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

Abstract

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 }.

Algebraic geometry in mixed characteristic

“…In mixed characteristic, while we do not know the full story, a positive answer at the associated graded level follows from the comparison result from [36] mentioned in Remark 2.14. Let us also remark that [168] has established the expected relationship of Milnor K-theory (extended as in [145]) and the (i, i)-part of the syntomic complexes, giving a p-adic analog of [185,225].…”

Ramified class field theory and duality over finite fields

“…The middle vertical arrow is an isomorphism by the Bloch-Gabber-Kato theorem for fields [5,Corollary 2.8] and the proof of the Gersten conjecture for Milnor K-theory ([27, Proposition 10]) and logarithmic Hodge-Witt sheaves ([13, Thèoréme 1.4]). The right vertical arrow is an isomorphism by [36,Theorem 0.3]. Since the rows are exact, it follows that the left vertical arrow is also an isomorphism.…”

“…It is easily seen that this map exists. One knows (e.g., see [36,Remark 1.6]) that this map in fact factors through dlog∶ KM r,X p m ↠ W m Ω r X,log . Moreover, this map is multiplicative.…”

Syntomic complexes and $p$-adic étale Tate twists

“…In general, the Kummer map (obtained from Example 1.5 and the cup product) induces a map (O × X ) ⊗i → H i (Z/p n (i) X ) which one can show to be surjective; see also [LM21] for more on the target; this determines the image of H i (Z/p n (i) X ) → R i j * (µ ⊗i p n ) as the subsheaf generated by O × X -symbols. 1 Theorem 1.8.…”

“…For more refined results about the connection of the {H i (Z/p n (i)(R))} to p-adic Milnor K-theory, cf. [LM21]. In the following, we use that for any ring R, we have a natural Kummer map R × → H 1 (Z p (1)(R)), cf.…”