The family of Thom spectra $y(n)$ interpolates between the sphere spectrum and the mod two Eilenberg--MacLane spectrum. Computations of Mahowald, Ravenel, Shick, and the authors show that the associative ring spectrum $y(n)$ has type $n$. Using trace methods, we give evidence that algebraic K-theory preserves this chromatic complexity. Our approach sheds light on the chromatic complexity of topological negative cyclic homology and topological periodic cyclic homology, which approximate algebraic K-theory and are of independent interest. Our main contribution is a homological approach that can be applied in great generality, such as to associative ring spectra $R$ without additional structure whose coefficient rings are not completely understood.
We give a simple argument to detect chromatic redshift in the algebraic $K$-theory of $\mathbb{E}_{\infty}$-ring spectra and give two applications: we show for $n\geq 1$ that $K(E_n)$, the algebraic $K$-theory of any height $n$ Lubin-Tate theory, has nontrivial $T(n+1)$-localization, and that $K^{(n)}(k)$, the $n$-fold iterated algebraic $K$-theory of a field $k$ of characteristic different from $p$, has nontrivial $T(n)$-localization.
We prove some $K$-theoretic descent results for finite group actions on stable $\infty$-categories, including the $p$-group case of the Galois descent conjecture of Ausoni-Rognes. We also prove vanishing results in accordance with Ausoni-Rognes's redshift philosophy: in particular, we show that if $R$ is an $\mathbb{E}_\infty$-ring spectrum with $L_{T(n)}R=0$, then $L_{T(n+1)}K(R)=0$. Our key observation is that descent and vanishing are logically interrelated, permitting to establish them simultaneously by induction on the height.
In this paper, we study the global structure of an algebraic avatar of the derived category of ind-coherent sheaves on the moduli stack of formal groups. In analogy with the stable homotopy category, we prove a version of the nilpotence theorem as well as the chromatic convergence theorem, and construct a generalized chromatic spectral sequence. Furthermore, we discuss analogs of the telescope conjecture and chromatic splitting conjecture in this setting, using the local duality techniques established earlier in joint work with Valenzuela.
We show that if $E$ is a $p$-local Landweber exact homology theory of height $n$ and $p > n^2+n+1$, then there exists an equivalence $h \mathcal{S}p_{E} \simeq h\mathcal{D}(E_{*}E)$ between homotopy categories of $E$-local spectra and differential $E_{*}E$-comodules, generalizing Bousfield's and Franke's results to heights $n > 1$.
We formulate a version of Hopkins' chromatic splitting conjecture for an arbitrary structured ring spectrum $R$, and prove it whenever $π_*R$ is Noetherian. As an application, these results provide a new local-to-global principle in the modular representation theory of finite groups.
We show that Lubin--Tate theories attached to algebraically closed fields are characterized among $T(n)$-local $\mathbb{E}_{\infty}$-rings as those that satisfy an analogue of Hilbert's Nullstellensatz. Furthermore, we show that for every $T(n)$-local $\mathbb{E}_{\infty}$-ring $R$, the collection of $\mathbb{E}_\infty$-ring maps from $R$ to such Lubin-Tate theories jointly detect nilpotence. In particular, we deduce that every non-zero $T(n)$-local $\mathbb{E}_{\infty}$-ring $R$ admits an $\mathbb{E}_\infty$-ring map to such a Lubin-Tate theory. As consequences, we construct $\mathbb{E}_{\inf…
Inspired by the Ax--Kochen isomorphism theorem, we develop a notion of categorical ultraproducts to capture the generic behavior of an infinite collection of mathematical objects. We employ this theory to give an asymptotic solution to the approximation problem in chromatic homotopy theory. More precisely, we show that the ultraproduct of the $E(n,p)$-local categories over any non-prinicipal ultrafilter on the set of prime numbers is equivalent to the ultraproduct of certain algebraic categories introduced by Franke. This shows that chromatic homotopy theory at a fixed height is asymptotically…
This text was written and used for a MAP Summer School at the University of Genova, August 28 to September 2, 2006. Available since then on the web site of the second author, it has been used and referenced by several colleagues working in Commutative Algebra and Algebraic Topology. To make safer such references, it was suggested to place it on the Arxiv repository. It is a relatively detailed exposition of the use of the Basic Perturbation Lemma to make constructive Homological Algebra (standard Homological Algebra is not constructive) and how this technology can be used in Commutative Algebr…
We recall the cohomological interpretation of the unipotent quotients of the fundamental groupoid of an algebraic complex variety (Beilinson, Deligne-Goncharov). We then give a construction of the resutting transition morphisms in terms of singular homology.
We define homological dimensions for S-algebras, the generalized rings that arise in algebraic topology. We compute the homological dimensions of a number of examples, and establish some basic properties. The most difficult computation is the global dimension of real K-theory KO and its connective version ko at the prime 2. We show that the global dimension of KO is 1, 2, or 3, and the global dimension of ko is 4 or 5.
We prove that the Farrell-Jones assembly map for connective algebraic K-theory is rationally injective, under mild homological finiteness conditions on the group and assuming that a weak version of the Leopoldt-Schneider conjecture holds for cyclotomic fields. This generalizes a result of Bökstedt, Hsiang, and Madsen, and leads to a concrete description of a large direct summand of $K_n(\mathbb{Z}[G])\otimes_{\mathbb{Z}}\mathbb{Q}$ in terms of group homology. In many cases the number theoretic conjectures are true, so we obtain rational injectivity results about assembly maps, in particular fo…
The celebrated 1999 Asynchronous Computability Theorem (ACT) of Herlihy and Shavit characterized the distributed tasks that are wait-free solvable, and thus uncovered a deep connection with algebraic topology. We present a novel interpretation of this theorem, through the notion of continuous task, defined by an input/output specification that is a continuous function. To do so, we introduce a chromatic version of a foundational result for algebraic topology: the simplicial approximation theorem. In addition to providing a different proof of the ACT, the notion of continuous task seems interes…
In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real valued map on the $n$-torus admits a fibre whose homological size is bounded below by some universal constant depending on $n$. He obtained similar estimates for maps with values in finite dimensional complexes, by a Lusternik--Schnirelmann type argument. We describe a new homological filling technique which enables us to derive sharp lower bounds in these theorems in certain situations. This partly realizes a programme envisaged by Gromov. In contra…
We study a family of subrings, indexed by the natural numbers, of the mod-p cohomology of a finite group G. These subrings are based on a family of v_n-periodic complex oriented cohomology theories and are constructed as rings of generalised characteristic classes. We identify the varieties associated to these subrings in terms of colimits over categories of elementary abelian subgroups of G, naturally interpolating between the work of Quillen on var(H^*(BG)), the variety of the whole cohomology ring, and that of Green and Leary on the variety of the Chern subring, var(Ch(G)). Our subrings giv…
Given an algebraic stack $X$, one may compare the derived category of quasi-coherent sheaves on $X$ with the category of dg-modules over the dg-ring of functions on $X$. We study the analogous question in stable homotopy theory, for derived stacks that arise via realizations of diagrams of Landweber-exact homology theories. We identify a condition (quasi-affineness of the map to the moduli stack of formal groups) under which the two categories are equivalent, and study applications to topological modular forms. In particular, we provide new examples of Galois extensions of ring spectra and van…
This thesis proposes a combinatorial generalization of a nilpotent operator on a vector space. The resulting object is highly natural, with basic connections to a variety of fields in pure mathematics, engineering, and the sciences. For the purpose of exposition we focus the discussion of applications on homological algebra and computation, with additional remarks in lattice theory, linear algebra, and abelian categories. For motivation, we recall that the methods of algebraic topology have driven remarkable progress in the qualitative study of large, noisy bodies of data over the past 15 year…
