Deformation Theory of the Chow Group of Zero Cycles

Lüders, Morten

doi:10.1093/qmathj/haaa004

Cited by 3 publications

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“…With V and X as in Corollary 4.4, the result implies by functoriality the existence of a natural restriction map CH j (X)/p r → H j Nis (X s , K M j,Xs /p r ) for each s ≥ 1, where X s := X ⊗ V V /m s is the corresponding thickening of the special fibre. When j = d is the relative dimension of X then these restriction maps are surjective by earlier work of the first author [32,33], answering a question of Kerz-Esnault-Wittenberg [28, Conj. 10.1] in the smooth case.…”

Section: An Application To the Gersten Conjecturementioning

confidence: 87%

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: An Application To the Gersten Conjecturementioning

confidence: 87%

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

Lüders¹,

Morrow²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Milnor K-theory of p-adic rings

Lüders

Morrow

2022

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We study the mod p r {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 {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.

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