Topological logarithmic structures

Rognes, John

doi:10.2140/gtm.2009.16.401

Cited by 26 publications

(67 citation statements)

References 66 publications

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“…Definition 1. 20. A monoid scheme X is pc (respectively, pctf) if for each x in X the stalk monoid A x is pc (respectively, pctf) in the sense of Definition 1.…”

Section: The Conductor Ideal For An Inclusionmentioning

confidence: 99%

The K -theory of toric varieties in positive characteristic

Cortiñas

Haesemeyer

Walker

et al. 2013

Journal of Topology

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“…Definition 1. 20. A monoid scheme X is pc (respectively, pctf) if for each x in X the stalk monoid A x is pc (respectively, pctf) in the sense of Definition 1.…”

Section: The Conductor Ideal For An Inclusionmentioning

confidence: 99%

The K -theory of toric varieties in positive characteristic

Cortiñas

Haesemeyer

Walker

et al. 2013

Journal of Topology

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“…With the rapid developments in derived algebraic geometry (e.g., Francis [34], Lurie [61,62,60], and Toën-Vezzosi [89,90]) and algebraic K-theory (e.g., Hesselholt-Madsen [51] and Rognes [82]), and the central nature of topological Quillen homology as a primary notion of a "homology" invariant that is sensitive to the algebraictopological structure, the development of these new tools for spectral algebras will have rich potential payoffs for any applications exploiting structured ring spectra.…”

Section: Introductionmentioning

confidence: 99%

Derived Koszul duality and TQ-homology completion of structured ring spectra

Ching

Harper

2019

Advances in Mathematics

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Working in the context of symmetric spectra, we consider any higher algebraic structures that can be described as algebras over an operad O. We prove that the fundamental adjunction comparing O-algebra spectra with coalgebra spectra over the associated comonad K, via topological Quillen homology (or TQ-homology), can be turned into an equivalence of homotopy theories by replacing O-algebras with the full subcategory of 0-connected Oalgebras. This resolves in the affirmative the 0-connected case of a conjecture of Francis-Gaitsgory.This derived Koszul duality result can be thought of as the spectral algebra analog of the fundamental work of Quillen and Sullivan on the rational homotopy theory of spaces, and the subsequent p-adic and integral work of Goerss and Mandell on cochains and homotopy type-the following are corollaries of our main result: (i) 0-connected O-algebra spectra are weakly equivalent if and only if their TQ-homology spectra are weakly equivalent as derived Kcoalgebras, and (ii) if a K-coalgebra spectrum is 0-connected and cofibrant, then it comes from the TQ-homology spectrum of an O-algebra. We construct the spectral algebra analog of the unstable Adams spectral sequence that starts from the TQ-homology groups TQ * (X) of an O-algebra X, and prove that it converges strongly to π * (X) when X is 0-connected.

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“…Our primary reference for logarithmic structures is [Ill02], supplemented by [Kat89,Rog09,Ogu]. Since our ultimate objects of study will be moduli of elliptic curves, we must also discuss this in the stack case.…”

Section: Logarithmic Structuresmentioning

confidence: 99%

“…Rognes has recently developed a closely related concept of topological logarithmic structures for applications in algebraic K-theory [Rog09]. The core construction in this paper is built on a map of E ∞ spaces from N to the multiplicative monoid of Tmf after completion at the cusp, modeling the logarithmic structure of M ell at the cusp.…”

Section: Introductionmentioning

confidence: 99%

Topological modular forms with level structure

Hill

Lawson

2015

Invent. math.

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The cohomology theory known as Tmf, for "topological modular forms," is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them. Along the way, we produce a natural way to restrict to the cusps, providing multiplicative maps from Tmf with level structure to forms of K-theory. In particular, this allows us to construct a connective spectrum tmf0(3) consistent with properties suggested by Mahowald and Rezk. This is accomplished using the machinery of logarithmic structures. We construct a presheaf of locally even-periodic elliptic cohomology theories, equipped with highly structured multiplication, on the log-étale site of the moduli of elliptic curves. Evaluating this presheaf on modular curves produces Tmf with level structure.

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Topological logarithmic structures

Abstract: We develop a theory of logarithmic structures on structured ring spectra, including constructions of logarithmic topological André-Quillen homology and logarithmic topological Hochschild homology.

Cited by 26 publications

References 66 publications

The K -theory of toric varieties in positive characteristic

The K -theory of toric varieties in positive characteristic

Derived Koszul duality and TQ-homology completion of structured ring spectra

Topological modular forms with level structure

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