<i>K</i>-theory of valuation rings

Kelly, Shane; Morrow, Matthew

doi:10.1112/s0010437x21007119

Cited by 12 publications

(19 citation statements)

References 44 publications

(59 reference statements)

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“…More precisely, given a base ring k and letting L sm K M j : k -algs loc → D(Z) be the left Kan extension of K M j from local, essentially smooth k-algebras to all local k-algebras, then show in Proposition 1.17 that the counit map H 0 (L sm K M j (A)) → K M j (A) is an isomorphism on any local k-algebra A. This left Kan extension observation also provides a new tool to control the Milnor K-theory of F palgebras; in particular, we prove the following Bloch-Kato-Gabber style isomorphisms, which are already known to hold [7,25,37] if improved Milnor K-theory is replaced by algebraic K-theory: Theorem 0.3. Let r, j ≥ 0.…”

Section: Introductionmentioning

confidence: 64%

“…We have reduced (i) and (ii) to proving that on the category CSm Fp , the functors Ω j log and RΓ ét (Spec −, Ω j log ) are left Kan extended from Sm Fp and from Sm Σ Fp respectively. But it was shown in [25] that both Ω j and dΩ j−1 , on the category CSm Fp , are left Kan extended from Sm Σ Fp , so the same is true of RΓ ét (Spec −, Ω j log ) using the second equivalence of part (ii). This equivalence also shows that there is a natural fibre sequence Ω j log → RΓ ét (Spec A, Ω j log )[j] → ν(j)(A)[−1] for all A ∈ CSm Fp ; the final term is rigid [7,Prop.…”

Section: An Application To the Gersten Conjecturementioning

confidence: 95%

“…4.30] hence left Kan extended from Sm Fp , and so we deduce the same for the first term. This allows us to complement the main theorem of [25], namely K j (A; Z/p r Z) ∼ = W r Ω j A,log , by also describing Milnor K-theory: Theorem 5.2 (Bloch-Kato-Gabber theorem for Cartier smooth algebras). For any local, Cartier smooth F p -algebra A, the symbol map K M j (A)/p r → W r Ω j A,log is an isomorphism for any r, j ≥ 0.…”

Section: An Application To the Gersten Conjecturementioning

confidence: 99%

“…We start by recalling the following terminology from [25]: an F p -algebra A is called Cartier smooth if it satisfies the following smoothness criteria:…”

Section: An Application To the Gersten Conjecturementioning

confidence: 99%

“…The following description of the étale-syntomic cohomology of Cartier smooth algebras was implicit in [25]:…”

Section: An Application To the Gersten Conjecturementioning

confidence: 99%

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Milnor $K$-theory of $p$-adic rings

Lüders¹,

Morrow²

2021

Preprint

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We study the mod p r Milnor K-groups of p-adically complete and p-henselian rings, establishing in particular a Nesterenko-Suslin style description in terms of the Milnor range of syntomic cohomology. In the case of smooth schemes over complete discrete valuation rings we prove the mod p r Gersten conjecture for Milnor K-theory locally in the Nisnevich topology. In characteristic p we show that the Bloch-Kato-Gabber theorem remains true for valuation rings, and for regular formal schemes in a pro sense.1 Formulation of main theorems for p-henselian, ind-smooth algebrasIn this section we state our main theorems concerning ind-smooth algebras over complete discrete valuation rings, and establish some relations between them as well as the key reductions to the special case which will then be proved in Section 2. Main theoremsWe adopt the usual definition of Milnor K-theory, even for non-local rings:Definition 1.1 (Milnor K-theory). Let R be a (always commutative) ring. We define the j th Milnor K-group K M j (R) to be the quotient of (R × ) ⊗j by the Steinberg relations, i.e. the subgroup of (R × ) ⊗j generated by elements of the form a 1 ⊗ • • • ⊗ a j where a l + a k = 1 for some 1 ≤ l < k ≤ j. As usual, the image of a 1 ⊗ • • • ⊗ a j in K M j (R) is denoted by {a 1 , . . . , a j }.

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Section: Introductionmentioning

confidence: 64%

Section: An Application To the Gersten Conjecturementioning

confidence: 95%

Section: An Application To the Gersten Conjecturementioning

confidence: 99%

“…We start by recalling the following terminology from [25]: an F p -algebra A is called Cartier smooth if it satisfies the following smoothness criteria:…”

Section: An Application To the Gersten Conjecturementioning

confidence: 99%

“…The following description of the étale-syntomic cohomology of Cartier smooth algebras was implicit in [25]:…”

Section: An Application To the Gersten Conjecturementioning

confidence: 99%

See 3 more Smart Citations

Milnor $K$-theory of $p$-adic rings

Lüders¹,

Morrow²

2021

Preprint

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K-theory of Prüfer domains

Banerjee

Sadhu

2022

Arch. Math.

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In this article, we show that for a quasicompact scheme X and n > 0, the n-th K-group K n (X) is a λ-module over a λ-ring K 0 (X) in the sense of Hesselholt. introductionIn [4], L. Hesselholt introduced the notion of module over λ-rings, i.e., λ-module. Let us first recall the definition from [4].such that the following axioms hold:(1) λ M,1 = id M ;(2) λ M,n λ M,m = λ M,nm for all m, n ≥ 1;(3) λ M,n (ax) = ψ n (a)λ M,n (x) for all a ∈ R and x ∈ M. Here ψ n is the n-th Adams operation associated to (R, λ R ).If we set M = R and λ M,n = ψ n , where ψ n are the Adams operations associated to (R, λ R ), then (R, ψ n ) is a (R, λ R )-module. For a quasicompact scheme X, K 0 (X) is a λ-ring with λ-operations defined by the usual exterior power on vector bundles. These exterior power operations have been extended to higher K-groups by several authors using homotopy theory (see [3], [6], [7] and [8]). Recently, a purely algebraic construction of the exterior power operations on higher K-groups of any quasicompact scheme is given in [2] using Grayson's description of higher K-groups in terms of binary complexes. In this article, we use the exterior power operations constructed in [2] to give a λ-module structure on K n (X) over K 0 (X) for n > 0 and any quasicompact scheme X.Here is our precise result: Theorem 1.2. For any quasicompact scheme X and n > 0, each K n (X) is a λ-module over K 0 (X).

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