Commutative Γ-rings do not model all commutative ring spectra

Lawson, Tyler

doi:10.4310/hha.2009.v11.n2.a9

Cited by 10 publications

(7 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…via Theorem 4.13) would prove that D-spaces satisfy the commutative monoid axiom. Secondly, because of the rectification axiom the rest of the work in [34] and [48] would prove that commutative D-rings were Quillen equivalent to E ∞ -algebras and this would contradict the main theorem of Tyler Lawson's paper [28].…”

Section: General Diagram Spectramentioning

confidence: 99%

Model structures on commutative monoids in general model categories

White

2017

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

We provide conditions on a monoidal model category M so that the category of commutative monoids in M inherits a model structure from M in which a map is a weak equivalence or fibration if and only if it is so in M. We then investigate properties of cofibrations of commutative monoids, rectification between E ∞ -algebras and commutative monoids, the relationship between commutative monoids and monoidal Bousfield localization functors, when the category of commutative monoids can be made left proper, and functoriality of the passage from a commutative monoid R to the category of commutative Ralgebras. In the final section we provide numerous examples of model categories satisfying our hypotheses.

show abstract

Section: General Diagram Spectramentioning

confidence: 99%

Model structures on commutative monoids in general model categories

White

2017

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

show abstract

“…For example, any Γ-space is canonically an S-module. A Γ-ring give rise a connective ring spectrum, although the two notions are not quite the same [21]. Important examples come from the Eilenberg-MacLane spectrum construction, which associates a spectrum HV to a simplicial abelian group V .…”

Section: 33mentioning

confidence: 99%

Derived Galois deformation rings

Galatius¹,

Venkatesh²

2018

Advances in Mathematics

View full text Add to dashboard Cite

We define a derived version of Mazur's Galois deformation ring. It is a pro-simplicial ring R classifying deformations of a fixed Galois representation to simplicial coefficient rings; its zeroth homotopy group π0R recovers Mazur's deformation ring.We give evidence that these rings R occur in the wild: For suitable Galois representations, the Langlands program predicts that π0R should act on the homology of an arithmetic group. We explain how the Taylor-Wiles method can be used to upgrade such an action to a graded action of π˚R on the homology.

show abstract

“…For instance, for our examples, we prove this below in Proposition 4.15. It is shown in [30] that the commutative ring spectra that come from commutative Γ-rings are rather special: they have trivial Dyer-Lashof operations of positive degree on their zeroth homology group. For instance, the free E ∞ -ring spectrum on a 0-cell cannot be modeled as a commutative Γ-ring.…”

Section: Stable Homotopical Algebramentioning

confidence: 99%

The homotopy types of free racks and quandles

Lawson,

Szymik

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We initiate the homotopical study of racks and quandles, two algebraic structures that govern knot theory and related braided structures in algebra and geometry. We prove analogs of Milnor's theorem on free groups for these theories and their pointed variants, identifying the homotopy types of the free racks and free quandles on spaces of generators. These results allow us to complete the stable classification of racks and quandles by identifying the ring spectra that model their stable homotopy theories. As an application, we show that the stable homotopy of a knot quandle is, in general, more complicated than what any Wirtinger presentation coming from a diagram predicts.

show abstract

Commutative Γ-rings do not model all commutative ring spectra

Abstract: We show that the free E ∞ -algebra on a zero-cell cannot be modeled by a commutative Γ-ring. The proof shows that DyerLashof operations of positive degree must vanish on the zero'th homology of such an object.

Cited by 10 publications

References 9 publications

Model structures on commutative monoids in general model categories

Model structures on commutative monoids in general model categories

Derived Galois deformation rings

The homotopy types of free racks and quandles

Contact Info

Product

Resources

About