of characteristic p. If p is a homogeneous nonmaximal prime ideal in H^∗(G,k), then there is an idempotent module κ_p which picks out the layer of the stable module category corresponding to p, and which was used by Benson, Carlson and Rickard [D.J. Benson, J.F. Carlson, J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997) 59-80] in their development of varieties for infinitely generated kG-modules. Our main theorem states that the Tate cohomology is a shift of the injective hull of H^∗(G,k)/p as a graded H^∗(G,k)-module. Since κ_p can be constructed using a version of the stable Koszul complex, this can be viewed as a statement of localized Gorenstein duality in modular representation theory. Various consequences of this theorem are given, including the statement that the stable endomorphism ring of the module κ_p is the p-completion of cohomology , and the statement that κ_p is a pure injective kG-module.In the course of proving the theorem, we further develop the framework introduced by Dwyer, Greenlees and Iyengar [W.G. Dwyer, J.P.C. Greenlees, S. Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006) 357-402] for translating between the unbounded derived categories and . We also construct a functor to the full stable module category, which extends the usual functor and which preserves Tate cohomology. The main theorem is formulated and proved in , and then translated to and finally to .The main theorem in can be viewed as stating that a version of Gorenstein duality holds after localizing at a prime ideal in H^∗(BG;k). This version of the theorem holds more generally for a compact Lie group satisfying a mild orientation condition. This duality lies behind the local cohomology spectral sequence of Greenlees and Lyubeznik for localizations of H^∗(BG;k).In a companion paper [D.J. Benson, Idempotent kG-modules with injective cohomology, J. Pure Appl. Algebra 212 (7) (2008) 1744-1746], a more recent and shorter proof of the main theorem is given. The more recent proof seems less natural, and does not say anything about localization of the Gorenstein condition for compact Lie groups.     

Moduli spaces of 2-stage Postnikov systems

Using the obstruction theory of Blanc, Dwyer and Goerss, we compute the moduli space of realizations of 2-stage Π-algebras concentrated in dimensions 1 and n or in dimensions n and n+1. The main technical tools are Postnikov truncation and connected covers of Π-algebras, and their effect on Quillen cohomology.     

Topological André-Quillen Cohomology and E∞ André-Quillen Cohomology

We show that the André-Quillen cohomology of an E_∞ simplicial algebra with arbitrary coefficients and the topological André-Quillen cohomology of an E_∞ ring spectrum with Eilenberg-Mac Lane coefficients may be calculated as the André-Quillen cohomology of an associated E_∞ differential graded algebra.     

Homotopy uniqueness of BG 2

We prove that the homotopy type of BG ₂ is determined by its mod 2 cohomology as well as by its Weyl group data. Received: 21 April 1997 / Revised version: 8 December 1997     

Structures de dérivabilité

We introduce a very general framework in which Quillen's theorems of existence, composition and adjunction for derived functors can be proved. We thus generalize and unify previous results by Dwyer, Hirschhorn, Kan and Smith, obtained in their formalism of “homotopical categories,” and by Radulescu-Banu in the context of Cisinski's “derivable categories.”     

Ghosts in modular representation theory

A ghost over a finite p-group G is a map between modular representations of G which is invisible in Tate cohomology. Motivated by the failure of the generating hypothesis—the statement that ghosts between finite-dimensional G-representations factor through a projective—we define the ghost number of kG to be the smallest integer l such that the composite of any l ghosts between finite-dimensional G-representations factors through a projective. In this paper we study ghosts and the ghost numbers of p-groups. We begin by showing that a weaker version of the generating hypothesis, where the target of the ghost is fixed to be the trivial representation k, holds for all p-groups. We then compute the ghost numbers of all cyclic p-groups and all abelian 2-groups with C₂ as a summand. We obtain bounds on the ghost numbers for abelian p-groups and for all 2-groups which have a cyclic subgroup of index 2. Using these bounds we determine the finite abelian groups which have ghost number at most 2. Our methods involve techniques from group theory, representation theory, triangulated category theory, and constructions motivated from homotopy theory.     

Algèbres et cogèbres de Gerstenhaber et cohomologie de Chevalley-Harrison

The fundamental example of Gerstenhaber algebra is the space Tpoly(Rd) of polyvector fields on Rd, equipped with the wedge product and the Schouten bracket. In this paper, we explicitely describe what is the enveloping G_∞ algebra of a Gerstenhaber algebra G. This structure gives us a definition of the Chevalley-Harrison cohomology operator for G. We finally show the nontriviality of a Chevalley-Harrison cohomology group for a natural Gerstenhaber subalgebra in Tpoly(Rd).     

A completion theorem for pro-G-spectra

We give a very general completion theorem for pro-spectra. We show that, if G is a compact Lie group, M[∗] is a pro-G-spectrum, and F is a family of (closed) subgroups of G, then the mapping pro-spectrum F(EF₊,M[∗]) is the F-adic completion of M[∗], in the sense that the map M[∗]→F(EF₊,M[∗]) is the universal map into an algebraically F-adically complete pro-spectrum. Here, F(EF₊,M[∗]) denotes the pro-G-spectrum , where runs over the finite subcomplexes of EF₊.     

Primitives and central detection numbers in group cohomology

Fix a prime p. Given a finite group G, let H^∗(G) denote its mod p cohomology. In the early 1990s, Henn, Lannes, and Schwartz introduced two invariants d₀(G) and d₁(G) of H^∗(G) viewed as a module over the mod p Steenrod algebra. They showed that, in a precise sense, H^∗(G) is respectively detected and determined by Hd(CG(V)) for d?d₀(G) and d?d₁(G), with V running through the elementary abelian p-subgroups of G.The main goal of this paper is to study how to calculate these invariants. We find that a critical role is played by the image of the restriction of H^∗(G) to H^∗(C), where C is the maximal central elementary abelian p-subgroup of G. A measure of this is the top degree e(G) of the finite dimensional Hopf algebra H^∗(C)_⊗H^∗(G)Fp, a number that tends to be quite easy to calculate.Our results are complete when G has a p-Sylow subgroup P in which every element of order p is central. Using the Benson-Carlson duality, we show that in this case, d₀(G)=d₀(P)=e(P), and a similar exact formula holds for d₁. As a bonus, we learn that He(G)(P) contains nontrivial essential cohomology, reproving and sharpening a theorem of Adem and Karagueuzian.In general, we are able to show that d₀(G)?max{e(CG(V))|V<G} if certain cases of Benson's Regularity Conjecture hold. In particular, this inequality holds for all groups such that the difference between the p-rank of G and the depth of H^∗(G) is at most 2. When we look at examples with p=2, we learn that d₀(G)?14 for all groups with 2-Sylow subgroup of order up to 64, with equality realized when G=SU(3,4).En route we study two objects of independent interest. If C is any central elementary abelian p-subgroup of G, then H^∗(G) is an H^∗(C)-comodule, and we prove that the subalgebra of H^∗(C)-primitives is always Noetherian of Krull dimension equal to the p-rank of G minus the p-rank of C. If the depth of H^∗(G) equals the rank of Z(G), we show that the depth essential cohomology of G is nonzero (reproving and extending a theorem of Green), and Cohen-Macauley in a certain sense, and prove related structural results.     

A note on Samelson products and mod p cohomology of classifying spaces of the exceptional Lie groups

Let G be an exceptional Lie group G₂, F₄, E₆, E₇ or E₈, and also set p is the corresponding prime 7, 13, 13, 19 or 31 respectively. If we localize spaces at p, G can be decomposed into a product of spheres. Using this decomposition, we take some elements in the homotopy groups of p-localized G, and we offer some non-zero 3-fold Samelson products of them. This implies that the nilpotency class of the localized self-homotopy group of G is greater than or equal to 3.The key lemma for these results is about a calculation on the cohomology operator P¹ in the cohomology of BG, where G and p are as above. During this calculation, we use some original ideas, which are also used in Kishimoto and Kaji (in press) [7] recently.     

Homotopy classification of braided graded categorical groups

For any group G, a certain cohomology theory of G-modules is developed. This cohomology arises from the homotopy theory of G-spaces and it is called the “abelian cohomology of G-modules”. Then, as the main results of this paper, natural one-to-one correspondences between elements of the 3^rd cohomology groups of G-modules, G-equivariant pointed simply-connected homotopy 3-types and equivalence classes of braided G-graded categorical groups are established. The relationship among all these objects with equivariant quadratic functions between G-modules is also discussed.     

Homotopical algebraic geometry I: topos theory

This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos.For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of ∞-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove in particular, that for an S-category T endowed with an S-topology, there exists a model category of stacks over T, generalizing the model category structure on simplicial presheaves over a Grothendieck site of Joyal and Jardine. We also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves, by proving that there exists a one-to-one correspondence between S-topologies on an S-category T, and certain left exact Bousfield localizations of the model category of pre-stacks on T. Based on the above results, we study the notion of model topos introduced by Rezk, and we relate it to our model categories of stacks over S-sites.In the second part of the paper, we present a parallel theory where S-categories, S-topologies and S-sites are replaced by model categories, model topologies and model sites. We prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over S-sites is provided by the simplicial localization construction of Dwyer and Kan. As an example of application, we propose a definition of étale K-theory of ring spectra, extending the étale K-theory of commutative rings.     

Calculus of functors and model categories

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure [B. Chorny, W.G. Dwyer, Homotopy theory of small diagrams over large categories, preprint, 2005]. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for a classification of polynomial and homogeneous functors. In the n-homogeneous model structure, the nth derivative is a Quillen functor to the category of spectra with Σn-action. After taking into account only finitary functors—which may be done in two different ways—the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T.G. Goodwillie [T.G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003) 645-711 (electronic)].     

A bilinear form of dilogarithm and motivic regulator map

The Bloch-Wigner function D₂ is a single-valued version of a dilogarithm function and is used by Bloch to describe the Borel regulator map from K₃(C) into R explicitly (c.f. [Bloch, Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves, American Mathematical Society, Providence, RI, 2000]). We introduce a new way to formulate a single-valued dilogarithm function and use it to explicitly define a motivic regulator map for , defined in terms of the motivic complex of Goodwillie and Lichtenbaum. We also detect certain explicit nonzero elements in the motivic cohomology group. Throughout this paper, a path will be a C¹-function from the unit interval [0,1] into C-{0}.