Tannaka duality and stable infinity-categories

Iwanari, Isamu

doi:10.1112/topo.12057

Cited by 4 publications

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References 43 publications

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“…It would be interesting to understand the precise relationship between these constructions. It would also be interesting to relate the results about universal graded-Tannakian categories to derived versions of Tannaka duality due to Iwanari [Iwa18] and Pridham [Pri18].…”

Section: Introductionmentioning

confidence: 99%

Graded-Tannakian categories of motives

Schäppi

2020

Preprint

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Given a rigid tensor-triangulated category and a vector space valued homological functor for which the Künneth isomorphism holds, we construct a universal graded-Tannakian category through which the given homological functor factors. We use this to (unconditionally) construct graded-Tannakian categories of pure motives associated to a fixed Weil cohomology theory, with a fiber functor realizing the given cohomology theory. Foradic cohomology and a ground field which is algebraic over a finite field, this category is Tannakian. In this case, we obtain in particular motivic Galois groups which act naturally on -adic cohomology without assuming any of the standard conjectures. We show that these graded-Tannakian categories are equivalent to Grothendieck's category of pure motives if the standard conjecture D holds. Contents 1. Introduction 1 Acknowledgements 3 2. Graded-Tannakian categories associated to Künneth functors 4 2.1. The underlying category 4 2.2. Day convolution 8 2.3. The universal property 11 3. Applications to Grothendieck's categories of motives 12 3.1. The basic Tannakian factorization 12 3.2. Tannakian categories associated to Weil cohomology theories 15 3.3. Homological standard conjectures 16 References 20

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Section: Introductionmentioning

confidence: 99%

Graded-Tannakian categories of motives

Schäppi

2020

Preprint

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“…• Iwanari [11] uses derived Tannaka duality to describe the stable ∞-category of motives generated by a Kimura finite Chow motives as a symmetric monoidal stable ∞-category of quasi-coherent complexes on a derived quotient stack. In particular, motives for an elliptic curve are Kimura finite.…”

Section: Introductionmentioning

confidence: 99%

Motives for an elliptic curve

Cao

2018

Math. Ann.

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In this paper we describe the rigid tensor triangulated subcategory of Voevodsky's triangulated category of motives generated by the motive of an elliptic curve as a derived category of dg modules over a commutative differential graded algebra in the category of representations over some reductive group.1 For the CM case, we only consider the complex multiplication is defined over k.Furthermore, the piecewhere V (n,0) ⊗ V ⊗i ⊗V ⊗j are pairwise non-isomorphic irreducible representations over a K-algebra End Res K/Q Gm⊗K ((F K ) ⊗n ) and V i,j are pairwise non-isomorphic irreducible representation over Res K/Q G m ⊗ K = T K . For simplicity, we delete V i,j and one may think that both V andV are endowed with the G m -action. In fact, End Res K/Q Gm⊗K ((F K ) ⊗n ) is a special case defined in the Section 3.9 of [1], which is called B n,K . Ancona's main result -Theorem 4.1 in [1] implies that the decomposition like Lemma 2.1 is holding for the CM elliptic motives:3 Here λ t is the transpose (or conjugate) of λ, which is defined by interchanging rows and columns in the Young diagram associated to λ. 4 We view cn as an idempotent in End((F K ) ⊗n ), which lies in End Res K/Q Gm⊗K ((F K ) ⊗n ).

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Regularity Of Spectral Stacks And Discreteness Of Weight-Hearts

Khan

Sosnilo

2021

The Quarterly Journal of Mathematics

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We study regularity in the context of connective ring spectra and spectral stacks. Parallel to that, we construct a weight structure on the category of compact quasi-coherent sheaves on spectral quotient stacks of the form $X=[\operatorname{Spec} R/G]$ defined over a field, where R is a connective ${{\mathcal{E}}_\infty}$-k-algebra and G is a linearly reductive group acting on R. Under reasonable assumptions, we show that regularity of X is equivalent to regularity of R. We also show that if R is bounded, such a stack is discrete. This result can be interpreted in terms of weight structures and suggests a general phenomenon: for a symmetric monoidal stable $\infty$-category with a compatible bounded weight structure, the existence of an adjacent t-structure satisfying a strong boundedness condition should imply discreteness of the weight-heart. We also prove a gluing result for weight structures and adjacent t-structures, in the setting of a semi-orthogonal decomposition of stable $\infty$-categories.

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Tannaka duality and stable infinity-categories

Cited by 4 publications

References 43 publications

Graded-Tannakian categories of motives

Graded-Tannakian categories of motives

Motives for an elliptic curve

Regularity Of Spectral Stacks And Discreteness Of Weight-Hearts

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