The telescope conjecture for algebraic stacks

Hall, Jack; Rydh, David

doi:10.1112/topo.12021

Cited by 9 publications

(11 citation statements)

References 30 publications

(87 reference statements)

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“…Theorem A, as well as the theory developed to establish it, have been used to classify the thick tensor ideals in [Hal16] (generalizing work of [Kri09, DM12]), to resolve the telescope conjecture for algebraic stacks [HR17] (extending [Ant14]), and for results on dg-enhancements [CS16, BLS16].…”

Section: Introductionmentioning

confidence: 99%

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Perfect complexes on algebraic stacks

Hall¹,

Rydh²

2017

Compositio Math.

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127

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Abstract. We develop a theory of unbounded derived categories of quasicoherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks that either have finite stabilizers or are local quotient stacks. We also extend Toën and Antieau-Gepner's results on derived Azumaya algebras and compact generation of sheaves on linear categories from derived schemes to derived DeligneMumford stacks. These are all consequences of our main theorem: compact generation of a presheaf of triangulated categories on an algebraic stack is local for the quasi-finite flat topology.

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Section: Introductionmentioning

confidence: 99%

“…Antieau [Ant14] has considered local-global results for the telescope conjecture. Some of these are generalized in [HR17].…”

Section: Introductionmentioning

confidence: 99%

Perfect complexes on algebraic stacks

Hall¹,

Rydh²

2017

Compositio Math.

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127

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“…We will prove below that D(Ind(A)) is rigidly compactly generated by D b (A) (that is, D(Ind(A)) is compactly generated, D b (A) is the full subcategory of compact objects, and these are moreover rigid). It is then a formal consequence that ω induces an injective map from the set of thick tensor ideals in D b (A) to those in D b (K ) [17,Corollary 4.2]. As the latter category is simple so is the former, as required.…”

Section: Lemma 26 Let a Be The Filtered Colimit Of Exact Categoriesmentioning

confidence: 95%

“…It therefore derives trivially to a conservative tensor triangulated functor preserving coproducts

ω : D false(prefixInd (A) false) \to D false(K^{'} false) .

We will prove below that

D (Ind (A))

is rigidly compactly generated by

{prefixD}^{prefixb} false(scriptA false)

(that is,

D (Ind (A))

is compactly generated,

{prefixD}^{prefixb} false(scriptA false)

is the full subcategory of compact objects, and these are moreover rigid). It is then a formal consequence that

ω

induces an injective map from the set of thick tensor ideals in

{prefixD}^{prefixb} false(scriptA false)

to those in

{prefixD}^{prefixb} false(K^{'} false)

[17, Corollary 4.2]. As the latter category is simple so is the former, as required.…”

Section: Simplicity Of (Derived) Tannakian Categoriesmentioning

confidence: 99%

“…† As Tannakian categories appear frequently in algebraic geometry (and elsewhere), this result is of independent interest. The proof of Theorem 2 proceeds by translating the problem into one about quasi-coherent sheaves on well-behaved stacks where one can exploit finite cohomological dimension in characteristic zero [13,16,17].…”

Section: Theorem 2 Let a Be A Tannakian Category In Characteristic Zmentioning

confidence: 99%

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A note on Tannakian categories and mixed motives

Gallauer

2020

Bull. London Math. Soc.

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We explain why every non‐trivial exact tensor functor on the triangulated category of mixed motives over a field double-struckF has zero kernel, if one assumes ‘all’ motivic conjectures. In other words, every non‐zero motive generates the whole category up to the tensor triangulated structure. Under the same assumptions, we also give a complete classification of triangulated étale motives over double-struckF with integral coefficients, up to the tensor triangulated structure, in terms of the characteristic and the orderings of double-struckF.

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