The derived category of complex periodic K-theory localized at an odd prime

Patchkoria, Irakli

doi:10.1016/j.aim.2017.01.023

Cited by 12 publications

(38 citation statements)

References 36 publications

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“…Here we just use that the homological dimension of the abelian category

{false(Grmod- π_{*} S false)}^{P}

is at most 3. Using the results of , one can see that these functors are compatible with the derivator structures. From this one obtains (see Section 4): Theorem Let

R

be a symmetric ring spectrum such that

π_{*} R

is concentrated in degrees divisible by a natural number

N ⩾ 5

and assume that the graded global homological dimension of

π_{*} R

is equal to two.…”

Section: Introductionmentioning

confidence: 91%

“…The aim of this paper is to extend the results of and . The following theorem is proved in Section 3: Theorem Let

R

be a symmetric ring spectrum such that

π_{*} R

is concentrated in degrees divisible by a natural number

N ⩾ 2

and assume that the graded global homological dimension of

π_{*} R

is less than

N - 1

.…”

Section: Introductionmentioning

confidence: 97%

“…It is an open problem in general to establish compatibility between the triangulated structures and the functor

scriptR

. The paper addresses this issue for homological dimension one. More precisely, in we establish the following result: Let

R

be a symmetric ring spectrum such that

π_{*} R

is concentrated in degrees divisible by a natural number

N ⩾ 4

and assume that the graded global homological dimension of

π_{*} R

is equal to one.…”

Section: Introductionmentioning

confidence: 99%

“…The paper addresses this issue for homological dimension one. More precisely, in we establish the following result: Let

R

be a symmetric ring spectrum such that

π_{*} R

is concentrated in degrees divisible by a natural number

N ⩾ 4

and assume that the graded global homological dimension of

π_{*} R

is equal to one. Then the equivalence

R : D false(π_{*} R false) \to Ho false(Mod- R false)

is triangulated.…”

Section: Introductionmentioning

confidence: 99%

“…Methods (which are due to Franke ) used in this paper as well as in are very general. In particular, they also apply to

E (1)

‐local spectra at a prime

p ⩾ 5

.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On exotic equivalences and a theorem of Franke

Patchkoria

2017

Bull. London Math. Soc.

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Abstract. Using Franke's methods we construct new examples of exotic equivalences. We show that for any symmetric ring spectrum R whose graded homotopy ring π * R is concentrated in dimensions divisible by a natural number N ≥ 5 and has homological dimension at most three, the homotopy category of R-modules is equivalent to the derived category of π * R. The Johnson-Wilson spectrum E(3) and the truncated Brown-Peterson spectrum BP 2 for any prime p ≥ 5 are our main examples. If additionally the homological dimension of π * R is equal to two, then the homotopy category of R-modules and the derived category of π * R are triangulated equivalent. Here the main examples are E(2) and BP 1 at p ≥ 5. The last part of the paper discusses a triangulated equivalence between the homotopy category of E(1)-local spectra at a prime p ≥ 5 and the derived category of Franke's model. This is a theorem of Franke and we fill a gap in the proof. IntroductionLet R be a symmetric ring spectrum and suppose that the homotopy ring π * R of R is concentrated in dimensions divisible by some natural number N ≥ 2. Further assume that the graded global homological dimension gl.dim π * R of π * R is less than N . Using Franke's methods from [5], the paper [10] constructs a functor R : D(π * R) → Ho(Mod -R). Here Ho(Mod -R) is the homotopy category of Mod-R, the model category of module spectra over R and D(π * R) is the derived category of π * R which is by definition the homotopy category of differential graded π * R-modules. The functor R has many interesting properties: It commutes with suspensions, is compatible with homology and homotopy groups, and restricts to an equivalence between the full subcategories of at most one-dimensional objects. In case when gl.dim π * R < N − 1, then R restricts to an equivalence between the full subcategories of at most two dimensional objects. As an application of the latter fact, one concludes that R : D(π * R) → Ho(Mod -R) is an equivalence of categories if N ≥ 4 and gl.dim π * R ≤ 2. These facts are proved in [10]. In particular, the latter result improves a classification theorem by Wolbert [14]. This paper extends the results of [10] to homological dimension three. More precisely we prove the following: Theorem 1.1.1. Let R be a symmetric ring spectrum such that π * R is concentrated in degrees divisible by a natural number N ≥ 2 and assume that the graded global homological dimension of π * R is less than N − 1. Then the functor R : D(π * R) → Ho(Mod-R) restricts to an equivalence of the full subcategories of at most three dimensional objects.As a consequence we obtain: Corollary 1.1.2. Let R be a symmetric ring spectrum such that π * R is concentrated in degrees divisible by a natural number N ≥ 5 and assume that the graded global homological dimension of π * R is less than or equal to three. Then the functor R : D(π * R) → Ho(Mod-R) is an equivalence of categories.Examples to which this corollary applies are the Johnson-Wilson spectrum E(3) and the truncated Brown-Peterson spectrum BP 2 at any pri...

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“…Here we just use that the homological dimension of the abelian category

{false(Grmod- π_{*} S false)}^{P}

is at most 3. Using the results of , one can see that these functors are compatible with the derivator structures. From this one obtains (see Section 4): Theorem Let

R

be a symmetric ring spectrum such that

π_{*} R

is concentrated in degrees divisible by a natural number

N ⩾ 5

and assume that the graded global homological dimension of

π_{*} R

is equal to two.…”

Section: Introductionmentioning

confidence: 91%

“…The aim of this paper is to extend the results of and . The following theorem is proved in Section 3: Theorem Let

R

be a symmetric ring spectrum such that

π_{*} R

is concentrated in degrees divisible by a natural number

N ⩾ 2

and assume that the graded global homological dimension of

π_{*} R

is less than

N - 1

.…”

Section: Introductionmentioning

confidence: 97%

“…It is an open problem in general to establish compatibility between the triangulated structures and the functor

scriptR

. The paper addresses this issue for homological dimension one. More precisely, in we establish the following result: Let

R

be a symmetric ring spectrum such that

π_{*} R

is concentrated in degrees divisible by a natural number

N ⩾ 4

and assume that the graded global homological dimension of

π_{*} R

is equal to one.…”

Section: Introductionmentioning

confidence: 99%

“…The paper addresses this issue for homological dimension one. More precisely, in we establish the following result: Let

R

be a symmetric ring spectrum such that

π_{*} R

is concentrated in degrees divisible by a natural number

N ⩾ 4

and assume that the graded global homological dimension of

π_{*} R

is equal to one. Then the equivalence

R : D false(π_{*} R false) \to Ho false(Mod- R false)

is triangulated.…”

Section: Introductionmentioning

confidence: 99%

“…Methods (which are due to Franke ) used in this paper as well as in are very general. In particular, they also apply to

E (1)

‐local spectra at a prime

p ⩾ 5

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On exotic equivalences and a theorem of Franke

Patchkoria

2017

Bull. London Math. Soc.

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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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Higher homotopy categories, higher derivators, and K-theory

Raptis

2022

Forum of Mathematics, Sigma

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For every $\infty $ -category $\mathscr {C}$ , there is a homotopy n-category $\mathrm {h}_n \mathscr {C}$ and a canonical functor $\gamma _n \colon \mathscr {C} \to \mathrm {h}_n \mathscr {C}$ . We study these higher homotopy categories, especially in connection with the existence and preservation of (co)limits, by introducing a higher categorical notion of weak colimit. Using homotopy n-categories, we introduce the notion of an n-derivator and study the main examples arising from $\infty $ -categories. Following the work of Maltsiniotis and Garkusha, we define K-theory for $\infty $ -derivators and prove that the canonical comparison map from the Waldhausen K-theory of $\mathscr {C}$ to the K-theory of the associated n-derivator $\mathbb {D}_{\mathscr {C}}^{(n)}$ is $(n+1)$ -connected. We also prove that this comparison map identifies derivator K-theory of $\infty $ -derivators in terms of a universal property. Moreover, using the canonical structure of higher weak pushouts in the homotopy n-category, we also define a K-theory space $K(\mathrm {h}_n \mathscr {C}, \mathrm {can})$ associated to $\mathrm {h}_n \mathscr {C}$ . We prove that the canonical comparison map from the Waldhausen K-theory of $\mathscr {C}$ to $K(\mathrm {h}_n \mathscr {C}, \mathrm {can})$ is n-connected.

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The derived category of complex periodic K-theory localized at an odd prime

Cited by 12 publications

References 36 publications

On exotic equivalences and a theorem of Franke

On exotic equivalences and a theorem of Franke

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

Higher homotopy categories, higher derivators, and K-theory

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