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Daniel Dugger

Publications and source records attributed to Daniel Dugger.

At least 37 records · Page 2Linked to original sources
arXiv2009

Postnikov extensions of ring spectra

We give a functorial construction of k-invariants for ring spectra and use these to classify extensions in the Postnikov tower of a ring spectrum.

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arXiv2009

The motivic Adams spectral sequence

We present some data on the cohomology of the motivic Steenrod algebra over an algebraically closed field. We discuss several features of the associated Adams spectral sequence, including the basic construction and convergence properties. The paper also deals with motivic versions of the May and Adams-Novikov spectral sequences. It is shown how these tools can be used to give new proofs of some classical results in algebraic topology. Also, the considerations reveal the existence of certain "exotic" motivic homotopy classes which have no classical analogues.

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arXiv2007

A curious example of two model categories and some associated differential graded algebras

The paper gives a new proof that the model categories of stable modules for the rings Z/(p^2) and (Z/p)[ε]/(ε^2) are not Quillen equivalent. The proof uses homotopy endomorphism ring spectra. Our considerations lead to an example of two differential graded algebras which are derived equivalent but whose associated model categories of modules are not Quillen equivalent. As a bonus, we also obtain derived equivalent dgas with non-isomorphic K-theories.

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arXiv2007

Enriched model categories and an application to additive endomorphism spectra

We define the notion of an additive model category, and we prove that any additive, stable, combinatorial model category has a natural enrichment over symmetric spectra based on simplicial abelian groups. As a consequence, every object in such a model category has a naturally associated endomorphism ring inside this spectra category. We establish the basic properties of this enrichment. We also develop some enriched model category theory. In particular, we have a notion of an adjoint pair of functors being a 'module' over another such pair. Such things are called "adjoint modules". We develop the general theory of these, and use them to prove a result about transporting enrichments over one symmetric monoidal model category to a Quillen equivalent one.

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arXiv2006

Etale homotopy and sums-of-squares formulas

This paper uses a relative of BP-cohomology to prove a theorem in characteristic p algebra. Specifically, we obtain some new necessary conditions for the existence of sums-of-squares formulas over fields of characteristic p > 2. These conditions were previously known in characteristic zero by results of Davis. Our proof uses a generalized etale cohomology theory called etale BP2.

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arXiv2006

Topological equivalences for differential graded algebras

We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an Eilenberg-Mac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasi-isomorphic dgas are topologically equivalent, but we produce explicit counter-examples of the converse. We also develop an associated notion of topological Morita equivalence using a homotopical version of tilting.

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arXiv2005

Large annihilators in Cayley-Dickson algebras

Cayley-Dickson algebras are an infinite sequence of non-associative algebras starting with the reals, complexes, quaternions, and octonions. We study the zero-divisors in the higher Cayley-Dickson algebras. In particular, we show that the annihilator of any element in the 2^n-dimensional Cayley-Dickson algebra has dimension at most 2^n-4n+4. Moroever, every multiple of four between 0 and this upper bound actually occurs as the dimension of some annihilator (a theorem of Moreno says that only multiples of four can occur). We completely describe all the zero-divisors whose annihilator has dimension 2^n-4n+4.

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arXiv2005

Motivic cell structures

An object in motivic homotopy theory is called cellular if it can be built out of motivic spheres using homotopy colimit constructions. We explore some examples and consequences of cellularity. We explain why the algebraic K-theory and algebraic cobordism spectra are both cellular, and prove some Kunneth theorems for cellular objects.

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arXiv2005

Spectral enrichments of model categories

We prove that every stable, combinatorial model category has a natural enrichment by symmetric spectra (or more precisely, a natural equivalence class of enrichments). This in some sense generalizes the simplicial enrichments of model categories provided by the Dwyer-Kan hammock localization. As one particular application, we are able to associate to every object of a stable, combinatorial model category a "homotopy endomorphism ring spectrum". The homotopy types of these ring spectra are preserved by Quillen equivalences, and so are an invariant for stable model categories.

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arXiv2004

Notes on the Milnor conjectures

These are some notes on the two Milnor conjectures and their proofs (due to Voevodsky, Orlov-Vishik-Voevodsky, and Morel).

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arXiv2004

Algebraic K-theory and sums-of-squares formulas

We prove a result about the non-existence of certain sums-of-squares formulas over a field. This generalizes an old theorem which used topological K-theory to obtain obstruction conditions when the field is the real numbers. Our result applies to arbitrary fields not of characteristic 2, making use of algebraic K-theory in place of topological K-theory.

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arXiv2003

The Hopf condition for bilinear forms over arbitrary fields

We settle an old question about the existence of certain "sums-of-squares" formulas over a field F (which are the simplest examples of composition formulas for quadratic forms). A classical theorem says that if such a formula exists over a field of characteristic 0, then certain binomial coefficients must be even. We use motivic cohomology to prove that the same result holds in characteristic p.

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arXiv2003

Hypercovers and simplicial presheaves

We prove that Jardine's model category of simplicial presheaves can be obtained by localizing the `discrete' version at the collection of all hypercovers. One consequence is that the fibrant objects can be explicitly identified in terms of a hypercover descent condition. Another is a very simple approach to change-of-site functors. In an appendix, we discuss how this hypercover localization compares to the more naive process of localizing at the Cech complexes; the two are not the same in general, but agree in some cases of interest.

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arXiv2003

Multiplicative structures on homotopy spectral sequences I

This mostly expository paper records some basic facts about towers of homotopy fiber sequences. We give a proof that a pairing of towers induces a pairing of associated spectral sequences, for towers of spaces and towers of spectra.

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arXiv2002

K-theory and derived equivalences

We show that if two rings have equivalent derived categories then they have the same algebraic K-theory. Similar results are given for G-theory, and for a large class of abelian categories.

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