Cdh descent, cdarc descent, and Milnor excision

Elmanto, Elden; Hoyois, Marc; Iwasa, Ryomei; Kelly, Shane

doi:10.1007/s00208-020-02083-5

Cited by 8 publications

(6 citation statements)

References 30 publications

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“…Note that both these sheaves commute with filtered colimits of rings, the latter because Ktheory is finitary and the former by [Sta18, Tag 0737]; furthermore, both these sheaves take values in D(Z/p r ) 0 , the latter by Theorem 1.1. Since both these cdh sheaves satisfy Milnor excision (the latter by Theorem 4.4), "(1) implies (3)" of the main theorem of [EHIK21] therefore implies that they are both in fact cdarc sheaves. This completes the proof of part (2).…”

Section: Characteristic Pmentioning

confidence: 87%

The K-theory of perfectoid rings

Antieau¹,

Mathew²,

Morrow³

2022

Preprint

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We establish various properties of the p-adic algebraic K-theory of smooth algebras over perfectoid rings living over perfectoid valuation rings. In particular, the p-adic K-theory of such rings is homotopy invariant, and coincides with the p-adic K-theory of the p-adic generic fibre in high degrees. In the case of smooth algebras over perfectoid valuation rings of mixed characteristic the latter isomorphism holds in all degrees and generalises a result of Nizioł.

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Section: Characteristic Pmentioning

confidence: 87%

The K-theory of perfectoid rings

Antieau¹,

Mathew²,

Morrow³

2022

Preprint

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“…If Y is quasi-compact, then there is a finite subcover refining Y ′ (see e.g. [EHIK,Lem. 2.1.2]), so in particular we may take Y ′ quasi-compact.…”

Section: (I)mentioning

confidence: 99%

Equivariant generalized cohomology via stacks

Khan¹,

Ravi²

2022

Preprint

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We prove a general form of the statement that the cohomology of a quotient stack can be computed by the Borel construction. It also applies to the lisse extensions of generalized cohomology theories like motivic cohomology and algebraic cobordism. As a consequence, we deduce the localization property for the equivariant algebraic bordism theory of Deshpande-Krishna-Heller-Malagón-López. We also give a Bernstein-Lunts-type gluing description of the ∞-category of equivariant sheaves on a scheme X, in terms of nonequivariant sheaves on X and sheaves on its Borel construction.

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“…Regarding the K-theory of perfect schemes themselves (rather than smooth schemes over them), we record the following calculation since it has not explicitly appeared previously; the cdarc topology, which is a completely decomposed analogue of the arc topology [BM21], is defined in [EHIK21].…”

Section: Characteristic Pmentioning

confidence: 99%

The $\text{K}$-theory of perfectoid rings

Antieau,

Mathew,

Morrow

2022

Doc. Math.

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We establish various properties of the p-adic algebraic Ktheory of smooth algebras over perfectoid rings living over perfectoid valuation rings. In particular, the p-adic K-theory of such rings is homotopy invariant, and coincides with the p-adic K-theory of the p-adic generic fibre in high degrees. In the case of smooth algebras over perfectoid valuation rings of mixed characteristic the latter isomorphism holds in all degrees and generalises a result of Nizioł.

show abstract

Cdh descent, cdarc descent, and Milnor excision

Cited by 8 publications

References 30 publications

The K-theory of perfectoid rings

The K-theory of perfectoid rings

Equivariant generalized cohomology via stacks

The $\text{K}$-theory of perfectoid rings

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