Equivariant chromatic localizations and commutativity

Hill, Michael A.

doi:10.1007/s40062-018-0226-2

Cited by 9 publications

(12 citation statements)

References 12 publications

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“…As a consequence of the latter and [20, Proposition 3.2], we see that the v 1 -localisation considered in this section is in fact a special case of the localisation considered in [20] (for E = ε * K ( p) ).…”

Section: Corollary 36 For Any Subgroup H Of G There Is An Isomorphisupporting

confidence: 59%

Rigidity and exotic models for $$v_1$$-local G-equivariant stable homotopy theory

Patchkoria

Roitzheim

2019

Math. Z.

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We prove that the v 1 -local G-equivariant stable homotopy category for G a finite group has a unique G-equivariant model at p = 2. This means that at the prime 2 the homotopy theory of G-spectra up to fixed point equivalences on K -theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for K -local spectra of the second author with the equivariant rigidity result for G-spectra of the first author. Further, when the prime p is at least 5 and does not divide the order of G, we provide an algebraic exotic model as well as a G-equivariant exotic model for the v 1 -local G-equivariant stable homotopy category, showing that for primes p ≥ 5 equivariant rigidity fails in general. B Constanze Roitzheim C.Roitzheim@kent.ac.uk Irakli Patchkoria

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Section: Corollary 36 For Any Subgroup H Of G There Is An Isomorphisupporting

confidence: 59%

Rigidity and exotic models for $$v_1$$-local G-equivariant stable homotopy theory

Patchkoria

Roitzheim

2019

Math. Z.

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“…Here EF is the cofibre of the natural map from a universal space EF + to S 0 . Recently, more examples have been discussed in [17,19].…”

Section: Rational Equivariant Cohomology Theoriesmentioning

confidence: 99%

An algebraic model for rational naïve-commutative ring SO(2)-spectra and equivariant elliptic cohomology

2020

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Equipping a non-equivariant topological $$\text {E}_\infty $$ E ∞ -operad with the trivial G-action gives an operad in G-spaces. For a G-spectrum, being an algebra over this operad does not provide any multiplicative norm maps on homotopy groups. Algebras over this operad are called naïve-commutative ring G-spectra. In this paper we take $$G=SO(2)$$ G = S O ( 2 ) and we show that commutative algebras in the algebraic model for rational SO(2)-spectra model rational naïve-commutative ring SO(2)-spectra. In particular, this applies to show that the SO(2)-equivariant cohomology associated to an elliptic curve C of Greenlees (Topology 44(6):1213–1279, 2005) is represented by an $$\text {E}_\infty $$ E ∞ -ring spectrum. Moreover, the category of modules over that $$\text {E}_\infty $$ E ∞ -ring spectrum is equivalent to the derived category of sheaves over the elliptic curve C with the Zariski torsion point topology.

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“…We will also follow [H] in writing N G H for the composite functor N G H ○ ↓ G H on G-spectra. More generally, by decomposing a finite G-set T into a disjoint union of orbits, the norm N T can be interpreted as the smash product of norms of the form N G Hi , as in [H,Definition 2.2].…”

Section: The Case P ≠ Qmentioning

confidence: 99%

“…Our goal is to understand π 0 L KU G S G , the equivariant generalization of the result of Adams-Baird and Ravenel mentioned above. As S G is an E ∞ -ring in G-spectra, [H,Corollary 3.12] implies L KU G S G is again E ∞ , and hence π 0 L KU G S G is a Green functor. In fact, we show in Proposition 10.2 that L KU G S G is a G-E ∞ ring.…”

Section: Introductionmentioning

confidence: 99%