aa r X i v : . [ m a t h . K T ] J a n Milnor K -theory of p -adic rings Morten L¨uders and Matthew Morrow
Abstract
We study the mod p r Milnor K -groups of p -adically complete and p -henselian rings, establishing inparticular 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 Gerstenconjecture for Milnor K -theory locally in the Nisnevich topology. In characteristic p we show thatthe Bloch–Kato–Gabber theorem remains true for valuation rings, and for regular formal schemes ina pro sense. In this article we study the mod p -power Milnor K -theory of p -adic rings by combining classical techniquesof Bloch–Kato [6] with recent progress in p -adic motivic cohomology, notably the syntomic cohomologyintroduced by Bhatt, Scholze, and the second author [3].Our fundamental results from which the others follow are the following “Gersten injectivity” andBloch–Kato isomorphism: Theorem 0.1.
Let V be a complete discrete valuation ring of mixed characteristic, and R a local, p -henselian, ind-smooth V -algebra with infinite residue field; let j, r ≥ . Then(i) the canonical map K Mj ( R ) /p r → K Mj ( R [ p ]) /p r is injective,(ii) and the Galois symbol K Mj ( R [ p ]) /p r → H j ´et ( R [ p ] , µ ⊗ jp r ) is an isomorphism. The hypothesis that R has infinite residue field may be weakened to it having big enough residue fieldso that b K Mj ( R ) → K Mj ( R ) is an isomorphism, where the domain denotes the improved Milnor K -theoryof Gabber and Kerz [27]. However, with no size hypotheses on the residue field of R we prove exactnessof the sequence 0 −→ b K Mj ( R ) /p r −→ H j ´et ( R [ p ] , µ ⊗ jp r ) −→ W r Ω j − R/πR, log −→ , (1)where the map to the group of de Rham–Witt dlog forms of the special fibre of R is Kato’s residue map;in the case of big enough residue field this is equivalent to Theorem 0.1.Combining Theorem 0.1 with existing Gersten results in Milnor K -theory due to Kerz [26, 27] and inmotivic cohomology due to Geisser [16], we then establish the mod p -power Gersten conjecture Nisnevichlocally on smooth V -schemes: Theorem 0.2.
Let V be a complete discrete valuation ring of mixed characteristic, X a smooth V -scheme, and R := O hX,x the henselian local ring at any point x ∈ X . Then the Gersten complex −→ b K Mj ( R ) /p r −→ K Mj (Frac R ) /p r −→ M x ∈ Spec R (1) K Mj − ( k ( x )) /p r −→ M x ∈ Spec R (2) K Mj − ( k ( x )) /p r −→ · · · is exact, and consequently the Bloch–Quillen isomorphism CH j ( X ) /p r ∼ = H j Nis ( X, b K Mj,X /p r ) holds. A new, albeit simple, observation which plays a key role in our arguments is that Milnor K -theory isright exact when left Kan extending from smooth algebras; this is inspired by Bhatt–Lurie’s observationthat connective K -theory is left Kan extended in this fashion, and by the subsequent on-going intro-duction of left Kan extensions from smooth algebras into the theory of motivic cohomology [12]. Moreprecisely, given a base ring k and letting L sm K Mj : k -algs loc → D ( Z ) be the left Kan extension of K Mj k -algebras to all local k -algebras, then show in Proposition 1.17 that thecounit map H ( L sm K Mj ( A )) → K Mj ( 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 p -algebras; in particular, we prove the following Bloch–Kato–Gabber style isomorphisms, which are alreadyknown to hold [7, 25, 37] if improved Milnor K -theory is replaced by algebraic K -theory: Theorem 0.3.
Let r, j ≥ .(i) Let A be a local, Cartier smooth F p -algebra (e.g., an essentially smooth, local algebra over a char-acteristic p valuation ring). Then the dlog map b K Mj ( A ) /p r → W r Ω jA, log is an isomorphism.(ii) Let A be an F-finite, regular, Noetherian, local F p -algebra, and I ⊆ A any ideal. Then the dlogmap induces an isomorphism of pro abelian groups { b K Mj ( A/I s ) /p r } s ≃ → { W r Ω jA/I s , log } s . To state our final main result and outline the proofs, we briefly recall the syntomic cohomologytheory Z p ( j )( A ) introduced in [3] for quasisyntomic rings and extended to arbitrary p -complete rings A in [2]. This weight- j syntomic cohomology Z p ( j )( A ) is a p -complete complex which is expectedto provide a general theory of p -adic ´etale motivic cohomology for A ; consequently, by a Beilinson–Lichtenbaum principle, one expects the truncation τ ≤ j Z p ( j )( A ) to share certain properties with Zariskimotivic cohomology. As part of these expectations we prove the following relation to Milnor K -theory,thereby providing a p -adic analogue of the Nesterenko–Suslin isomorphism K Mj ( k ) ∼ = H j Mot ( k, Z ( j )) forfields k [40]. Theorem 0.4.
For any local, p -complete ring A , there are natural isomorphisms b K Mj ( A ) /p r ≃ → H j ( Z p ( j )( A ) /p r ) for all r, j ≥ . We finish the introduction by sketching the proofs of the main theorems, ignoring completely thedifficulties caused by possible small residue fields.Theorems 0.1 and 0.4 are proved by a convoluted reduction to a special case which we treat by hand.Let R be as in Theorem 0.1. First we use a forthcoming result of Bhatt–Clausen–Mathew identifying thesyntomic cohomology Z p ( j )( R ) with older approaches to p -adic ´etale motivic cohomology (see Theorem1.14 for details), a consequence of which is that the kernel of Kato’s residue map in (1) identifies with H j ( Z p ( j )( R ) /p r ). In other words, for such R , Theorems 0.1 and 0.4 are equivalent. Independently,rigidity results of Antieau–Mathew–Nikolaus joint with the second author [2] imply that τ ≤ j Z p ( j )( − ) isleft Kan extended from smooth algebras in a suitable sense (see Proposition 1.16 for details); combinedwith the right exactness property of Milnor K -theory mentioned above, this reduces Theorem 0.4 to thecase in which the ring is ind-smooth over Z p . Putting together these two steps (and making a standardtype of argument by adding roots of unity to reduce mod p r assertions to the mod p case), it is finallyenough to prove Theorem 0.1 when R is ind-smooth over Z p and r = 1.The main difficulty is then to show that the the composition K Mj ( R ) /p → K Mj ( R [ p ]) → H j ´et ( R [ p ] , µ ⊗ jp )is injective. Using Bloch–Kato’s filtration on p -adic nearby cycles, this reduces when p = 2 to showingthat their differential map Ω j − R/pR → H j ´et ( R [ p ] , µ ⊗ jp ) factors through the aforementioned composition;we construct this factoring, which we suspect is known to experts but whose proof does not appear inthe literature, using Dennis–Stein symbols. When p = 2 (but still over base dvr Z p , so that epp − = 2)a further step of the Bloch–Kato filtration, corresponding to differential maps out of the cokernel ofthe Artin–Schreier maps C − − j − εR/pR → Ω j − εR/pR /d Ω j − ε − R/pR for ε = 1 ,
2, must be analysed. Here themanipulations of the Dennis–Stein symbols are more involved, complicated further by the fact that therestriction of the unit filtration on K Mj ( R [ p ]) to K Mj ( R ) had not been previously identified; see § Remark 0.5 (Unit group filtrations) . Let V be a complete discrete valuation ring of mixed characteristic,with uniformiser π and field of fractions K ; although the observations of this remark continue to holdfor suitably smooth local V -algebras, notably as in Theorem 0.1, we restrict to V itself for the sake ofconcreteness. 2ilnor K -theory of p -adic ringsThe i th step U i K Mj ( K ) of the unit group filtration on K Mj ( K ) is defined to be the subgroup generatedby symbols { a , . . . , a j } where at least one of a , . . . , a j ∈ K × belongs to 1 + π i V . The graded piecesof this filtration have been extensively studied, by Bloch–Kato [6], Kurihara [31], Nakamura [39], andothers; see [38] for a survey.Filtering K Mj ( V ) in the analogous way (i.e., using a , . . . , a j ∈ V × with at least one belonging to1 + π i V ) results in pathological behaviour: this filtration is not multiplicative and the canonical map K Mj ( V ) → K Mj ( K ) is not injective on graded pieces. The general goal of § K Mj ( V ) (namely the restriction of the unit group filtration on K Mj ( K ), assuming injectivity of K Mj ( V ) → K Mj ( K )) in the special case that V is absolutely unramified:in fact, the pathological behaviour in this case only occurs when p = 2, in which case one must enlargethe second step of the filtration by including symbols { ap, bp } where a, b ∈ V , or equivalently byadding certain Dennis–Stein symbols. See Definition 2.3 and Remark 2.6.This suggests an alternative approach to our proof of the injectivity of K Mj ( V ) /p r → H j ´et ( K, µ ⊗ jp r ).Firstly enlarge the pathological filtration on K Mj ( V ), probably by modifying it in degrees i whenever i is divisible by p and in the range 1 ≤ i ≤ epp − (c.f., the Bloch–Kato description of the graded pieces inthese degrees [6, Corol. 1.4.1], noting that the graded step of degree epp − might not vanish as we do notwork ´etale locally). Secondly, reduce to the case r = 1 and V containing a primitive p th -root of unity.Thirdly, refine the arguments of § H j ´et ( K, µ ⊗ jp ) [6, Thm. 6.7] factor through the graded pieces of K Mj ( V ) /p .We finish the sketches of the main theorems. Theorem 0.2 follows from Theorem 0.1 and knownGersten results in motivic and ´etale cohomology, so we refer the reader directly to § p is a consequence of the classicalBloch–Kato–Gabber theorem and our left Kan extension arguments) to describing H j ( Z ( j )( − ) /p r ) ofvaluation rings and of regular Noetherian rings modulo powers of an ideal. This in turn reduces tocalculations in derived de Rham cohomology, which in the first case are due to Gabber–Ramero [15,Thm. 6.5.8(ii) & Corol. 6.5.21] and Gabber [29, App.], and in the second case may be found in work ofthe second author [37]. Acknowledgements
We thank Bhargav Bhatt, Dustin Clausen, and Akhil Mathew for discussions about their work and ShujiSaito for related correspondence.The first author is supported by the DFG Research Fellowship LU 2418/1-1. The second author waspartly supported by the ANR JCJC project
P´eriodes en G´eom´etrie Arithm´etique et Motivique (ANR-18-CE40-0017). p -henselian,ind-smooth algebras In this section we state our main theorems concerning ind-smooth algebras over complete discrete valu-ation rings, and establish some relations between them as well as the key reductions to the special casewhich will then be proved in Section 2.
We 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 MilnorK-group K Mj ( 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 ⊗ · · · ⊗ a j where a l + a k = 1 for some 1 ≤ l < k ≤ j . As usual, theimage of a ⊗ · · · ⊗ a j in K Mj ( R ) is denoted by { a , . . . , a j } .3orten L¨uders and Matthew MorrowFor any ring R and natural number p invertible in R , we let h jp : K Mj ( R ) /p −→ H j ´et ( R, µ ⊗ jp )denote Tate’s cohomological symbol, also known as the Galois symbol or norm residue homomorphism.For a proof of the existence of the Galois symbol we refer to [47] where it is proven for R being a field.The proof in the above generality is analogous. Remark 1.2 (Big residue fields and improved Milnor K -theory) . Suppose in this remark that R is alocal ring of arbitrary residue characteristic, and consider the restriction of the Galois symbol for R [ p ]to the Milnor K -theory of R , namely K Mj ( R ) −→ K Mj ( R [ p ]) h jp −→ H j ´et ( R [ p ] , µ ⊗ jp ) . This symbol factors through Gabber–Kerz’ improved Milnor K -theory b K Mj ( R ) [27] (which we recall isa quotient of K Mj ( R ) by [27, Thm. 13]). Indeed, letting R ( t ) and R ( t , t ) denote the rational functionrings of [27, Lem. 8] (which have infinite residue field), we have b K Mj ( R ) = ker( K Mj ( R ( t )) i ∗ − i ∗ −−−−−→ K Mj ( R ( t , t ))by definition, and H j ´et ( R [ p ] , µ ⊗ jp ) = ker( H j ´et ( R ( t )[ p ] , µ ⊗ jp ) i ∗ − i ∗ −−−−−→ H j ´et ( R ( t , t )[ p ] , µ ⊗ jp ))by [27, Prop. 9] since ´etale cohomology has transfers for finite ´etale extensions of rings; naturality of theGalois symbol now proves the claim.Now let S be a local, finite ´etale R -algebra. Then there exist norm maps N : b K M ∗ ( S ) → b K M ∗ ( R ) onimproved Milnor K -theory [27, §
1] (depending, for example, on a chosen presentation of S ( t ) as a finite´etale R ( t )-algebra), and we expect that these are compatible with the transfer maps on ´etale cohomologyin the sense that the following diagram commutes b K Mj ( S ) / / N (cid:15) (cid:15) H j ´et ( S [ p ] , µ ⊗ jp ) N (cid:15) (cid:15) b K Mj ( R ) / / H j ´et ( R [ p ] , µ ⊗ jp ) . We will prove this compatibility in the special case of interest to us in Lemma 1.10. (Here is a possiblemethod of proof in general, which in the case of fields can be found in [18, pg. 237]. Since the norm mapson improved Milnor K -theory are induced by those on the K -theory of K Mj ( − ( t )) and K Mj ( − ( t , t )),this compatibility immediately reduces to the case that R has infinite residue field. Now the norm mapson for Milnor K -theory for local rings infinite residue field are defined in [27, Def. 5.5] using an analogueof the Milnor–Bass–Tate sequence for local rings with infinite residue field. Similarly one should be ableto prove an analogue of Faddeev’s exact sequence for ´etale cohomology for local rings with infinite residuefield to define the norm map in that setting. Then one checks the compatibility of the two sequences.)We recall also that the canonical quotient map K Mj ( R ) → b K Mj ( R ) is an isomorphism if the residuefield of R has > M j elements [27, Prop. 10(5)], where M j is a certain bound independent of R (andimplicitly chosen to satisfy 1 = M ≤ M ≤ M ≤ · · · ). Since j will be clear from the context, we willsay in this case simply that R has “big residue field”.The following is the first formulation of our main result; by p -henselian we mean that the ring ishenselian along the ideal generated by p . Theorem 1.3.
Let V be a complete discrete valuation ring of mixed characteristic, and R a local, p -henselian, ind-smooth V -algebra; let j, r ≥ and assume R has big residue field. Then K -theory of p -adic rings (i) the cohomological symbol h jp r : K Mj ( R [ p ]) /p r −→ H j ´et ( R [ p ] , µ ⊗ jp r ) is an isomorphism;(ii) the canonical map K Mj ( R ) /p r −→ K Mj ( R [ p ]) /p r is injective. Remark 1.4 ( j = 2) . The arguments of Dennis–Stein [11] may be used to prove Theorem 1.3(ii) in thecase j = 2: take the exact sequence of [35, Corol. 4.3] (with t being a uniformiser of V ) mod p r . Thisspecial case will actually be used in the course of our proof of the general case (see the end of the proofof Proposition 1.13).In order to avoid the assumption that R has big residue field, we now reformulate Theorem 1.3 toavoid any reference to the Milnor K -theory of R [ p ]. So let V and R be as in Theorem 1.3 but drop theassumption that R has big residue field. Letting m ⊆ V be the maximal ideal (notation which will beused throughout the paper), note that m R is a prime ideal of R and the localisation R m R is a discretevaluation ring (since it is a filtered colimit of dvrs, all of whose maximal ideals are generated by m ); let F be the field of fractions of its henselisation R h m R , and note that the residue field of F is Frac( R/ m R ).Then F is a henselian discrete valuation field of mixed characteristic, and so the cohomological symbol h jp r : K Mj ( F ) /p r → H j ´et ( F, µ ⊗ jp r ) is an isomorphism by Bloch–Kato [6, Thm. 5.12]. This may be used todefine Kato’s residue map ∂ : H j ´et ( R [ p ] , µ ⊗ jp r ) −→ W r Ω j − R/ m R, log (2)as usual: Lemma 1.5.
With notation as in the previous paragraph, the composition H j ´et ( R [ p ] , µ ⊗ jp r ) −→ H j ´et ( F, µ ⊗ jp r ) h jpr ∼ = K Mj ( F ) /p r ∂ −→ K Mj − (Frac( R/ m R )) /p r dlog −−−→ W r Ω j − R/ m R ) , log lands inside W r Ω j − R/ m R, log , where ∂ is the boundary map in Milnor K -theory for the discrete valuationfield F .Proof. Throughout the paper we adopt the following notation when given a V -scheme X : the inclusionof the special and generic fibres are denoted by i : X × V V / m ֒ → X and j : X [ p ] → X respectively.By taking a filtered colimit we may reduce to the same assertion in which R is replaced by a local,essentially smooth V -algebra S , at which point the claim follows by taking global sections on Spec( S/ m S )of Bloch–Kato’s residue map of ´etale sheaves ∂ : i ∗ R j j ∗ µ ⊗ jp r → W r Ω j − S/ m S ) , log [6, (6.6)]. We note thatBloch–Kato’s construction of this map can also be replaced by an argument through Gersten complexes[44, Lem. 3.2.4]. Remark 1.6 (Symbolic generation of W r Ω j log ) . Let A be a local F p -algebra. We denote by W r Ω jA, log the subgroup of W r Ω jA generated by dlog [ f ] ∧ · · · ∧ dlog [ f j ] for f , . . . , f j ∈ A × , where [ f ] ∈ W r ( A ) isthe Teichm¨uller lift of any f ∈ A × and dlog [ f ] := d [ f ][ f ] . This coincides with the more common definition,often denoted by ν r ( j )( A ), in terms of forms which are ´etale locally generated by such dlog forms by [37,Corol. 4.2(i)].The symbol map K Mj ( A ) → W r Ω jA, log descends to improved Milnor K -theory. Indeed, the map W r Ω jA, log → Q p ∈ Spec A W r Ω jA sh p , log is injective, where A sh p is the strict henselisation of A p , so it is enoughto show that the symbol map descends for each A sh p ; but then there is nothing to prove as K Mj ( A sh p ) ≃ → b K Mj ( A sh p ).If A is moreover assumed to be regular Noetherian (or, more generally, ind smooth over F p ) then thesymbol map dlog : b K Mj ( A ) /p r → W r Ω jA, log is an isomorphism by the Bloch–Kato–Gabber theorem [6]and the Gersten conjectures for both sides [20, 26] (see [37, Thm. 5.1] for a more detailed discussion ofthe proof). 5orten L¨uders and Matthew MorrowThe second formulation of Theorem 1.3 eliminates the hypothesis that R has big residue field: Theorem 1.7.
Let V be a complete discrete valuation ring of mixed characteristic, and R a local, p -henselian, ind-smooth V -algebra; let j, r ≥ . Then the sequence −→ b K Mj ( R ) /p r −→ H j ´et ( R [ p ] , µ ⊗ jp r ) ∂ −→ W r Ω j − R/ m R, log −→ is exact. We will see in the next section that Theorem 1.3 and Theorem 1.7 are indeed equivalent if R has bigresidue field. Remark 1.8.
The assumption that V is complete in Theorems 1.3 and 1.7 is actually redundant. Indeed,all terms in the conclusions of the theorems are unaltered if we replace R by the p -henselisation of R ⊗ V b V (c.f., the first paragraph of the proof of Theorem 3.1). p , absolutely unramified, big residue field case In this section we reduce Theorems 1.3 and 1.7 to the special case of Theorem 1.3 where V has big residuefield, is absolutely unramified, and r = 1 (see Corollary 1.19) which will then be proved in Section 2 (seeTheorem 2.17).We begin by reducing Theorem 1.7 to the case of big residue field: Proposition 1.9.
Theorem 1.7 reduces to the case that V has big residue field (i.e., its residue field hasmore than M j elements).Proof. Obviously we only need to worry about the case that V has finite residue field k . By taking afiltered colimit we reduce to the case that R is the p -henselisation of a local, essentially smooth V -algebra;let K denote its residue field, which is a finitely generated, separable field extension of k . So we mayrealise K as a finite separable extension of a rational function field k ( t ) := k ( t , . . . , t d ).Pick any integer ℓ ≥ M j which is coprime to both p and | K : k ( t ) | , and let k ′ be the unique degree ℓ extension of the finite field k . Note that K ⊗ k k ′ = K ⊗ k ( t ) k ′ ( t ) is the tensor product of finite fieldextensions of coprime degree, hence is a field.Let V ′ be the finite ´etale extension of V corresponding to the extension k ′ of k , and set R ′ := R ⊗ V V ′ .Note that R ′ is still p -henselian, since it is a finite extension of R , and that it is local since R ′ / m R R ′ = K ⊗ k k ′ is a field. In particular, we are allowed to assume that Theorem 1.7 holds for R ′ .But now we obtain Theorem 1.7 for R by an easy norm argument as follows. There is a map ofcomplexes 0 / / b K Mj ( R ′ ) /p r / / N (cid:4) (cid:4) ✸ ✤☛ H j ´et ( R ′ [ p ] , µ ⊗ jp r ) / / N (cid:4) (cid:4) ✸ ✤☛ W r Ω j − R ′ /πR ′ , log / / / / b K Mj ( R ) /p r O O / / H j ´et ( R [ p ] , µ ⊗ jp r ) / / O O W r Ω j − R/πR, log O O / / N is multiplication by ℓ , so we deduce that b K Mj ( R ) /p r → H j ´et ( R [ p ] , µ ⊗ jp r ) is injective.Then exactness at the middle of the bottom row follows from the analogous exactness of the top rowand compatibility of the norm maps (see Lemma 1.10). Since Kato’s residue map is surjective (see theproof of Proposition 1.12) we have proved exactness of the bottom row, as desired.As promised in Remark 1.2 and used in the previous proof, here is the required compatibility of thenorm maps on improved Milnor K -theory and ´etale cohomology: Lemma 1.10.
Let R be a local, p -henselian, ind-smooth algebra over a complete discrete valuation ringof mixed characteristic; let S be a local, finite ´etale R -algebra. Then the diagram b K Mj ( S ) / / N (cid:15) (cid:15) H j ´et ( S [ p ] , µ ⊗ jp r ) N (cid:15) (cid:15) b K Mj ( R ) / / H j ´et ( R [ p ] , µ ⊗ jp r )6ilnor K -theory of p -adic rings commutes for any j, r ≥ .Proof. To reduce the compatibility to the case of fields we let F := Frac( R h m R ) be as in the paragraphbefore Lemma 1.5 and set L := S ⊗ R F , which we claim is a field. Since L is finite ´etale over the field F , it is enough to show that it is an integral domain, which will follow from showing that S ⊗ R R h m R isan integral domain. But this is finite ´etale over the henselian discrete valuation ring R h m R , so it is anintegral domain if and only if its special fibre S ⊗ R R m R / m R m R is local. But this is finite ´etale overthe field R m R / m R m R so (conversely to above) it is enough to check it is an integral domain; but it is alocalisation of S/ m S , which is an integral domain since it is local and ind-smooth over the field V / m .There is a cube of maps b K Mj ( L ) / / N (cid:15) (cid:15) H j ´et ( L, µ ⊗ jp r ) N (cid:15) (cid:15) b K Mj ( S ) ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ / / N (cid:15) (cid:15) H j ´et ( S [ p ] , µ ⊗ jp r ) ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ N (cid:15) (cid:15) b K Mj ( F ) / / H j ´et ( F, µ ⊗ jp r ) b K Mj ( R ) / / ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ H j ´et ( R [ p ] , µ ⊗ jp r ) ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ in which the goal is to prove that the front face commutes. The back left face commutes by the compat-ibility of Kerz’ norm maps for R → S and F → L ; the back right face commutes by compabibility of thenorm maps on Milnor K -theory and ´etale cohomology in the case of fields [18, Prop. 7.5.5]. Since thetop, bottom, and front right faces clearly commute, the desired commutativity of the front right face nowfollows formally from the fact that the map H j ´et ( R [ p ] , µ ⊗ jp r ) → H j ´et ( F, µ ⊗ jp r ) is injective: this is a result ofGabber which we recall as Theorem 5.8.To compare the two main theorems, we will use the following localisation sequence; we record it ingreater generality than immediately necessary: Lemma 1.11.
Let R be a ring additively generated by units and t ∈ R a non-zero-divisor in the Jacobsonradical such that tR is a prime ideal, R tR is a discrete valuation ring, and R [ t ] ∩ R tR = R . Then thereare unique homomorphisms ∂ and ∂ t fitting into a commutative diagram with exact row K Mj ( R ) (cid:15) (cid:15) (cid:15) (cid:15) / / K Mj ( R [ t ]) ∂ t x x ♣♣♣♣♣♣♣♣♣♣ ∂ / / K Mj − ( R/tR ) / / K Mj ( R/tR ) and satisfying ∂ ( { a , . . . , a j − , t } ) = { a , . . . , a j − } , ∂ t ( { a , . . . , a j − , t } ) = 0 for all a , . . . , a j − ∈ R × .Proof. The hypotheses ensure that any unit of R [ t ] may be written uniquely as t i u for some i ∈ Z and u ∈ R × . The maps ∂ and ∂ t may then be constructed via the usual argument due to Serre which iswell-known in the case of a discrete valuation ring with uniformiser t ; e.g., see [18, Prop. 7.1.4].To see that the row is exact one considers the map K Mj − ( R/tR ) −→ K Mj ( R [ t ]) / Im( K Mj ( R ) → K Mj ( R [ t ])) , { a , . . . , a j − } 7→ { e a , . . . , e a j − , t } , We record the following criterion: given a ring R and non-zero-divisor t ∈ R , then R tR / T r ≥ t r R tR is a discretevaluation ring (so, in particular, if R tR is t -adically separated then it is a discrete valuation ring). Indeed, R tR is a localring with maximal ideal generated by non-zero-divisor π , and then it is easily to check π is still a non-zero-divisor in thequotient R ′ := R tR / ∩ r ≥ t r R tR . So the latter is a local ring, with maximal ideal generated by a single non-zero-divisor t ,such that max { r ≥ x ∈ t r R ′ } exists for all non-zero x ∈ R ′ ; it easily follows that R ′ is a dvr. e a i ∈ R × are arbitrary lifts of a i ∈ ( R/tR ) × . It is enough to show that this is well-defined, asit will then provide the desired inverse to ∂ modulo the image of K Mj ( R ); the well-definedness reducesto checking that { bt, t } ∈ Im( K M ( R ) → K M ( R [ t ]) for all b ∈ R . But R is additively generatedby units, so the group 1 + tR is generated multiplicatively by its subset 1 + tR × (Proof: by inductionon the length of the expression for b as a sum of units, noting that if b = b ′ + u with u a unit then1 + bt = (1 + b ′ t )(1 + u b ′ t t ).) and therefore we may assume b is a unit; then { bt, t } = { bt, − b } indeed lies in the image. Proposition 1.12.
For any fixed V , R , j , r , where R has big residue field, Theorems 1.3 and 1.7 areequivalent.Proof. We have a commutative diagram of complexes0 / / b K Mj ( R ) /p r / / H j ´et ( R [ p ] , µ ⊗ jp r ) / / W r Ω j − R/πR, log / / / / K Mj ( R ) /p r / / ∼ = O O K Mj ( R [ p ]) /p rh jpr O O ∂ / / O O K Mj − ( R/πR ) /p r / / ∼ = O O R and R/πR have big residue field (for theisomorhism on the right, note that
R/πR is an ind-smooth, local F p -algebra and see Remark 1.6). Froma diagram chase we now see that the top complex is exact if and only if (1) the bottom complex is alsoinjective at the left and (2) the Galois symbol is an isomorphism.The following type of reduction to the mod p case is well-known, though we are forced to modify theusual argument since the Milnor K -theory of the non-local ring R [ p ] does not a priori admit norm maps: Proposition 1.13.
Let V be a complete discrete valuation ring of mixed characteristic; letting F denoteits field of fractions, set F ′ := F ( ζ p ) and let V ′ be the ring of integers of F ′ . Assume that f ( F ′ /F ) = 1 ,i.e., that no residue field extension occurs. Let R be a local, p -henselian, ind-smooth V -algebra whoseresidue field has ≥ M J elements for some fixed J ≥ , and put R ′ := R ⊗ V V ′ . Then“Theorem 1.3 for V → R , for r = 1 , and for ≤ j ≤ J andTheorem 1.3 for V ′ → R ′ , for r = 1 , and for ≤ j ≤ J ”impliesTheorem 1.3 for V → R , for all r ≥ , and for ≤ j ≤ J .Proof. Let R be a local, p -henselian, ind-smooth V -algebra, and set R ′ := R ⊗ V V ′ . Note that R ′ isind-smooth over V ′ , that it is p -henselian (since it is a finite R -algebra), and consequently that it is local(since R ′ / m ′ R ′ = R/ m R ⊗ V/ m V ′ / m ′ = R/ m R is local, where we crucially use our hypothesis that V and V ′ have the same residue field).Assuming Theorem 1.3 for r = 1, j ≥
0, and both V → R and V ′ → R ′ , we will actually proveTheorem 1.3 for r ≥
1, for 0 ≤ j ≤ J , and for both V → R and V ′ → R ′ ; this looks stronger than statedin the proposition but is actually equivalent, since the statement of the proposition can also be appliedto the case V = V ′ ). We do this by induction on r .We begin by proving part (i) of Theorem 1.3. We will use the commutative diagram in which therows are exact: · · · δ / / H j ´et ( R [ p ] , µ ⊗ jp r − ) ρ / / H j ´et ( R [ p ] , µ ⊗ jp r ) / / H j ´et ( R [ p ] , µ ⊗ jp ) / / K Mj ( R [ p ]) /p r − ∼ = O O p / / K Mj ( R [ p ]) /p r O O / / K Mj ( R [ p ]) /p / / ∼ = O O K -theory of p -adic ringsWriting δ ′ for the boundary map in the analogous diagram for R ′ , there is a commutative diagram H j − ( R ′ [ p ] , µ ⊗ jp ) δ ′ / / H j ´et ( R ′ [ p ] , µ ⊗ jp r − ) µ p ⊗ F p K Mj − ( R ′ [ p ]) /p [ ] ∪ h j − pr − ∼ = O O γ / / K Mj ( R ′ [ p ]) /p r − h jpr − ∼ = O O (3)where γ denotes multiplication and [ ] : µ p ( V ′ ) → H ( R ′ [ p ] , µ p ) is the canonical map, where we write µ p ( V ′ ) for the group of p th roots of unity in V ′ . The commutativity of this diagram is proved exactly asin the case of a field, for which we refer to [18, Lem. 7.5.10].The vertical arrows in (3) are isomorphisms by the inductive hypothesis and the fact that the topleft term can equivalently be written µ p ( V ′ ) ⊗ F p H j − ( R ′ [ p ] , µ ⊗ j − p ). A diagram chase now proves theinductive step in the case of V ′ → R ′ Now let ∆ be the Galois group of the extension Frac V ⊆ Frac V ′ , and replace the commutativesquare (3) by its ∆ invariants; the existence of trace maps on ´etale cohomology and the coprimeness of p, | ∆ | imply that H j ´et ( R ′ [ p ] , µ ⊗ jp r − ) ∆ = H j ´et ( R [ p ] , µ ⊗ jp r − ) (and similarly for the top left term), whence theresulting commutative square is H j − ( R [ p ] , µ ⊗ jp ) δ / / H j ´et ( R [ p ] , µ ⊗ jp r − )( µ p ⊗ F p K Mj − ( R ′ [ p ]) /p ) ∆ ∼ = O O γ / / K Mj ( R [ p ]) /p r − ∼ = O O (4)We emphasise that we are not using the a priori existence of any trace maps on Milnor K -theory toobtain the identification ( K Mj ( R ′ [ p ]) /p r − ) ∆ = K Mj ( R [ p ]) /p r − , but rather the inductive hypothesis toidentify both terms with ´etale cohomology (although these identifications in fact define the trace mapswhich could alternatively be used). The surjectivity of the left arrow, combined with our initial diagramand inductive hypothesis, shows at once that the middle symbol map in the initial diagram is injective,as required to complete the proof of the reduction.Part (ii): We claim first that the map K Mj ( R ) → K Mj ( R ′ ) is injective modulo any power of p .Indeed, by a filtered colimit argument we may assume that R is in addition regular Noetherian, in whichcase it satisfies the conditions of Dahlhausen’s version of Kerz’ norm map [9, Prop. C.1]: indeed, R isfactorial by Auslander–Buchsbaum, and V ′ = V [ X ] /π for some monic irreducible polynomial π ∈ V [ X ]which remains irreducible in R [ X ] (since otherwise R ′ = R [ X ] /π would not be a domain). So there is anorm map b K Mj ( R ′ ) N −→ b K Mj ( R ) such that the composition K Mj ( R ) = b K Mj ( R ) → b K Mj ( R ′ ) N −→ b K Mj ( R ) = K Mj ( R ) is multiplication by an integer coprime to p . This proves the claimed injectivity, thereby reducingthe desired injectivity of K Mj ( R ) /p → K Mj ( R [ p ]) /p r to that of K Mj ( R ′ ) /p → K Mj ( R ′ [ p ]) /p r , which wewill check by induction on r .To do this we will use the commutative diagram µ p ⊗ F p K Mj − ( R ′ [ p ]) /p γ / / K Mj ( R ′ [ p ]) /p r − p / / K Mj ( R ′ [ p ]) /p r / / K Mj ( R ′ [ p ]) /p / / K Mj ( R ′ ) /p r − ?(cid:31) O O p / / K Mj ( R ′ ) /p r O O / / K Mj ( R ′ ) /p / / ?(cid:31) O O α ∈ K Mj ( R ′ ) /p r is an element which vanishes in K Mj ( R ′ [ p ]) /p r . Adiagram chase immediately shows the following (bearing in mind that any element of K Mj − ( R ′ [ p ]) maybe written as β + { π } β ′ for some β ∈ K Mj − ( R ′ ), β ′ ∈ K Mj − ( R ′ )): the element α may be written as pα ′ , where α ′ ∈ K Mj ( R ′ ) /p r − has image { ζ p } β + { ζ p , π } β ′ in K Mj ( R ′ [ p ] /p r − ). But ζ p = 1 + πa forsome a ∈ V ′ , so the second term may be rewritten h a, π i β ′ and the injectivity of the left vertical arrow9orten L¨uders and Matthew Morrowshows that α ′ = { ζ p } β + h a, π i β ′ . Multiplication by p , i.e., the lower left horizontal map, now maps α ′ to α = p ( { ζ p } β + h a, π i β ′ ) = p h a, π i β ′ . But this term is zero since p h a, π i is zero in K M ( R ′ ) /p r : indeedcertainly p h a, π i = p { ζ p , π } vanishes in K M ( R ′ [ p ]), and K M ( R ′ ) /p r → K M ( R ′ [ p ]) /p r is injective byRemark 1.4.It remains to reduce further to the case that V is absolutely unramified. This is achieved via a left Kanextension argument using motivic cohomology. For any p -adically complete ring A , or more generallyderived p -complete simplicial ring, we denote by Z p ( j )( A ) its weight- j ´etale-syntomic cohomology inthe sense of [3]; more precisely, this was defined in [3] for quasisyntomic rings, then extended via leftKan extension to derived p -complete simplicial rings in [2, § Z p ( j )( A ) is a p -complete complex supported in cohomological degree ≤ j + 1. To avoidconstantly needing to include additional completions, it will be convenient to write Z p ( j )( A ) := Z p ( j )(Rlim s A ⊗ LZ Z /p s Z )for any p -henselian ring, where Rlim s A ⊗ LZ Z /p s Z is the derived p -adic completion of A ; if A has bounded p -power torsion, in particular if A is Noetherian, this coincides with the usual p -adic completion b A .For smooth algebras over complete discrete valuation rings, forthcoming work of Bhatt–Clausen–Mathew concerning the motivic filtration on Selmer K -theory will include the following identificationof Z p ( j ) with the original approach to p -adic ´etale motivic cohomology studied by Geisser, Sato, andSchneider [16, 44, 45]: Theorem 1.14 (Bhatt–Clausen–Mathew) . Let V be a mixed characteristic complete discrete valuationring and R a p -henselian, ind-smooth V -algebra. Then there are natural equivalences Z p ( j )( R ) /p r ≃ hofib (cid:0) R Γ ´et ( R/ m R, i ∗ τ ≤ j R j ∗ µ ⊗ jp r ) −→ R Γ ´et ( R/ m R, W r Ω j − )[ − j ] (cid:1) for all r, j ≥ , where the arrow is Bloch–Kato’s residue map as in the proof of Lemma 1.5. Corollary 1.15.
Let V be a mixed characteristic complete discrete valuation ring and R a p -henselian,ind-smooth V -algebra. Then there are natural isomorphisms H j ( Z p ( j )( R ) /p r ) ∼ = ker( H j ´et ( R [ p ] , µ ⊗ jp r ) ∂ −→ W r Ω j − R/ m R, log ) for all r, j ≥ .Proof. This follows from Theorem 1.14 and Gabber’s affine analogue of the proper base change theoremimplying that R Γ ´et ( R/ m R, i ∗ τ ≤ j R j ∗ µ ⊗ jp r ) ≃ R Γ ´et ( R, τ ≤ j R j ∗ µ ⊗ jp r ).The second property of the Z p ( j )( − ) which we need to recall is that they are suitably left Kanextended: Proposition 1.16.
For each j, r ≥ , the functor { p -henselian rings } → D ( Z /p r Z ) , A τ ≤ j ( Z p ( j )( A ) /p r ) is left Kan extended from p -henselian, ind-smooth Z ( p ) -algebras.Proof. Dropping the truncation initially, the functor Z p ( j )( − ) /p r itself is left Kan extended from p -henselisations of finitely generated Z ( p ) -polynomial algebras (hence a fortiori from p -henselian, ind-smooth Z ( p ) -algebras) thanks to the analogous result in the p -complete case [2, Thm. 5.1(2)].It therefore remains to check that τ >j ( Z p ( j )( − ) /p r ) is left Kan extended from p -henselian, ind-smooth Z ( p ) -algebras. For any fixed p -henselian A , it is equivalent by cofinality to show that the canonical maphocolim R → A τ >j ( Z p ( j )( R ) /p r ) → τ >j ( Z p ( j )( A ) /p r ) is an equivalence, where the colimit is taken over p -henselian, ind-smooth Z ( p ) -algebras R equipped with morphism R → A which is a henselian surjection.But this follows by noting the much stronger fact that each map τ >j ( Z p ( j )( R ) /p r ) → τ >j ( Z p ( j )( A ) /p r )is an equivalence: indeed, by taking a filtered colimit it is enough to check that τ >j ( Z p ( j )( R ) /p r ) ∼ → τ >j ( Z p ( j )( A ) /p r ) whenever R → A is a henselian surjection of p -henselian rings with bounded p -powertorsion; but then b R → b A is also a henselian surjection (we leave this simple verification to the reader)and indeed τ >j ( Z p ( j )( b R ) /p r ) ∼ → τ >j ( Z p ( j )( b A ) /p r ) by the rigidity property of Z p ( j ) [2, Thm. 5.2].10ilnor K -theory of p -adic ringsWe will combine this with the following left Kan extension property of Milnor K -theory; we refer thereader to [12, App. A] for further information about left Kan extending from smooth algebras. Proposition 1.17.
Let k be a commutative ring and ℓ ≥ an integer; let L sm ( K Mj /ℓ ) : Alg loc k → D ≤ ( Z ) be the left Kan extension of K Mn ( − ) /ℓ from local, ind-smooth k -algebras. Then, for any local k -algebra A , the co-unit map H ( L sm ( K Mj /ℓ )( A )) → K Mj ( A ) /ℓ is an isomorphism; i.e., Milnor K -theory is “rightexact”.Proof. We use a standard type of argument going back to Quillen, showing that functors with suitableuniversal properties are right exact. To simplify the notation we write k Mj = K Mj /ℓ . We will exploit thefact that the left Kan extension is lax symmetric monoidal, so that L j ≥ H ( L sm k Mj ( A )) naturally hasthe structure of a graded commutative ring.We first handle the case j = 1. In that case more is known: the functor K M = G m is left Kanextended from local, ind-smooth K -algebras by [12, Prop. A.0.1], so taking H is not even necessary if ℓ = 0. When ℓ > K M ℓ −→ K M → k M → loc k induces aright exact sequence of abelian groups L sm K M ( A ) ℓ −→ L sm K M ( A ) → L sm k M ( A ) →
0; from the previoussentence we deduce that H ( L sm k M ( A )) = A × /ℓ , as required.Let R • → A be a simplicial resolution by ind-smooth, local k -algebras where, for each simplicialdegree q ≥
0, the kernel of the surjection R q → A is a henselian ideal. Such a resolution exists andcalculates our desired left Kan extension; namely, given any functor F : Sm loc k → D ( Z ) commuting withfiltered colimits, the value of its left Kan extension L sm F on the arbitrary local k -algebra A is given bythe geometric realisation of the simplicial object q F ( R q ). In particular, L sm k Mj ( A ) ∈ D ( Z ) can berepresented by (the complex associated to) the simplicial abelian group k Mj ( R • ) : q k Mj ( R q ).We will argue using the following commutative diagram of graded commutative rings L j ≥ k Mj ( R ) (cid:15) (cid:15) ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ L j ≥ H ( L sm k Mj ( R • )) / / L j ≥ k Mj ( A )in which all arrows are surjective (the vertical as H ( k Mj ( R • )) is by definition the equaliser of k Mj ( R ) ⇒ k Mj ( R ); the diagonal as R → A is surjective on units). The goal is to show that the horizontal arrowis an isomorphism.The horizontal arrow is an isomorphism in degrees 0 (as k M is the constant functor Z /ℓ Z ) and 1(explained above). Therefore L j ≥ H ( L sm k Mj ( R • )) is the quotient of L j ≥ k Mj ( R ) by a graded idealwhich is contained in I := ker( L j ≥ k Mj ( R ) → L j ≥ k Mj ( A )) (by the existence of the commutativediagram) and contains the ideal J generated by the degree one elements ker( R × → A × ) (since thehorizontal map is an isomorphism in degree 1). But a standard argument in Milnor K -theory shows that I = J , so in fact the three ideals coincide and the bottom horizontal map is an isomorphism.We may now state and prove the final reduction: Proposition 1.18.
Fix j, r ≥ , and let V ′ ⊆ V be an extension of mixed characteristic complete discretevaluation rings, both having big residue field. ThenTheorem 1.7 for all local, p -henselian, ind-smooth V ′ -algebrasimpliesTheorem 1.7 for all local, p -henselian, ind-smooth V -algebras.Proof. Given a local, p -henselian, ind-smooth V -algebra, the description of Corollary 1.15 shows thatTheorem 1.7 is exactly the assertion that the symbol map defines a natural isomorphism K Mj ( R ) /p r ∼ = H j ( Z /p r Z ( j )( R )) = ker( H j ´et ( R [ p ] , µ ⊗ jp r ) → W r Ω j − R/πR, log ) . (5)11orten L¨uders and Matthew MorrowAssuming that this is true for all local, p -henselian, ind-smooth V ′ -algebras then it follows by left Kanextending from these, using Propositions 1.16 and 1.17, that there are natural isomorphisms K Mj ( A ) /p r ≃ → H j ( Z /p r Z ( j )( A )) (6)for all local, p -henselian V ′ -algebras A .But when A = R is a local, p -henselian, ind-smooth V -algebra then the right side is the kernel ofKato’s residue map by another application of Theorem 1.14. That is, assuming that (5) is an isomorphismfor all local, p -henselian, ind-smooth V ′ -algebras, we have proved it is an isomorphism for all local, p -henselian, ind-smooth V -algebras, as required.To summarise the reductions: Corollary 1.19.
Theorems 1.3 and 1.7 in general reduce to the special case of Theorem 1.3 where V has big residue field, is absolutely unramified, and r = 1 .Proof. We suppose Theorem 1.3 is known whenever V has big residue field, is absolutely unramified, and r = 1. It follows from Proposition 1.12 that Theorem 1.7 is true under the same conditions (obviously if V has big residue field then so does R ). For general V with big residue field there exists an absolutelyunramified complete discrete valuation ring V ′ ⊆ V with the same residue field (it is the ring of Wittvector if the residue field is perfect; in the imperfect case it still exists but is not unique [13, Chap. 2 § V has bigresidue field and r = 1. So Theorem 1.3 is true under the same conditions, again by Proposition 1.12; forany V with big residue field satisfying the hypothesis of Proposition 1.13, that proposition then lets usextend Theorem 1.3 to all r ≥
1. But this hypothesis holds in particular if V is absolutely unramified;so appealing yet again to Proposition 1.12, we have shown that Theorem 1.7 is true whenever V isabsolutely unramified and has big residue field.By again using Proposition 1.18 as in the previous paragraph we eliminate the hypothesis that V isabsolutely unramified, and then by Proposition 1.9 we obtain Theorem 1.7 in general. A final applicationof Proposition 1.12 implies Theorem 1.3 in general. Remark 1.20 (Replacing ind-smoothness by quasismoothness) . We can actually offer the followingweakening of the ind-smooth hypotheses in Theorem 1.3(ii). Let V be a complete discrete valuation ringof mixed characteristic, and A a local, p -henselian V -algebra with big residue field satisfying the following“quasismoothness” conditions: πA is a prime ideal, A πA is a discrete valuation ring, A [ π ] ∩ A πA = A ,and the F p -algebra A/πA is Cartier smooth (see § K Mj ( A ) /p r → K Mj ( A [ p ]) /p r is injectivewith cokernel K Mj − ( A/πA ) /p r ∼ = W r Ω jA/πA, log .We prove this as follows. Theorem 1.3(ii) and Lemma 1.11 provide an exact sequence0 −→ K Mj ( − ) /p r −→ K Mj ( − [ p ]) /p r −→ K Mj − ( − /π − ) /p r −→ p -henselian, ind-smooth V -algebras; we may left Kan extend this toall local p -henselian V -algebras and evaluate on A to obtain a fibre sequence L sm ( K Mj /p r )( A ) −→ L sm ( K Mj ( − [ p ]) /p r )( A ) −→ L sm ( K Mj − ( − /π − ) /p r )( A ) . The H of the outer two terms are respectively K Mj ( A ) /p r and K Mj ( A/πA ) /p r by Lemma 1.17. Mean-while, a modification of the argument of that lemma, crucially using that A [ p ] × = A × π Z (and similarlyfor any ind-smooth V -algebra R in place of A ), shows that H of the middle term is K Mj ( A [ p ]) /p r , i.e., K Mj ( − [ p ]) /p r is “right exact at A ”. Finally, Proposition 5.1(i) shows that the rightmost term identifieswith W r Ω j − A, log , and in particular it has no H . The H s of the fibre sequence therefore provide thedesired short exact sequence.Under the hypotheses on A , it is conceivable that the cohomological symbol K Mj ( A [ p ]) /p r → H j ´et ( A [ p ] , µ ⊗ jp r ) is also an isomorphism, but we have not seriously tried to prove it.12ilnor K -theory of p -adic rings p case In this section we prove Theorem 1.3 in the special case that V has big residue field, is absolutelyunramified, and r = 1 (see Theorem 2.17). Thanks to Corollary 1.19, this will complete the proof ofTheorems 1.3 and 1.7. K ( R ) Given a ring R , and elements a, b ∈ R such that 1 + ab ∈ R × , let h a, b i ∈ K ( R ) denote the correspondingDennis–Stein symbol defined in [10]. These symbols have the following properties:(D1) h a, b i = −h− b, − a i for a, b ∈ R such that 1 + ab ∈ R × .(D2) h a, b i + h a, c i = h a, b + c + abc i for a, b, c ∈ R such that 1 + ab, ac ∈ R × .(D3) h a, bc i = h ab, c i + h ac, b i for a, b, c ∈ R such that 1 + abc ∈ R × . We recall also the following relations to Steinberg symbols:(D4) h a, b i = { ab, b } for a, b ∈ R such that b, ab ∈ R × .(D5) h a, b i = n − a − b , ab − b o for a, b ∈ R such that 1 + a, − b, ab ∈ R × .For a deduction of (D5) from the relations (D1)–(D3), we refer to [35, Rem. 3.14]. Note that weuse the older sign convention of [10] for these symbols; this changed around 1980 and our h a, b i became h− a, b i (see [51, after III, Def. 5.11]). Definition 2.1.
Let k ≥ R is said to be k -fold stable if and only if whenever a , b , ..., a k , b k ∈ R are given such that ( a , b i ) = · · · = ( a k , b k ) = R , then there exists r ∈ R such that a + rb , . . . , a k + rb k are units.Following [35], we will say that R is weakly k -fold stable if and only if whenever a , b , ..., a k , b k − ∈ R are given, then there exists u ∈ R such that 1 + ub , ..., ub k − are units. Note that weak k -foldstability follows from k -fold stability (using the pairs (1 , b ) , ..., (1 , b k − ) , (0 , Remark 2.2 (Reminder on presentations of K ) . (i) If R is local or three-fold stable then K ( R ) isgenerated by the Dennis–Stein symbols subject to relations (D1)–(D3) [50, Thm. 1].(ii) If R is local and has residue field = F , then K ( R ) is generated by the Steinberg symbols subjectto bilinearity, Steinberg relation, and the alternating relation { x, y } = −{ y, x } [30, Corol. 3.2].(iii) If R is weakly five-fold stable, then bilinearily and the Steinberg relation imply the skew symmetryrelation { x, − x } = 0 (e.g., [35, Lem. 3.6], which itself implies the alternating relation; so from(ii) we deduce that if R is local with residue field having > K M ( R ) ≃ → K ( R )(even better, this isomorphism actually holds for any five-fold stable ring by combining (i) with[49, Thm. 8.4]).(iv) Since (D4) shows that any Steinberg symbol can be written as a Dennis–Stein symbol, we see from(i) that if R is local or three-fold stable then K ( R ) is generated by Steinberg symbols.For the rest of this subsection we fix a ring R which is additively generated by its units and whoseJacobson radical contains the prime number p . Definition 2.3.
For i ≥
0, let U i K ( R ) ⊆ K ( R ) denote the subgroup generated by Steinberg symbols { a, b } where a or b belongs to 1 + p i R . To give uniform statements, we also set V i K ( R ) := ( U K ( R ) + hh pc, p i : c ∈ R i i = 2 = pU i K ( R ) else.It follows from part (ii) of the next lemma that V K ( R ) ⊆ U K ( R ), so this really is a descendingfiltration. 13orten L¨uders and Matthew Morrow Lemma 2.4.
Let i ≥ , and fix an element π ∈ p i R . Then:(i) h π, a i + h π, b i ≡ h π, a + b i mod U i K ( R ) for all a, b ∈ R ;(ii) If R is additively generated by units then h π, b i ∈ U i K ( R ) for all b ∈ R .Proof. (i): Set c = ab/ (1 + π ( a + b )), so that (1 + πa )(1 + πb ) = (1 + π ( a + b ))(1 + π c ) and a + b + πab =( a + b ) + πc + π ( a + b ) πc . Then h π, a i + h π, b i (D2) = h π, a + b + πab i = h π, ( a + b ) + πc + π ( a + b ) πc i . Applying (D2) again lets us rewrite the right side h π, a + b i + h π, πc i . The second symbol lies in U i K ( R )by (D5).(ii) By part (i) and the fact that R is additively generated by units, we reduce to the case that b isa unit; but then the claim is clear from (D4). Lemma 2.5.
Let i ≥ . Then(i) { ap i , bp } ∈ V i +1 K ( R ) for all a, b ∈ R ;(ii) { ap i , − } ∈ V i +1 K ( R ) for all a ∈ R .(iii) h p i − a, p i , h p i − , pa i ∈ V i K ( R ) for all a ∈ R .Proof. (i): All congruences below are with respect to U i +1 K ( R ). One has { p i a, pb } ≡ { p i a (1 + pb ) , pb } (D5) = h p i a, pb i . (7)Next, a trivial induction using (D2) shows that, for any element x ∈ R such that 1+ x ∈ R × , and n ≥
1, wehave n h x, i = h x, f ( x ) i for some polynomial f with integer coefficients such that f (0) = n ; note that thissymbol equals 0 since h x, i = { x, } . In particular 0 = ( p − h p i a, i = h p i a, p − p i c i for some c ∈ R ,which we add to the right side of (7), using (D2), to obtain h p i a, p − p i c +1+ pb + p i a ( p − p i c )(1+ pb ) i = h p i a, pd i with d = 1 + b + p i − ( c + a ( p − p i c )). But this equals h p i +1 a, d i + h p i ad, p i by (D3), wherethe first term lies in U i +1 K ( R ) by Lemma 2.4(ii); meanwhile the second term is h p i ad, p i (D5) = (cid:26) − p i ad − p , p i +1 ad − p (cid:27) ≡ −{ p i ad, − p } . (8)Assuming i ≥
2, then the right side is ≡ { p i a (1 + b ) , − p } thanks to the value of d , and so inconclusion we have shown that { p i a, pb } ≡ −{ p i a (1 + b ) , − p } for all a, b ∈ R ; but replacing the pair a, b by the pair a (1 + b ) , p is odd and i = 1. Returning to line (8), itis enough to show that h pe, p i ∈ U K ( R ) for all e ∈ R . But2 h pe, p i (D3) = h e, p i (D1) = −h− p , − e i (D5) ∈ U K ( R ) , and this is enough since gr K ( R ) is killed by p and so has no 2-torsion.The second special case is when p = 2 and i = 1. But we showed above that { p i a, pb } ≡ h p i ad, p i ,which lies in V K ( M ) by definition.(ii): When p is odd this again follows from the observation that gr i K ( R ) has no 2-torsion. When p = 2 it follows from part (i) by noting that { ap i , − } = { ap i , − } .(iii): When i ≥ h p i − a, p i (D5) = (cid:26) − p i − a − p , p i a − p (cid:27) ≡ −{ p i − a, − p } mod U i K ( R ) , which belongs to V i K M ( R ) by part (i). Meanwhile, when i = 1, the claim is special case of Lemma2.4(ii) (using (D1) to swap the order of the symbol).14ilnor K -theory of p -adic rings Remark 2.6.
As far as we are aware, Lemma 2.5(i) is not true in the case p = 2 = i + 1 if wereplace V K ( R ) by U K ( R ). In fact, it follows from (D5) that V K ( R ) could equivalently be definedas U K ( R ) + h{ ap, bp } : a, b ∈ R i . Indeed, we showed in the proof of Lemma 2.5(i) that { p i a, pb } ≡ h p i ad, p i mod U i +1 K ( R ) for some d ∈ R . The V -filtration is thus the minimalmodification of the U -filtration which is multiplicative; see also Lemma 2.11. Corollary 2.7.
Let i ≥ , and fix an element π ∈ p i R . Then h π, a i + h π, b i ≡ h π, a + b i mod V i +1 K ( R ) for all a, b ∈ R .Proof. Repeating the proof of Lemma 2.4(i), we must show that h π, πc i ∈ V i +1 K ( R ) for all c ∈ R .Writing π = p i b for some b ∈ R , we have h π, πc i = h p i b, p i bc i (D3) = h p i b, p i bc i + h p i b c, p i ; the first termlies in U i K ( R ) ⊆ V i +1 K ( R ) by Lemma 2.4(ii), while the second term lies in V i +1 K ( R ) by Lemma2.5(iii). Lemma 2.8.
Assume in addition that R is p -henselian. If p is odd then V K ( R ) ⊆ pK ( R ) . If p > then V K ( R ) ⊆ pK ( R ) ; if p = 2 and the Artin–Schreier map x x + x is surjective on R/pR , then V K ( R ) ⊆ pK ( R ) .Proof. If p is odd then any element of 1 + p R can be written as the p th -power of an element of 1 + pR ;this is also true when p = 2 if the Artin–Schreier map is surjective, in which case we also use the identity − h− p, − x i = h p ( x + x ) , p i (by (D1) and (D2)) to see that V K ( R ) ⊆ K ( R ).If p = 2 but we drop any assumption on the Artin–Schreier map, then it is still true that any elementof 1 + p R can be written as the p th -power of an element of 1 + p R . The next step is to define the standard differential maps onto the graded pieces, for which we recall thefollowing description of differential forms:
Lemma 2.9.
Let R be a ring which is additively generated by its units. Then the map R ⊗ Z R ×⊗ Z j −→ Ω jR , a ⊗ b ⊗ · · · ⊗ b j a db b ∧ · · · ∧ db j b j is surjective and its kernel is generated by elements of the following types:- a ⊗ b ⊗ · · · ⊗ b j , where b i = b j for some i = j , and- P ni =1 a i ⊗ a i ⊗ b ⊗ · · · ⊗ b j − − P mi =1 a ′ i ⊗ a ′ i ⊗ b ⊗ · · · ⊗ b j − where a i , a ′ i ∈ R × are elementssatisfying P ni =1 a i = P mi =1 a ′ i .Proof. Used in [4, Lem. 4.3], a detailed proof may be found in the appendix of [23].To define the differential maps in a wide degree of generality, it is helpful to adopt the followingdefinitions (in practice our ring R will be local and K ∗ ( R ) will either be its improved Milnor K -theoryor a quotient thereof): Definition 2.10.
Let R be a ring which is additively generated by units and whose Jacobson radicalcontains p . Let K ∗ ( R ) be a graded quotient of K M ∗ ( R ) such that the quotient map K M ( R ) → K ( R )kills Ker( K M ( R ) → K ( R )).For i, j ≥
0, we let U i K j ( R ) ⊆ K j ( R ) denote the subgroup generated by Steinberg symbols { a , . . . , a j } where at least one of a , . . . , a j ∈ R × belongs to 1 + p i R . Similarly to Definition 2.3,we also introduce V i K j ( R ) := ( V K ( R ) · K j − ( R ) p = 2 = i and j ≥ ,U i K j ( R ) else.whose graded pieces will be denoted by gr iV K j ( R ) := V i K j ( R ) /V i +1 K j ( R ).Although it is not necessary for our main results, one justification for the V -filtration is the followingmultiplicativity: 15orten L¨uders and Matthew Morrow Lemma 2.11.
Under the same hypotheses as the previous definition, the V -filtration is descending andmultiplicative, i.e., V i K j ( R ) · V i ′ K j ′ ( R ) ⊆ V i + i ′ K j + j ′ ( R ) for all i, i ′ , j, j ′ ≥ .Proof. We only treat the case where p = 2, i = 2, i ′ = 1. All other cases follow from a modification ofthe arguments in the proof of Lemma 2.5, and in any case will not be needed for the main results of thearticle.Given any a, c ∈ R , we need to show that h pa, p i{ pc } ∈ V K M ( R ). We have that h pa, p i{ pc } (D5) = { pa, − (1 + p a ) , pc } = { pa, p a, pc } + { pa, − , pc } Lem. 2.5(i) ≡ { pa, − , pc } mod V K ( R )Now applying the congruence in Remark 2.6 to see that this is ≡ h pd, p i mod U K ( R ) {− } for some d ∈ R ; but U K ( R ) {− } ⊆ V K ( R ) by Lemma 2.5(ii), so we have reduced the problem to showingthat h pd, p i{− } ∈ V K M ( R ). Rewriting this using (D5) and again using Lemma 2.5(ii), it is equivalentto check that { pd, − , − } ∈ V K M ( R ). By the same argument as in the proof of Lemma 1.11 wemay assume that d ∈ R × .Next note that we can write 1 + pd = (1 − pd )(1 + p d ′ ) for some d ′ ∈ R , and therefore we have { pd, − , − } = { pd, − , − d } − { (1 − pd )(1 + p d ′ ) , − , d } . But { pd, − , − d } and { − pd, − , d } vanish by Lemma 2.14 below, and { p d ′ , − , − d } lies in V K M ( R ) by Lemma 2.5(ii).We next prove that Bloch–Kato’s maps to the graded pieces of p -adic ´etale cohomology factor throughthe K -groups of R itself. In the case of the map ρ ij when p = 2, a similar result is stated by Kuriharain special cases [31, Lem. 2.2], though he omits the manipulations of Dennis–Stein symbols which webelieve are necessary. Definition 2.12.
Let R be an F p -algebra. We define e ν ( j )( R ) := coker(1 − C − : Ω jR → Ω jR /d Ω j − R ),where C − is the inverse Cartier operator. Equivalently e ν ( j )( R ) = H (Spec R, Ω j log ), as follows from theshort exact sequence 0 → Ω j log → Ω j − C − −−−−→ Ω j /d Ω j − → R [37, Corol. 4.2(iii)]. Proposition 2.13.
Let R and K ∗ ( R ) be as in Definition 2.10, and fix j ≥ , i ≥ . Then there arewell-defined homomorphisms ρ ij : Ω j − R/pR → gr iV K j ( R ) , a db b ∧ · · · ∧ db j − b j −
7→ { e ap i , e b , . . . , e b j − } and σ ij : Ω j − R/pR → gr iV K j ( R ) , a db b ∧ · · · ∧ db j − b j −
7→ h e ap i − , p i{ e b , . . . , e b j − } where tilde denotes arbitrary lifts of elements from R/pR to R . Moreover:(i) When p = 2 = i and j ≥ , the sum ρ ij ⊕ σ ij : Ω j − R/pR ⊕ Ω j − R/pR → gr i K Mj ( R ) is surjective; in allother cases ρ ij itself is surjective.(ii) Assume p = 2 = i , that j ≥ , and K ∗ ( R ) is killed by p . Then ρ ij factors through e ν ( j − R/pR ) ;assuming in addition that R is weakly -fold stable, then also σ ij factors through e ν ( j − R/pR ) .Proof. First note that { e ap i , e b , . . . , e b j − } and h e ap i − , p i{ e b , . . . , e b j − } do belong to V i K j ( R ); in thefirst case this is by definition of the U -filtration, in the second case it is by Lemma 2.5(iii). In otherwords, we at least have well-defined maps e ρ ij : R × ( R × ) × j − −→ gr iV K j ( R ) , a db b ∧ · · · ∧ db j − b j −
7→ { ap i , b , . . . , b j − } (and similarly e σ ij ), and our goal is to show that these descend to Ω j − R/pR (resp. Ω j − R/pR ).Firstly, multilinearity of Steinberg symbols shows that e ρ ij and e σ ij are multilinear away from the firstcoordinate. In the case of the first coordinate, e ρ ij is multilinear by definition of the U -filtration, while16ilnor K -theory of p -adic ringsfor e σ ij we argue as follows: for a, a ′ ∈ R , we first use (D1) to swap the other of the necessary relationfor simplicity and then argue as in Lemma 2.4 to see that h p, ap i − i + h p, a ′ p i − i = h p, ( a + a ′ ) p i − i + h p, p i − aa ′ p i − ( a + a ′ ) i , where the second term lies in V i +1 K ( R ) by Lemma 2.5(iii).We next claim that the induced map on R ⊗ Z ( R × ) ⊗ Z j − descends further to R × ⊗ Z ( R × ) ⊗ Z j − (resp. j − j − σ ij ), where we write R := R/pR for simplicity. For ρ ij each ofthe necessary relations follows immediately either from the definition of the U -filtration or Lemma 2.5(i),so we will focus on the slightly more complicated case of σ ij : firstly, if a ∈ pR then h ap i − , p i ∈ V i +1 K ( R )by Lemma 2.5(iii); secondly, if b ∈ pR then h ap i − , p i{ b } ∈ V i +1 K ( R ) by Lemmas 2.5(iii) & 2.11.Our desired maps ρ ij and σ ij have therefore been shown to be well-defined after pre-composition withthe surjection R ⊗ Z R ×⊗ Z j − → Ω j − R , a ⊗ b ⊗ · · · ⊗ b j a db b ∧ · · · ∧ db j − b j − (resp. j − j − iV K j ( R ).For the first type of relation, if b i = b j in R for some i = j then we are free to pick the same lift e b i = e b j ,and the desired relation follows from the fact that { e ap i , e b , e b j } = { e ap i , e b i , − } ∈ V i +1 K ( R ) byLemma 2.5.For the second type of relation, it is enough to check that the maps R × → gr iV K ( R ), a
7→ { ap i , a } ,and R × → gr iV K ( R ), a
7→ h ap i − , p i{ a } , extend (necessarily uniquely) to additive maps defined on allof R . In the first case we observe that { ap i , a } = h p i , a i , where the latter symbol makes sense forany a ∈ R and is additive by Corollary 2.7. In the second case, when i = 1 the problem is vacuous as h a, p i{ a } = −{ ap, − a, a } = 0 (first equality by (D1) and (D4), second equality since {− a, a } = 0 in K ( R )); when i > h ap i − , p i{ a } (D5) = {− ap i − − p , ap i − p , a }≡ −{− ap i − − p , − p, a } mod V i +1 K ( R )= { ap i − , a, − p } + {− (1 − p ) , − p, a } (D4) = h p i − , a i{ − p } where the congruence holds by Lemma 2.5, and the final equality again using vanishing of {− (1 − p ) , − p } ;the final term makes sense for any a ∈ R and is additive modulo V i K ( R ) { − p } ⊆ V i +1 K ( R ) byCorollary 2.7 and Lemma 2.11.This completes the proof that the map ρ ij and σ ij are well-defined. Claim (i) about surjectivity isimmediate from the definition of the V -filtration, so it remains to establishing claim (ii). So in the restof the proof we assume p = 2 = i and j ≥
2, and that K ∗ ( R ) is killed by p .First we show that ρ ij vanishes on closed forms; it is enough (since R is additively generated byunits) to check closed forms da ∧ db b ∧ · · · ∧ db j − b j − = d ( a ∧ db b ∧ · · · ∧ db j − b j − ) where a, b , . . . , b j − ∈ R × .Such a form is sent to { e ap , e a, e b , . . . , e b j − } ∈ gr V K ( R ). But u := e a is a unit and we have { up , u } ≡ { up , − u } mod V K ( R ) by Lemma 2.5(ii), and then { up , − u } (D4) = h− p , − u i (D1) = −h u, p i (D3) = − h up, p i ∈ V K ( R ), which therefore vanishes in gr V K ( R ). To show that ρ ij descendsfurther to e ν ( j − R/pR ), it is similarly enough to observe that { e a − e a ) p , e b , . . . , e b j − } is a multipleof 2 mod V K Mj ( R ), which is true since 1 + ( e a p − e a ) p = (1 − e ap ) .Since the final sentence can also be applied to σ ij , it remains only to show that σ ij vanishes on closedforms; similarly to the previous paragraph, it is enough to check on closed forms − da ∧ db b ∧ · · · ∧ db j − b j − (the minus sign is included to simplify the following manipulations; in any case − R/pR ), andthis reduces to showing that h− up, p i{ u } ∈ V K ( R ) for all u ∈ R × . We have h− up, p i{ u } (D5) = { − up, − (1 − up ) , u } = { up, − up , − u } + { up, − up , − } + { up, − , u } (note that for the first equation we also use that 1 − p = −
1) where we have already shown in theprevious paragraph that { up , u } ∈ V K ( R ), and in Lemma 2.5(ii) that { − up , − } ∈ V K ( R ).Therefore, to complete the proof that h− up, p i{ u } ∈ V K ( R ) it is more than enough to show that17orten L¨uders and Matthew Morrow { up, − , u } vanishes; but since − − p , this is a special case of Lemma 2.14 (although we assumeweak 5-fold stability in the statement of that lemma, the proof only uses weak 4-fold stability and skewsymmetry; but the latter follows in the current context from the fact that K M ( R ) → K ( R ) factorsthrough Im( K M ( R ) → K ( R )), where skew symmetry holds).We record the following general lemma which was required: Lemma 2.14.
Let R be a weakly -fold stable ring and π ∈ R an element of the Jacobson radical. Then { − uπ, − π, u } = 0 in K M ( R ) for all u ∈ R × .Proof. Firstly, weak 5-fold stability implies that skew symmetry { x, − x } = 0 holds for all x ∈ R × . So itis equivalent to prove vanishing of the symbol { − uπ, − u (1 − π ) , u } , or even of { − uπ, u, u ( π − } .Morally the reason that this symbol vanishes is that its terms sum to 1 [46, Corol. 1.8], but we carefullyunfold Suslin–Yarosh’s proof to show that our stability condition suffices.By 4-fold stability, there exists a unit a ∈ R × such that 1 + a , 1 − au − , and 1 + a (1 + u − ) are allunits; so u − a is a unit, and we set b := u a − uπa − u ∈ R × . Using that π is in the Jacobson radical, it is easilychecked that a + b and 1 + b are both units; note also that aba + b − uπ = u .It now follows from [46, Lem. 1.5] (with s := uπ , whence s − a , s − b , and s − a − b are all units)that { a, uπ − a } + { b, uπ − b } = { u, u ( π − } . So it is enough to show that { − uπ, x, uπ − x } = 0 forall x ∈ R × such that 1 + x is also a unit (namely, x = a, b ). To prove this we first recall the standardidentity { y, x } = { y + x, − xy } for all x, y ∈ R × for which x + y is also a unit (to check this, just expand0 = { yx + y , xx + y } ). Applying this with y = 1 − uπ we deduce that, for all x ∈ R × for which 1 + x is a unit, { − uπ, x, uπ − x } = { − uπ + x, − x − uπ , uπ − x } , which vanishes as desired by the Steinberg relation. Corollary 2.15.
Let R be a p -henselian ring which is additively generated by units; assume that K M ( R ) ≃ → K ( R ) . Set k Mj ( R ) := K Mj ( R ) /p .(i) When p is odd, there is an exact sequence Ω j − R/pR ρ j −→ k Mj ( R ) ∂ −→ k Mj ( R/pR ) −→ (ii) When p = 2 and R is weakly -fold stable, then the map ρ j ⊕ σ j : e ν ( j − R/pR ) ⊕ e ν ( j − R/pR ) −→ k Mj ( R ) is well-defined and its cokernel fits into an exact sequence Ω j − R/pR ρ j −→ coker( ρ j ⊕ σ j ) ∂ −→ k Mj ( R/pR ) → Proof.
First note that V k Mj ( R ) = Ker( k Mj ( R ) → k Mj ( R/pR ). This follows from the usual construc-tion of an inverse map k Mj ( R/pR ) → k Mj ( R ) /V k Mj ( R ), { a , . . . , a j } 7→ { e a , . . . , e a j } . If p is odd then V k Mj ( R ) = 0, and if p is even then V k Mj ( R ) = 0 (see Lemma 2.8). The statements now follow fromProposition 2.13. In this section we complete the proof of the main theorems of 1.1. We will need the following theoremwhich is a special case of [6, Corol. 1.4.1] and which calculates the sheaf of p -adic vanishing cycles i ∗ R j ∗ µ ⊗ jp (for the notation see the proof of Lemma 1.5) on a smooth scheme over a complete discretevaluation ring V of mixed characteristic. We cite their result only in the special case when r = 1 and V is absolutely unramified, when the filtration calculating i ∗ R j j ∗ µ ⊗ jp is very short.18ilnor K -theory of p -adic rings Theorem 2.16 (Bloch–Kato) . Let V be an absolutely unramified, complete discrete valuation ring ofmixed characteristic, and X a smooth V -scheme. Then there is a natural short exact sequence of ´etalesheaves on Y −→ Ω j − Y ρ j −→ i ∗ R j j ∗ µ ⊗ jp ∂ ⊕ ∂ p −−−→ Ω j − Y, log ⊕ Ω jY, log −→ . We may finally prove Theorem 1.3 in the unramified, mod p case: Theorem 2.17.
Let V be an absolutely unramified, complete discrete valuation ring of mixed charac-teristic, and R a local, p -henselian, ind-smooth V -algebra; let j ≥ and assume R has big residue field.Then(i) the cohomological symbol h jp : K Mj ( R [ p ]) /p −→ H j ´et ( R [ p ] , µ ⊗ jp ) is an isomorphism;(ii) the canonical map K Mj ( R ) /p −→ K Mj ( R [ p ]) /p is injective.Proof. Although X := Spec R is not smooth over V , the conclusions of Theorem 2.16 remain valid bytaking a filtered colimit. In particular, by taking cohomology we obtain a short exact sequence0 −→ Ω j − R/pR ρ j −→ H (Spec( R/pR ) , i ∗ R j j ∗ µ ⊗ jp ) ∂ ⊕ ∂ p −−−→ Ω j − R/pR, log ⊕ Ω jR/pR, log −→ ∂ ⊕ ∂ p : H (Spec( R/pR ) , i ∗ R j j ∗ µ ⊗ jp ) ≃ → e ν ( j − R/pR ) ⊕ e ν ( j )( R/pR ) (10)for each j ≥ p is odd. Then we have the following commutative diagram0 / / Ω j − R/pR ρ j / / H (Spec( R/pR ) , i ∗ R j j ∗ µ ⊗ jp ) ∂ ⊕ ∂ p / / Ω j − R/pR, log ⊕ Ω jR/pR, log / / H j ´et ( R [ p ] , µ ⊗ jp ) α O O k Mj ( R [ p ]) h j O O Ω j − R/pR ρ j / / k Mj ( R ) O O / / k Mj ( R/pR ) / / (0 , dlog) O O ρ j in the bottom row is injective. Next, the map α is surjective: indeed, it fits into a shortexact sequence0 −→ H (Spec( R/pR ) , i ∗ R j − j ∗ µ ⊗ jp ) β −→ H j ´et ( R [ p ] , µ ⊗ jp ) α −→ H (Spec( R/pR ) , i ∗ R j j ∗ µ ⊗ jp ) → . (11)This follows from the spectral sequence E s,t = H s ´et (Spec( R/pR ) , i ∗ R t j ∗ ( µ ⊗ jp )) ⇒ H s + t ´et ( R [ p ] , µ ⊗ jp )and the fact that Spec( R/pR ) has ´etale cohomological dimensional ≤ p -torsion ´etale sheaves (weremark that (11) remains valid when p = 2 and will be used later in that case).The composition α ◦ h j is an isomorphism: indeed, there is an exact sequenceΩ j − R/pR −→ k Mj ( R [ p ]) ∂ ⊕ ∂ p −−−→ k Mj − ( R/pR ) ⊕ k Mj ( R/pR ) −→ t = p ) and Corollary 2.15(i), and this may be compared to the top row ofthe diagram to prove the isomorphism. 19orten L¨uders and Matthew MorrowThe hypotheses of the proposition, and hence the diagram, remain valid if we replace R by O F :=the henselisation of the discrete valuation ring R pR ; as indicated, we write F := Frac O F = O F [ p ]. Thenthe cohomological symbol h j : k Mj ( F ) ≃ → H j ´et ( F, µ ⊗ jp ) is an isomorphism by Bloch–Kato [6, Thm. 5.12];combined with the isomorphism of the previous paragraph (for O F rather than R ), we deduce that α O F : H j ´et ( F, µ ⊗ jp ) → H (Spec( O F /p O F ) , i ∗ R j j ∗ µ ⊗ jp ) is an isomorphism.Two results of Gabber, which we recall in Theorem 5.8, imply that H j ´et ( R [ p ] , µ ⊗ jp ) → H j ´et ( F, µ ⊗ jp ) isinjective. So from the injectivity of α O F and the surjectivity of α (and their compatibility), it followsthat α is an isomorphism. Therefore h j is an isomorphism and a diagram chase reveals that k Mj ( R ) → k Mj ( R [ p ]) is injective. Diagrammatically, this last argument can be summarised as follows: k Mj ( F ) ∼ = h j / / H j ´et ( F, µ ⊗ jp ) ∼ = α O F / / H (Spec( O F /p O F ) , i ∗ R j j ∗ µ ⊗ jp ) k Mj ( R [ p ]) h j / / H j ´et ( R [ p ] , µ ⊗ jp ) ?(cid:31) O O α / / / / H (Spec( R/pR ) , i ∗ R j j ∗ µ ⊗ jp ) . O O It remains to treat the case p = 2. In that case we have a diagram (ignoring the dashed arrow forthe moment)0 / / Ω j − R/pR ρ j / / H (Spec( R/pR ) , i ∗ R j j ∗ µ ⊗ jp ) ∂ ⊕ ∂ p / / Ω j − R/pR, log ⊕ Ω jR/pR, log / / H j ´et ( R [ p ] , µ ⊗ jp ) α O O k Mj ( R [ p ]) h j ❤❤❤❤❤❤❤❤❤❤❤❤❤❤ e ν ( j − R/pR ) ⊕ e ν ( j − R/pR ) ρ j ⊕ σ j (cid:15) (cid:15) β ◦ γ O O k Mj ( R ) k k ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ (cid:15) (cid:15) Ω j − R/pR = : : ρ j / / coker( ρ j ⊕ σ j ) / / h h ♦ ♣ s ① ✁ ☛ ✤✸❂❋❑◆❖ k Mj ( R/pR ) / / (0 , dlog) O O ρ j ⊕ σ j is thatof Corollary 2.15(ii). The top half of the middle of the diagram consists of the exact sequence0 −→ e ν ( j − R/pR ) ⊕ e ν ( j − R/p ) β ◦ γ −−→ H j ´et ( R [ p ] , µ ⊗ jp ) α −→ H (Spec( R/pR ) , i ∗ R j j ∗ µ ⊗ jp ) −→ γ : e ν ( j − R/pR ) ⊕ e ν ( j − R/pR ) (10) ∼ = H (Spec( R/pR ) , i ∗ R j − j ∗ µ ⊗ j − p ) ζ p ∼ = H (Spec( R/pR ) , i ∗ R j − j ∗ µ ⊗ jp ) , where we use (10) for j − ζ p denotes theisomorphism given by cupping with the primitive p th -root of unity ζ p = − ∈ R .It is clear that the diagram commutes, except for identifying the two maps e ν ( j − R/pR ) ⊕ e ν ( j − R/pR ) → H j ´et ( R [ p ] , µ ⊗ jp ) . But by functoriality and Gabber’s aforementioned injectivity H j ´et ( R [ p ] , µ ⊗ jp ) ֒ → H j ´et ( F, µ ⊗ jp ) this reducesto the analogous commutativity for O F in place of R ; in that case the commutativity is implicit in [6,(5.15)] and mentioned explicitly in [8, § ρ j ⊕ σ j is injective.Secondly, it implies that the map k Mj ( R ) → H (Spec( R/pR ) , i ∗ R j j ∗ µ ⊗ jp ) factors through coker( ρ j ⊕ σ j );the dashed map therefore exists and the diagram continues to commute.20ilnor K -theory of p -adic ringsWith the dashed map in place, it follows immediately from a simple diagram chase that the ρ j in thebottom row is injective. A slightly longer diagram chase shows that k Mj ( R ) → k Mj ( R [ p ]) is injective.The map ∂ ⊕ ∂ p : k Mj ( R [ p ]) → Ω j − R/pR, log ⊕ Ω jR/pR, log is surjective and its kernel is contained in theimage of k Mj ( R ) → k Mj ( R [ p ]) by Lemma 1.11. Diagram chasing now shows that h j is an isomorphism. p -adic Nesterenko–Suslin isomorphism The Nesterenko–Suslin isomorphism [40], reproved by Totaro [48], states that for any field k there isa natural isomorphism K Mj ( k ) ≃ → H j ( k, Z ( j )) where the target denotes the weight j , degree j motiviccohomology of k . This was extended by Kerz to all regular local rings containing a field [26] [27,Prop. 10(11)] (replacing Milnor K -theory by improved Milnor K -theory in the small residue field case).In this section we prove the following p -adic analogue in mixed characteristic; the generality in which Z p ( j )( A ) is defined allows us to avoid any regularity hypotheses: Theorem 3.1.
The Galois cohomological symbol induces, for any local, p -henselian ring A , naturalisomorphisms b K Mj ( A ) /p r ≃ → H j ( Z /p r ( j )( A )) for all r, j ≥ .Proof. We begin by explaining the existence of a natural symbol map K Mj ( A ) → H j ( Z /p r ( j )( A )),induced by the Galois symbol in the smooth case. So suppose first that A is a local, p -henselian,ind-smooth Z ( p ) -algebra; to put ourselves in the context of our main theorems (which are stated overcomplete discrete valuation rings), let A ′ be the p -henselisation of A ⊗ Z ( p ) Z p and note that A ′ is a local, p -henselian, ind-smooth Z p -algebra. The Galois symbol therefore induces K Mj ( A ) → K Mj ( A ′ ) → Ker( H j ´et ( A ′ [ p ] , µ ⊗ jp r ) → W r Ω j − A ′ /pA ′ , log )) Corol. 1.15 ∼ = H j ( Z p ( j )( A ′ ) /p r ) ≃ ← H j ( Z p ( j )( A ) /p r ) , where the final isomorphism holds since A and A ′ have the same p -adic completions (from which it alsofollows that the first arrow is an isomorphism modulo any power of p , which will be relevant later).This defines the desired map K Mj ( A ) → H j ( Z p ( j )( A ) /p r ) whenever A is a local, p -henselian, ind-smooth Z ( p ) -algebra.For general local, p -henselian rings A , we define K Mj ( A ) → H j ( Z p ( j )( A ) /p j ) via left Kan extensionfrom the ind-smooth case, using Propositions 1.16 and 1.17. Concretely, this means that if we pick any lo-cal, p -henselian, ind-smooth Z ( p ) -algebra R surjecting onto A , then the map K Mj ( R ) → H j ( Z p ( j )( R ) /p r )defined in the previous paragraph descends (necessarily uniquely) to a map K Mj ( A ) → H j ( Z p ( j )( A ) /p r ).Our goal is to show that this descends further to an isomorphism b K Mj ( A ) /p r ≃ → H j ( Z p ( j )( A ) /p r ).We have already established this isomorphism in two cases:(i) “smooth case”, namely whenever A = R is a local, ind-smooth p -henselian algebra over a completediscrete valuation ring: indeed, this is precisely Theorem 1.7;(ii) and “big residue field case”, namely whenever A is a local, p -henselian algebra over a completediscrete valuation ring with big residue field: this is precisely line (6) of the proof of Proposition 1.18,which was conditional at the time on the now-established Theorem 1.7.To establish the isomorphism in general we may reduce, by taking a filtered colimit, to the case that A is the p -henselisation of a local, essentially finite type Z ( p ) -algebra. Applying the same trick as in theproof of Proposition 1.9, we realise the residue field K of A as a finite extension of F p ( t , . . . , t d ) andthen pick a big enough finite field k ′ such that | k ′ : F p | is coprime to both p and | K : F p ( t ) | . Let A ′ be the p -henselisation of A ⊗ Z ( p ) W ( k ′ ), which is a local, p -henselian, W ( k ′ )-algebra, and consider the21orten L¨uders and Matthew Morrowcommutative diagram K Mj ( A ′ ) /p r ∼ = / / H j ( Z p ( j )( A ′ ) /p r ) ∼ = H j ( Z p ( j )( c A ′ ) /p r ) K Mj ( A ) /p r / / O O H j ( Z p ( j )( A ) /p r ) O O ∼ = H j ( Z p ( j )( b A ) /p r ) O O The symbol map in the top row is an isomorphism by the big residue field case. The right vertical arrowis injective since c A ′ = b A ⊗ Z p W ( k ′ ) is a finite ´etale extension of b A of degree prime to p : see Corollary 3.4.Since K Mj ( A ′ ) /p r ≃ → b K Mj ( A ′ ) /p r ⊇ b K Mj ( A ) /p r , it now follows formally from a diagram chase that thesymbol map for A descends to an injection b K Mj ( A ) /p r ֒ → H j ( Z p ( j )( A ) /p r ).To prove surjectivity, pick an p -henselian, ind-smooth Z ( p ) -algebra surjecting onto A and henseliseit along the kernel; the result R is an p -henselian, ind-smooth Z p -algebra equipped with a henseliansurjection R → A . As at the end of the proof of Proposition 1.16, the induced map on completions b R → b A is still a henselian surjection, and so rigidity for the Z p ( j ) [2, Thm. 5.2] implies that H j ( Z p ( j )( R ) /p r ) → H j ( Z p ( j )( A ) /p r ) is surjective. This reduces the necessary surjectivity to the case of R in place of A ; butthat is covered by the smooth case mentioned above. Remark 3.2.
Theorem 3.1 implies, in particular, that the maps H j ( Z /p r +1 ( j )( A )) → H j ( Z /p r ( j )( A ))are surjective for all r ≥
1; equivalently H j +1 ( Z p ( j )( A )) is p -torsion-free, whence H j ( Z p ( j )( A )) /p r ≃ → H j ( Z /p r ( j )( A )) for all r ≥ p : H j +1 ( Z p ( j )( R ) /p r ) → H j +1 ( Z p ( j )( R ) /p r +1 ) is injective; as at the end of the proof of theprevious theorem we may use rigidity to reduce to the case that A = R is a local, p -henselian, ind-smooth Z p -algebra, then we use rigidity again to replace R by R := R/pR .The exact sequence 0 → W r − Ω j log p −→ W r Ω j log → Ω j log → R induces upontaking cohomology a sequence0 → W r − Ω jR, log p −→ W r Ω jR, log → Ω jR, log → e ν r − ( j )( R ) p −→ e ν r ( j )( R ) → e ν ( j )( R ) → e ν r ( j )( R ) := H (Spec R, W r Ω j log ).But the projection map W r Ω jR, log → Ω jR, log is surjective since both sides are quotients of K Mj ( R ) (seeRemark 1.6), so we have proved exactness of 0 → e ν r − ( j )( R ) p −→ e ν r ( j )( R ) → e ν ( j )( A ) →
0. Recallingthat H j +1 ( Z /p r ( j )( A )) = e ν r ( j )( A ) [3, Corol. 8.19], this is precisely the desired result.To treat the passage from Milnor K -theory to its improved variant, we unsurprisingly needed normmaps on Z p ( j )( − ). Although these should exist in much greater generality (say, for arbitrary finitequasisyntomic maps), the case of finite ´etale maps is sufficient for our purposes, namely Corollary 3.4. Lemma 3.3.
To any finite ´etale map A → B of p -adically complete rings there are associated normmaps N = N B/A : Z p ( j )( B ) → Z p ( j )( A ) for all j ≥ satisfying the following properties:(i) For any map A → A ′ where A ′ is another p -adically complete ring, the diagram Z p ( j )( A ) (cid:15) (cid:15) Z p ( j )( B ) N o o (cid:15) (cid:15) Z p ( j )( A ′ ) Z p ( j )( A ′ ⊗ A B ) N o o commutes.(ii) When A → B has constant degree d , then the composition Z p ( j )( A ) → Z p ( j )( B ) N −→ Z p ( j )( A ) ismultiplication by d . K -theory of p -adic rings Proof.
Suppose first that A is quasisyntomic, in which case B is automatically also quasisyntomic.Letting A → S be a quasisyntomic cover where S is quasiregular semiperfectoid, then quasisyntomicdescent of Z p ( j ) implies that Z p ( j )( A ) is equivalent to the totalisation of Z p ( j )( − ) of the p -completedˇCech nerve of this cover, i.e., Z p ( j )( A ) ≃ | S / / / / S b ⊗ A S / / / / / / S b ⊗ A S b ⊗ A S / / / / / / / / · · ·| The base change S B := S ⊗ A B is a quasiregular semiperfectoid ring providing a quasisyntomic cover of B , and so similarly Z p ( j )( B ) ≃ | Z p ( j )( S b ⊗ B • B ) | .Since all terms in the ˇCech nerves are themselves quasiregular semiperfectoid rings [], there arenatural equivalences Z p ( j )( S b ⊗ A n ) ≃ τ [2 j − , j ] K ( S ⊗ A n ; Z p ) for all n ≥
0, and similarly for S B , andtherefore the K -theory norm map for the finite ´etale extension S b ⊗ A n → S b ⊗ B nB = S b ⊗ A n ⊗ A B induces N S b ⊗ BnB /S b ⊗ An : Z p ( j )( S b ⊗ B nB ) → Z p ( j )( S b ⊗ A n ). These are moreover compatible in the sense that thediagram Z p ( j )( S b ⊗ A n ) f (cid:15) (cid:15) Z p ( j )( S b ⊗ B nB ) N o o f (cid:15) (cid:15) Z p ( j )( S b ⊗ A m ) Z p ( j )( S b ⊗ B mB ) N o o commutes for any map f : [ m ] → [ n ], since the analogous diagram commutes in K -theory; so we maytotalise to induce a norm map N SB/A : Z p ( j )( B ) → Z p ( j )( A ) which appears to depend on S .But this dependence on S is superficial: given any quasisymtomic cover of A by another quasiregularsemiperfectoid T , which we assume receives a map from S since otherwise we replace it by S ⊗ A T , thecompatible commutative diagrams Z p ( j )( S b ⊗ A n ) (cid:15) (cid:15) Z p ( j )( S b ⊗ B nB ) N o o (cid:15) (cid:15) Z p ( j )( T b ⊗ A n ) Z p ( j )( T b ⊗ B nB ) N o o (12)(again, since the analogous diagrams in K -theory commute and are compatible over n ) totalise to acommutative diagram Z p ( j )( A ) ≃ (cid:15) (cid:15) Z p ( j )( B ) ≃ (cid:15) (cid:15) | Z p ( j )( S b ⊗ A • ) | ≃ (cid:15) (cid:15) | Z p ( j )( S b ⊗ B • B ) | N o o ≃ (cid:15) (cid:15) | Z p ( j )( T b ⊗ A • ) | | Z p ( j )( T b ⊗ B • B ) | N o o i.e., informally N SB/A = N S ′ B/A . More precisely, we therefore define N B/A be be the limit of N SB/A overall covers S of A , viewing Z p ( A ) as the limit of | Z p ( j )( S b ⊗ A • ) | over all such covers. But in practice inwhat follows, we will just pick a particular cover S .We next check functoriality (i), assuming that both A and A ′ are quasisyntomic so that the norm mapshave been defined. Let A → S be a quasisyntomic cover with S quasiregular semiperfectoid, and thenlet S b ⊗ A A ′ → T a quasisyntomic cover with T quasiregular semiperfectoid; note that the composition A ′ → T is a quasisyntomic cover. Calculating the norms for A → B and A ′ → A ′ ⊗ A B using the covers S and T respectively, the desired functoriality follows from totalising over the commutative diagramswhich are almost identical to (12), just replacing A and B in the bottom row by A ′ and B ′ = A ′ ⊗ A B .We next extend the norm map to finite ´etale maps A → B between arbitrary p -complete rings; theidea is to compatibly simpicialy resolve A and B to reduce to the case already treated. Similarly to the23orten L¨uders and Matthew Morrowproof of Proposition 1.17 we pick a simplicial resolution R • → A by ind-smooth Z p -algebras such thatthe kernel of each surjection R q → A is a henselian ideal. Then R • /p s R • ∼ → A ⊗ LZ Z /p s Z for each s ≥ b R • ∼ → A ; here b R • is the simplicial ring obtainedby p -adically completing each R q (which represents the derived p -adic completion of R • ), and we recallthat A is derived p -adically complete (as it is p -adically complete, which includes separated under ourconventions). The kernel of each surjection b R q → A is henselian, as follows from a straightforward seriesof manipulations which we leave to the reader.In conclusion, we have constructed a simplicial resolution b R • ∼ → A along henselian surjections, whereeach ring in the resolution is quasisyntomic. The finite ´etale map A → B lifts uniquely to a finite ´etalemap b R q → Q q for each q ≥
0, and these assemble to form a simplicial ring Q • → B . Observe that each Q q is also quasisyntomic over Z p , that Q q → B is a henselian surjection (as henselian surjections arepreserved under base change along integral maps), and that Q • → B is an equivalence (if B were a finitefree A -module then Q • would be a finite free b R • -module and this would be clear; the general case is adirect summand of such a free case).The already constructed norm maps in the quasisyntomic case therefore define a map N : Z p ( j )( b R • ) → Z p ( j )( Q • ) of simplicial complexes. Since Z p ( j )( − ) commutes with p -completed sifted colimits [2, Thm. 5.1(2)],we may then geometrically realise and p -complete to define N B/A : Z p ( j )( B ) → Z p ( j )( A ).The independence of N B/A on the chosen resolution is proved similarly to the independence on S inthe first part of the proof, as is its functoriality (i); since we do not need these results for Corollary 3.4,we omit the details of the proofs.Property (ii) reduces via the definitions to the case that A and B are quasiregular semiperfectoid, inwhich case it follows from the analogous property of the K -theory norm map. Corollary 3.4.
Let A → B be a finite ´etale map of p -adically complete rings, of constant degree notdivisible by p . Then Z p ( j )( A ) → Z p ( j )( B ) is split injective.Proof. Letting d be the degree, the splitting is provided by d N B/A thanks to Lemma 3.3(ii).As an application of the p -adic Nesterenko–Suslin theorem, we describe the p -adic Milnor K-groupslocally on smooth formal schemes over the rings of integers of perfectoid fields: Theorem 3.5.
Let C be a perfectoid field of characteristic containing all p -power roots of unity, and X a smooth, p -adic formal O C -scheme; let x ∈ X and let R := O X,x be the corresponding local ring; let j ≥ .(i) The Galois symbol b K Mj ( R ) /p r −→ H j ´et ( R [ p ] , µ ⊗ jp r ) is an isomorphism; if R has big residue field then the canonical map K Mj ( R ) /p r → K Mj ( R [ p ]) /p r is also an isomorphism.(ii) The p -adic completion of b K Mj ( R ) is p -power-free.Proof. We begin by recalling the structure of the ring R . Picking an affine open neighbourhood Spf S of x , with x corresponding to the prime ideal q ⊆ S , then R = lim −→ f ∈ S \ q d S [ f ] where the hat denotes p -adiccompletion. In particular, R is a p -henselian local ring having residue field k ( q ).(i) The Galois symbol is an isomorphism thanks to Theorem 3.1 and the known identification H j ( Z /p r ( j )( R )) ∼ = H j ´et ( R [ p ] , µ ⊗ jp r ) [3, Thm. 10.1]. When R has big residue field the Galois symbolfactors through b K Mj ( R ) /p r = K Mj ( R ) /p r → K Mj ( R [ p ]) /p r , so it remains only to prove that this map issurjective: but that follows from the existence of u, π ∈ O C , with u being a unit, such that p = uπ p r .(ii) In the first diagram of the proof of Proposition 1.13 the boundary map δ is injective since theprevious map in the sequence H j − ( R [ p ] , µ ⊗ jp r ) → H j − ( R [ p ] , µ ⊗ jp ) may be identified (using (i) for j − b K Mj ( R ) /p r → b K Mj ( R ) /p . Again using (i) (this timefor j ), we deduce that the multiplication map p : b K Mj ( R ) /p r − → b K Mj ( R ) /p r is injective, from whichthe p -torsion-freeness claim follows. 24ilnor K -theory of p -adic rings Remark 3.6.
In [22] Izhboldin proves that the Milnor K-groups of a field of characteristic p > p -torsion free. Combined with the Gersten conjecture for Milnor K-theory in equal characteristic, thisimplies that for a smooth scheme X over a field k of characteristic p > b K Mj,X is p -torsion free. Theorem 3.5(ii) may be considered to be an analogue of this statement in mixedcharacteristic. The Gersten conjecture in Milnor K -theory predicts that, for any regular Noetherian local ring R , theGersten complex0 −→ b K Mj ( R ) −→ K Mj (Frac R ) −→ M x ∈ Spec R (1) K Mj − ( k ( x )) −→ M x ∈ Spec R (2) K Mj − ( k ( x )) −→ · · · is exact for each j ≥
0; here we write Spec R ( i ) for the set of codimension i points of Spec R . If R contains a field then the Gersten complex is known to be universally exact by Kerz [26, 27], but themixed characteristic case is open. p -henselian, ind-smooth case By combining Theorem 1.3(ii) with Kerz’ and other existing Gersten results in motivic cohomology, weprove the Gersten conjecture in mod p -power Milnor K -theory for p -henselian, ind-smooth algebras overcomplete discrete valuation rings: Theorem 4.1.
Let V be a complete discrete valuation ring of mixed characteristic, and R a p -henselian,regular, Noetherian, local V -algebra such that R/ m R is ind-smooth over V / m . Then the mod p -powerGersten conjecture holds for R , i.e., for any r, j ≥ , the complex −→ b K Mj ( R ) /p r −→ K Mj (Frac R ) /p r −→ M x ∈ Spec R (1) K Mj − ( k ( x )) /p r −→ M x ∈ Spec R (2) K Mj − ( k ( x )) /p r −→ · · · is exact.Proof. As mentioned before the theorem, this will follow from combining our main injectivity theoremwith existing results; there are various ways to carry out the argument, among which we propose thefollowing (whose advantage is that it circumvents any new use of Panin’s trick [42] to reduce the ind-smooth case to the smooth case). First note that the hypotheses imply that V → R is geometricallyregular, therefore ind-smooth by N´eron–Popescu; so our earlier results do apply.Let Z = Spec( R/ m R ) and X η = Spec( R [ p ]) be the special and generic fibres, so that we have Gerstencomplexes g j ( X ) = 0 −→ K Mj (Frac R ) /p r → M x ∈ X (1) K Mj − ( x ) /p r → · · · g j ( Z ) = 0 −→ K Mj (Frac R ) /p r → M x ∈ Z (1) K Mj − ( x ) /p r → · · · g j ( X η ) = 0 −→ K Mj (Frac R ) /p r → M x ∈ X (1) η K Mj − ( x ) /p r → · · · . fitting into a short exact sequence 0 → g j − ( Z )[ − → g j ( X ) → g j ( X η ) →
0. Thanks to the Gerstenconjecture in Milnor K -theory over fields [27], the canonical map b K Mj ( R ) /p r → g j ( Z ) is an equivalenceand g j ( X η ) calculates the Zariski cohomology of the improved Milnor K -theory sheaf b K Mj /p r on X η (which is typically not local).From the short exact sequence of Gersten complexes we therefore obtain an exact sequence0 −→ H ( g j ( X )) −→ H ( X η , b K Mj /p r ) −→ b K Mj − ( A/ m A ) /p r −→ H ( g j ( X )) −→ H n ( g j ( X )) ≃ → H n Zar ( X η , b K Mj /p r ) for n ≥
2. But by passage to a filtered colimit overLemma 4.2(ii) below we know that H n Zar ( X π , b K Mj /p r ) = ( H j ´et ( X η , µ ⊗ jp r ) if n = 0 , n > . This proves the desired acyclicity in degrees ≥ b K Mj ( R ) /p r ≃ → H ( g j ( X )).We required the following (presumably well-known) result: Lemma 4.2.
Let V be a complete discrete valuation ring of mixed characteristic, and S an essentiallysmooth, local V -algebra; let r, j ≥ . Then(i) H n Zar (Spec S [ p ] , R i ε ∗ µ ⊗ jp r ) = 0 for all i ≤ j and all n > , where ε : Spec S [ p ] ´et → Spec S [ p ] Zar isthe projection map of sites.(ii) H n Zar (Spec S [ p ] , b K Mj /p r ) = ( H j ´et (Spec S [ p ] , µ ⊗ jp r ) if n = 0 , if n > . Proof.
Part (ii) is follows from (i) via the change of topology spectral sequence, using the Bloch–Katoisomorphism b K Mj /p r ≃ → R j ε ∗ µ ⊗ jp r on Spec S [ p ].To prove part (i) we use known Gersten results in motivic cohomology. For any essentially finitetype scheme Y over V , let Z ( j ) mot ( Y ) := z j ( Y, • )[ − j ] denote its motivic cohomology complex definedvia Bloch’s higher Chow groups. Similarly to the previous proof above with Milnor K -theory, there areassociated Gersten complexes for any i, j ≥ g i ( j ) mot (Spec S ) = 0 −→ H i ( Z ( j ) mot (Frac S ) /p r ) −→ M x ∈ Spec S (1) H i − ( Z ( j − mot ( k ( x )) /p r ) −→ · · · g i ( j ) mot (Spec S/ m S ) = 0 −→ H i ( Z ( j ) mot (Frac( S/ m S )) /p r ) −→ M x ∈ Spec S/ m S (1) H i − ( Z ( j − mot ( k ( x )) /p r ) −→ · · · g i ( j ) mot (Spec S [ p ]) = 0 −→ H i ( Z ( j ) mot (Frac( S )) /p r ) −→ M x ∈ Spec S [ 1 p ] (1) H i − ( Z ( j − mot ( k ( x )) /p r ) −→ · · · fitting into short exact sequences0 → g i − ( j ) mot (Spec S/ m S )[ − → g i ( j ) mot (Spec S ) → g i ( j ) mot (Spec S [ p ]) → . However, unlike the previous case of Milnor K -theory, we may now appeal to the fact that the Gerstenconjecture in motivic cohomology is known not only for essentially smooth algebras over fields, but alsofor the essentially smooth V -algebra S [16, Corol. 4.5] (at least with mod p r -coefficients). So from thecorresponding long exact sequence we deduce that H n Zar (Spec S [ p ] , H i ( Z ( j ) mot /p r )) = 0 for n >
0, where H i ( Z ( j ) mot /p r ) denotes the Zariski sheafification of U H i ( Z ( j ) mot ( U ) /p r ). The proof is completedby appealing to the Beilinson–Lichtenbaum isomorphism H i ( Z ( j ) mot /p r ) ≃ R i ε ∗ µ ⊗ jp r on Spec S [ p ] when i ≤ j . Remark 4.3.
In the context of Theorem 4.1, the injectivity at the beginning of the Gersten complex,namely b K Mj ( R ) /p r ֒ → K Mj (Frac R ) /p r , can be deduced more directly. Indeed, letting F be as in thestatement of Theorem 5.8, it is enough to check that the composition b K Mj ( R ) /p r −→ K Mj (Frac R ) /p r −→ K Mj ( F ) /p r h jpr −−→ H j ´et ( F, µ ⊗ jp r )is injective. But this composition coincides with b K Mj ( R ) /p r −→ H j ´et ( R [ p ] , µ ⊗ jp r ) −→ H j ´et ( F, µ ⊗ jp r ) , where the first arrow is injective by Theorem 1.7 and the second by Theorem 5.8.26ilnor K -theory of p -adic ringsWe note that, in particular, we have proved the mod p -power Gersten conjecture Nisnevich locallyon smooth V -schemes: Corollary 4.4.
Let V be a complete discrete valuation ring of mixed characteristic and X a smooth V -scheme. Then the Gersten sequence of Nisnevich sheaves on X −→ b K Mj,X /p r −→ M x ∈ X (0) i x ∗ ( K Mj ( x ) /p r ) −→ M x ∈ X (1) i x ∗ ( K Mj − ( x ) /p r ) −→ · · · (which will be explained further in the course of the proof ) is exact, and consequently there is a naturalBloch–Quillen isomorphism CH j ( X ) /p r ∼ = H j Nis ( X, b K Mj,X /p r ) . Proof.
For each point x ∈ X , let K Mj ( x ) /p r denote the Nisnevich sheaf on Spec k ( x ) defined by sendingeach ´etale k ( x )-algebra L to K Mj ( L ) /p r ; recall that this is indeed a Nisnevich sheaf because it is additive[41, 1.2]. Let i x : Spec k ( x ) Nis → X Nis denote the canonical map of sites, and i x ∗ ( K Mj ( x ) /p r ) theresulting pushforward of this sheaf to X Nis .In other words, for each ´etale f : U → X we have i x ∗ ( K Mj ( x ) /p r )( U ) = L y ∈ f − ( x ) K Mj ( k ( y )) /p r ,and the Nisnevich Gersten complex M x ∈ X (0) i x ∗ ( K Mj ( x ) /p r ) −→ M x ∈ X (1) i x ∗ ( K Mj − ( x ) /p r ) −→ · · · is characterised by the fact that its restriction to U Zar is the usual Zariski Gersten complex for all such U .Moreover, as sheaves on the site Spec k ( x ) Nis have no higher cohomology on any object, the same is trueof their pushforwards to X Nis ; in particular, each Nisnevich sheaf i x ∗ ( K Mj ( x ) /p r ) on X has no highercohomology. To prove that the aforementioned Nisnevich Gersten complex is a resolution of b K Mj,X /p r (which denotes the Nisnevich sheafification of U b K Mj ( O U ( U )) /p r ), we must check the exactness ofthe Gersten complexes0 −→ b K Mj ( A ) /p r −→ M x ∈ Spec A (0) K Mj ( k ( x )) /p r −→ M x ∈ Spec A (1) K Mj − ( k ( x )) /p r −→ · · · where A runs over the henselian local rings attached to all points of X . When the point lies in thegeneric fibre of X , so that A contains a field, we appeal to Kerz [27]; when the point lies in the specialfibre, so that A is p -henselian, we instead appeal to Theorem 4.1.The Bloch–Quillen formula follows as the Gersten resolution allows us to calculate the cohomologyof b K Mj,X /p r as H j Nis ( X, b K Mj,X /p r ) = coker (cid:0) M x ∈ X ( j − k ( x ) × /p r → M x ∈ X ( j ) Z /p r (cid:1) = CH j ( X ) /p r Remark 4.5.
With V and X as in Corollary 4.4, the result implies by functoriality the existence ofa natural restriction map CH j ( X ) /p r → H j Nis ( X s , b K Mj,X s /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 thenthese restriction maps are surjective by earlier work of the first author [32, 33], answering a question ofKerz–Esnault–Wittenberg [28, Conj. 10.1] in the smooth case. K M /p r of (incomplete) discrete valuation rings Using the relative Gersten arguments pioneered by Bloch [5] and Gillet–Levine [19], adapted to Milnor K -theory by the first author [34], (and Panin’s trick [42] to reduce the ind-smooth case to the essentiallysmooth case) the Gersten conjecture for ind-smooth algebras over a discrete valuation ring V mostlyreduces to case of the discrete valuation rings obtained by localising smooth V -algebras at the genericpoint of their special fibre. We therefore record here a case of the Gersten conjecture for discrete valuationrings which we were surprised not to find in the literature; it is independent from our main results andthe style of argument is not new. 27orten L¨uders and Matthew MorrowWe first review some classical results; let O be a discrete valuation ring of residue characteristic p > F and its residue field by k . Then the map K ( O ) → K ( F ) is injectiveby Dennis–Stein [11], with p -torsion-free cokerel k × , whence K ( O ) /p r → K ( F ) /p r is also injective forany r ≥ O is equi-characteristic. Then the map K ( O ) → K ( F ) is injective by Quillen [43], and thecokernel K ( k ) is p -torsion-free by Izhboldin [22] (identifying it with K M ( k ) by Matsumoto), whence K ( O ) /p r → K ( F ) /p r is again injective. Secondly, if O has infinite residue field then K M ( O ) → K ( F )was shown to be injective by Suslin–Yarosh [46] (though this is of course superseded by Kerz [26]).Suppose instead that O has mixed characteristic (0 , p ). Then the injectivity of K ( O ) → K ( F )remains open, but that of K ( O ; Z /p r Z ) → K ( F ; Z /p r Z ) is a theorem of Geisser–Levine [17, Thm. 8.2].If p > b K ( O ) /p r injects into K ( O ) /p r ⊆ K ( O ; Z /p r Z ) [26, Prop. 10(6)] and so we deduce that b K M ( O ) /p r → K M ( F ) /p r is also injective. Here we observe that the argument of Suslin–Yarosh mayalso be used to prove the latter injectivity, without the hypothesis that p > Proposition 4.6.
Let O be a discrete valuation ring of mixed characteristic, with field of fractions F and residue field k . Then there is an exact sequence −→ b K M ( O ) /p r −→ K M ( F ) /p r ∂ −→ K M ( k ) /p r −→ for any r ≥ .Proof. The canonical map K ( O , π O ; Z /p r Z ) → K ( O ; Z /p r Z ) is injective since K ( k ; Z /p r Z ) is gen-erated by symbols (which lift to K ( O ; Z /p r Z )) by Geisser–Levine, and this restricts to an injection K ( O , π O ) /p r → K ( O ) /p r . From the exact sequence K ( O , π O ) → K ( O ) → K ( k ) → K ( k ) is generated by symbols, which lift) we therefore obtain a short exactsequence 0 −→ K ( O , π O ) /p r −→ K ( O ) /p r −→ K ( k ) /p r −→ . Assuming that O has infinite residue field, the proof is completed by recalling Suslin–Yarosh’s argu-ment, which is itself a K M -version of Dennis–Stein’s argument: taking [46, Thms. 3.9 & 4.1] modulo p r implies the existence of a cocartesian square K ( O , π O ) /p r / / (cid:15) (cid:15) K M ( O ) /p r (cid:15) (cid:15) K M ( O ) /p r ·{ π } / / K M ( F ) /p r which indeed completes the proof when combined with the above exact sequence and the identity b K M ( O ) = K ( O ).When O has finite residue field, the injectivity of b K ( O ) /p r → K ( F ) /p r then follows from a standardnorm trick by picking a tower of finite ´etale extensions O ⊆ O ⊆ O ⊆ · · · of discrete valuation ringswhere each extension has degree prime to p . Remark 4.7.
The focus of this subsection is in the incomplete case, but we mention that b K Mj ( O ) → K Mj ( F ) is known to be injective for all j ≥ O is compelte and has finite residue field [9]. p -adic Milnor K -theory of local F p -algebras In this section we present two variants of the Bloch–Kato–Gabber theorem, describing the mod p -power Milnor K -theory of certain local F p -algebras. The analogous isomorphisms for algebraic K -theory are already known by earlier work of the second author and collaborators [7, Thm. 5.27] [25,Thm. 2.1]; the new tool which allows us to treat Milnor K -theory is the left Kan extension observationof Proposition 1.17. 28ilnor K -theory of p -adic rings F p -algebras We start by recalling the following terminology from [25]: an F p -algebra A is called Cartier smooth if itsatisfies the following smoothness criteria:- Ω A is a flat A -module;- H j ( L A/ F p ) = 0 for all j > C − : Ω jA → H j (Ω • A ) is an isomorphism for all j ≥ F p denote the category of Cartier smooth F p -algebras, which includes the category of smooth F p -algebras Sm F p ; to simplify notation in the following proof, let Sm Σ F p be the subcategory of finitelygenerated polynomial F p -algebras.As a more interesting example, results of Gabber–Ramero [15, Thm. 6.5.8(ii) & Corol. 6.5.21] andGabber [29, App.] state that any valuation ring of characteristic p is Cartier smooth.The following description of the ´etale-syntomic cohomology of Cartier smooth algebras was implicitin [25]: Proposition 5.1.
Let r, j ≥ .(i) The functor W r Ω j log : CSm F p → D ( Z ) is left Kan extended from Sm F p .(ii) For any A ∈ CSm F p , there are natural equivalences Z /p r Z ( j )( A )[ j ] ≃ R Γ ´et (Spec A, W r Ω j log ) ≃ [ W r Ω jA F − −−−→ W r Ω jA /dV r − Ω j − A ] . Proof.
The second equivalence in (ii) actually holds for any F p -algebra, since on any F p -scheme X thereis a short exact sequence of ´etale sheaves 0 → W r Ω jX, log → W r Ω jX F − −−−→ W r Ω jX /dV r − Ω j − X → A ∈ CSm F p , there is a natural short exact sequence 0 → W r − Ω jA, log p −→ W r Ω jA, log → Ω jA, log → r = 1. Similarly, since this short exactsequence equally holds for any ´etale A -algebra (since it is also Cartier smooth), there is an induced shortexact sequence of sheaves on the ´etale site of A and a corresponding fibre sequence of cohomology R Γ ´et (Spec A, W r − Ω j log ) p −→ R Γ ´et (Spec A, W r Ω j log ) → R Γ ´et (Spec A, Ω j log ) . So the claim “ R Γ ´et (Spec − , W r Ω j log ) : CSm F p → D ( Z ) is left Kan extended from Sm Σ F p ” also reduces tothe case r = 1. But this claim would imply the first equivalence in (ii): it is even equivalent to it asthe first equivalence in (ii) is already known for all finitely generated polynomial F p -algebras (even allsmooth F p -algebras) [3, Corol. 8.19] and Z /p r Z ( j )( − ) is left Kan extended from Sm Σ F p (either by theformula Z /p j ( j )( − ) = hofib( ϕp j − N ≥ j LW Ω → LW Ω A ) /p r of [3], or just by quoting [2, Thm. 5.1(ii)]).We have reduced (i) and (ii) to proving that on the category CSm F p , the functors Ω j log and R Γ ´et (Spec − , Ω j log )are left Kan extended from Sm F p and from Sm Σ F p respectively.But it was shown in [25] that both Ω j and d Ω j − , on the category CSm F p , are left Kan extendedfrom Sm Σ F p , so the same is true of R Γ ´et (Spec − , Ω j log ) using the second equivalence of part (ii). Thisequivalence also shows that there is a natural fibre sequence Ω j log → R Γ ´et (Spec A, Ω j log )[ j ] → e ν ( j )( A )[ − A ∈ CSm F p ; the final term is rigid [7, Prop. 4.30] hence left Kan extended from Sm F p , and so wededuce 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 jA, log , by alsodescribing 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 b K Mj ( A ) /p r → W r Ω jA, log is an isomorphism for any r, j ≥ . Proof.
Although this follows from Theorem 3.1 and Proposition 5.1(ii), we prefer to give a proof whichavoids the passage through mixed characteristic.The symbol map is surjective by Remark 1.6 so it remains to treat its injectivity, for which we mayassume A has big residue field (otherwise take a finite ´etale extension A ′ ⊇ A of degree coprime to p ,with A ′ also local, so that b K Mj ( A ) /p r → K Mj ( A ′ ) /p r is injective by existence of the norm map). We view K Mj ( − ) /p r → W r Ω j log as a map of functors on SCm loc F p . It is an isomorphism on any essentially smooth,local F p -algebra with big residue field by the Bloch–Kato–Gabber theorem (see Remark 1.6 again); byleft Kan extending using Propositions 1.17 and 5.1 we obtain the desired isomorphism for A . Note herethat, in the diagram of essentially smooth, local F p -algebras mapping to A , those with big residue fieldform a cofinal system (alternatively, computing the left Kan extension by the type of simplicial resolutionwhich appeared in Proposition 1.17, just note that each R q has the same big residue field as A ). We prove a version of the Bloch–Kato–Gabber theorem for pro rings, and an associated continuitytheorem for Milnor K -theory. The initial two lemmas are very much in the style of [37], to which werefer for further background on arguments with pro abelian groups (which we denote by curly brackets { A s } s , rather than “ lim ←− ” s A s ) and the role of F-finiteness. Lemma 5.3.
Let A be an F-finite, regular, Noetherian, local F p -algebra, and I ⊆ A an ideal. Then thereis a natural short exact sequence of pro abelian groups → { W r − Ω jA/I s , log } s p −→ { W r Ω jA/I s , log } s → { Ω jA/I s , log } s → for each j ≥ .Proof. It was shown in [37, Corol. 4.8] that the diagonally indexed pro abelian group { W s Ω jA/I s , log } s is p -torsion-free and the canonical map { W s Ω jA/I s , log /p r } s → { W r Ω jA/I s , log } s is an isomorphism. So the de-sired short exact sequence is 0 → { W s Ω jA/I s , log /p r − } s p −→ { W s Ω jA/I s , log /p r } s → { W s Ω jA/I s , log /p } s → F : F p -algs → Ab, let L F : F p -algs → D ≤ ( Z ) denote the left Kan extension from Sm Σ F p of the restriction of F . For the application to Theorem 5.5, note that for each of the three functors in thelemma it would be equivalent to left Kan extend from Sm F p (as follows from the Cartier isomorphism,or alternatively apply the following lemma with I = 0). Lemma 5.4.
Let A be a F-finite, regular, Noetherian F p -algebra and I ⊆ A an ideal. Then the canonicalmaps of complexes L Ω jA/I s −→ Ω jA/I s L B Ω jA/I s −→ B Ω jA/I s L Z Ω jA/I s −→ Z Ω jA/I s become equivalences of pro complexes after applying { } s .Proof. By “equivalences of pro complexes” we simply mean in each case that { H n ( − ) } s = 0 for n > { H ( − ) } s identifies with target pro abelian group.Since H ( L Ω jA/I s ) = Ω jA/I s for each s , in the first case it remains to prove that { H n ( L Ω jA/I s ) } = 0for n >
0. From the transitivity sequence for F p → A → A/I s one knows that L Ω jA/I s has a naturalfiltration with graded pieces L Ω iA ⊗ A L Ω j − i ( A/I s ) /A for i = 1 , . . . , j ; but a classical argument of M. Andr´e[1, Prop. X.12] (see also [36, Thm. 4.4(i)]) shows that { H n ( L Ω j − i ( A/I s ) /A ) } s = 0 unless n = j − i = 0. Sincealso L Ω iA ≃ Ω iA as A is geometrically regular over F p , the spectral sequence associated to the filtration(i.e., the spectral sequence of Kassel–Sletsjøe [24]) now yields the desired vanishing.The pro equivalences for B Ω j and Z Ω j now follow by the usual increasing induction on j , just as inthe Cartier smooth case (see the first three paragraphs of the proof of [25, Prop. 2.5]), since the inverseCartier map C − : { Ω jA/I s } s → { H j (Ω • A/I s ) } s is known to be an isomorphism of pro abelian groups forall j ≥ K -theory of p -adic ringsWe may now present the main theorem of the subsection, describing the pro system of Milnor K -groups of the successive quotients of a regular, local F p -algebra: Theorem 5.5.
Let A be an F-finite, regular, Noetherian, local F p -algebra, and I ⊆ A any ideal. Thenthe symbol map induces an isomorphism of pro abelian groups { b K Mj ( A/I s ) /p r } s ≃ → { W r Ω jA/I s , log } s forany r, j ≥ .Proof. Morally one would like to say that this theorem is a special case of Theorem 5.2, as the resultsof [37, §
2] implicitly show that the pro ring { A/I s } s satisfies a pro analogue of the definition of Cartiersmoothness. But we provide the details of the proof.The symbol map is automatically surjective (even for each fixed level s ), and for injectivity we performthe same trick as in Theorem 5.2 to henceforth assume that A has big residue field. For any essentiallysmooth, local F p -algebra R with big residue field, the symbol map K Mj ( R ) /p r → W r Ω jR, log is an isomor-phism as recalled in Remark 1.6. Left Kan extending from such R and appealing to Proposition 1.17,the theorem reduces to checking that the co-unit map H (( L sm W r Ω j log )( A/I s )) → W r Ω jA/I s , log inducesan isomorphism of pro abelian groups over s .This in turn reduces to the case r = 1 by comparing the fibre sequences( L sm W r − Ω j log )( A/I s ) p −→ ( L sm W r Ω j log )( A/I s ) → ( L sm Ω j log )( A/I s )(induced by Illusie’s short exact sequence 0 → W r − Ω jR, log p −→ W r Ω jR, log → Ω jR, log → F p -algebra [21, § I.5.7]) to the short exact sequence of Lemma 5.3. We then consider thefibre sequence Ω j − , log → [Ω j − C − − −−−−→ Ω j − /d Ω j − − ] → e ν ( j )( − )[ −
1] on arbitrary F p -algebras: left Kanextending it from essentially smooth, local F p -algebras, evaluating on A/I s , comparing to the originalsequence for A/I s itself, and using that e ν ( j )( − ) is left Kan extended from smooth algebras, the problemis reduced to checking that the co-unit maps L sm Ω jA/I s → Ω jA/I s [0] and L sm B Ω jA/I s → B Ω jA/I s [0] induceequivalences of pro complexes. But this is exactly what we checked in Lemma 5.4. Corollary 5.6.
Let A be an F-finite, regular, Noetherian, local F p -algebra, and I ⊆ A an ideal such that A is I -adically complete. Then the canonical map K Mj ( A ) /p r → lim ←− s K Mj ( A/I s ) /p r is an isomorphismfor any r ≥ .Proof. In light of the classical Bloch–Kato–Gabber theorem and Theorem 5.5, it is equivalent to provethat the map W r Ω jA, log → lim ←− s W r Ω jA/I s , log is an isomorphism. This was proved in [37, Corol. 4.11]. Remark 5.7.
Note that, combined with [7, Thm. 5.27], Theorem 5.5 shows that the canonical map { b K Mj ( A/I s ) /p r } s ≃ → { K j ( A/I s ) /p r } s is an isomorphism.Moreover, just as explained in [37, Rem. 5.8], the implicit bounds appearing in the isomorphisms ofpro systems are uniform when localising (or more generally passing to ´etale algebras). Therefore, forany F-finite, regular, Noetherian F p -scheme X , and closed subscheme Y ֒ → X , the canonical maps of(Zariski, Nisnevich, or ´etale) sheaves on Y K j,Y s /p r ←− b K Mj,Y s /p s dlog −−−→ W r Ω jY s , log becomes an isomorphism of pro sheaves after applying {} s . Appendix: A Gersten injectivity result of Gabber
The following injectivity result of Gabber was required in the proof of Theorem 2.17:
Theorem 5.8 (Gabber) . Let V be a mixed characteristic discrete valuation ring and R a local, ind-smooth p -henselian V -algebra; let R h p R be the henselisation of the discrete valuation ring R p R , and F := R h p R [ p ] its field of fractions. Let F be a torsion ´etale sheaf on Spec R [ p ] which is pulled back from Spec V [ p ] . Then the canonical map H n ´et ( R [ p ] , F ) −→ H n ´et ( F, F ) is injective for all n ≥ . Proof.
Since both sides commute with filtered colimits in R , we may assume that R is the henselisationalong p S q of S q , where S is a smooth V -algebra and q ⊆ S is some prime ideal containing p S ; note thatthen R h p R = S h p S , where the latter denotes the henselisation of the discrete valuation ring S p S .The beginning of Gabber’s Gersten resolution [14, Eqn. ( ∗ )], at the point q of Spec S (Gabber’sscheme M ), asserts that the map H n ´et (Spec S q / p S q , i ∗ Rj ∗ F ) −→ H n ´et (Spec S p A / p S p A , i ∗ Rj ∗ F )is injective, where i, j are the usual closed and open inclusions Spec S/ p S i ֒ → Spec S j ←− Spec S [ p ], and wesuppress the additional pullbacks along Spec S p R / p S p S → Spec S q / p S q → Spec S/ p S from the notation.Noting that R/ p R = S q / p S q , Gabber’s affine analogue of the proper base change theorem [] impliesthat the canonical map H n ´et (Spec R [ p ] , F ) = H n ´et (Spec R, Rj ∗ F ) −→ H n ´et (Spec S q / p S q , i ∗ Rj ∗ F )is an isomorphism. The analogous assertion is equally true for the henselian surjection R h p R → R p A / p R p A = S p A / p S p A , thereby completing the proof. References [1]
Andr´e, M.
Homologie des alg`ebres commutatives . Springer-Verlag, Berlin, 1974. Die Grundlehrender mathematischen Wissenschaften, Band 206.[2]
Antieau, B., Mathew, A., Morrow, M., and Nikolaus, T.
On the Beilinson fiber square. arXiv:2003.12541 (2020).[3]
Bhatt, B., Morrow, M., and Scholze, P.
Topological Hochschild homology and integral p -adicHodge theory. Publ. Math. Inst. Hautes ´Etudes Sci. 129 (2019), 199–310.[4]
Bloch, S. p -adic ´etale cohomology. In Arithmetic and geometry, Vol. I , vol. 35 of
Progr. Math.
Birkh¨auser Boston, Boston, MA, 1983, pp. 13–26.[5]
Bloch, S.
A note on Gersten’s conjecture in the mixed characteristic case. In
Applications ofalgebraic K -theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) ,vol. 55 of Contemp. Math.
Amer. Math. Soc., Providence, RI, 1986, pp. 75–78.[6]
Bloch, S., and Kato, K. p -adic ´etale cohomology. Inst. Hautes ´Etudes Sci. Publ. Math. , 63(1986), 107–152.[7]
Clausen, D., Mathew, A., and Morrow, M. K -theory and topological cyclic homology ofhenselian pairs. Journals of the AMS, to appear (2018).[8]
Colliot-Th´el`ene, J.-L.
Cohomologie galoisienne des corps valu´es discrets henseliens, d’apr`es K.Kato et S. Bloch. In
Algebraic K -theory and its applications (Trieste, 1997) . World Sci. Publ., RiverEdge, NJ, 1999, pp. 120–163.[9] Dahlhausen, C.
Milnor K-theory of complete discrete valuation rings with finite residue fields.
J.Pure Appl. Algebra 222 , 6 (2018), 1355–1371.[10]
Dennis, R. K., and Stein, M. R. K of radical ideals and semi-local rings revisited. In Algebraic K -theory, II: “Classical” algebraic K -theory and connections with arithmetic (Proc. Conf., BattelleMemorial Inst., Seattle, Wash., 1972) . Springer, Berlin, 1973, pp. 281–303. Lecture Notes in Math.Vol. 342.[11] Dennis, R. K., and Stein, M. R. K of discrete valuation rings. Advances in Math. 18 , 2 (1975),182–238.[12]
Elmanto, E., Hoyois, M., Khan, A. A., Sosnilo, V., and Yakerson, M.
Modules overalgebraic cobordism.
Forum. Math. Pi, to appear (2019).32ilnor K -theory of p -adic rings[13] Fesenko, I. B., and Vostokov, S. V.
Local fields and their extensions , second ed., vol. 121 of
Translations of Mathematical Monographs . American Mathematical Society, Providence, RI, 2002.With a foreword by I. R. Shafarevich.[14]
Gabber, O.
Gersten’s conjecture for some complexes of vanishing cycles.
Manuscripta Math. 85 ,3-4 (1994), 323–343.[15]
Gabber, O., and Ramero, L.
Almost ring theory , vol. 1800 of
Lecture Notes in Mathematics .Springer-Verlag, Berlin, 2003.[16]
Geisser, T.
Motivic cohomology over Dedekind rings.
Math. Z. 248 , 4 (2004), 773–794.[17]
Geisser, T., and Levine, M.
The K -theory of fields in characteristic p . Invent. Math. 139 , 3(2000), 459–493.[18]
Gille, P., and Szamuely, T.
Central simple algebras and Galois cohomology , vol. 101 of
Cam-bridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2006.[19]
Gillet, H., and Levine, M.
The relative form of Gersten’s conjecture over a discrete valuationring: the smooth case.
J. Pure Appl. Algebra 46 , 1 (1987), 59–71.[20]
Gros, M., and Suwa, N.
Application d’Abel-Jacobi p -adique et cycles alg´ebriques. Duke Math.J. 57 , 2 (1988), 579–613.[21]
Illusie, L.
Complexe de de Rham-Witt et cohomologie cristalline.
Ann. Sci. ´Ecole Norm. Sup. (4)12 , 4 (1979), 501–661.[22]
Izhboldin, O. On p -torsion in K M ∗ for fields of characteristic p . In Algebraic K -theory , vol. 4 of Adv. Soviet Math.
Amer. Math. Soc., Providence, RI, 1991, pp. 129–144.[23]
Izhboldin, O. p -primary part of the Milnor K -groups and Galois cohomologies of fields of charac-teristic p . In Invitation to higher local fields (M¨unster, 1999) , vol. 3 of
Geom. Topol. Monogr.
Geom.Topol. Publ., Coventry, 2000, pp. 19–41. With an appendix by Masato Kurihara and Ivan Fesenko.[24]
Kassel, C., and Sletsjøe, A. B.
Base change, transitivity and K¨unneth formulas for the Quillendecomposition of Hochschild homology.
Math. Scand. 70 , 2 (1992), 186–192.[25]
Kelly, S., and Morrow, M. K -theory of valuation rings. Compositio Mathematica, to appear (2018).[26]
Kerz, M.
The Gersten conjecture for Milnor K -theory. Invent. Math. 175 , 1 (2009), 1–33.[27]
Kerz, M.
Milnor K -theory of local rings with finite residue fields. J. Algebraic Geom. 19 , 1 (2010),173–191.[28]
Kerz, M., Esnault, H., and Wittenberg, O.
A restriction isomorphism for cycles of relativedimension zero.
Camb. J. Math. 4 , 2 (2016), 163–196.[29]
Kerz, M., Strunk, F., and Tamme, G.
Algebraic K -theory and descent for blow-ups. Invent.Math. 211 , 2 (2018), 523–577.[30]
Kolster, M. K of noncommutative local rings. J. Algebra 95 , 1 (1985), 173–200.[31]
Kurihara, M.
Abelian extensions of an absolutely unramified local field with general residue field.
Invent. Math. 93 , 2 (1988), 451–480.[32]
L¨uders, M.
Algebraization for zero-cycles and the p -adic cycle class map. Math. Res. Lett. 26 , 2(2019), 557–585.[33]
L¨uders, M.
Deformation theory of the Chow group of zero cycles.
Q. J. Math. 71 , 2 (2020),677–701. 33orten L¨uders and Matthew Morrow[34]
L¨uders, M.
On the relative Gersten conjecture for Milnor K -theory in the smooth case. arXiv:2010.02622 (2020).[35] Morrow, M. K of localisations of local rings. Journal of Algebra 399 (2014), 190–204.[36]
Morrow, M.
Pro unitality and pro excision in algebraic K -theory and cyclic homology. J. ReineAngew. Math. 736 (2018), 95–139.[37]
Morrow, M. K -theory and logarithmic Hodge-Witt sheaves of formal schemes in characteristic p . Ann. Sci. ´Ec. Norm. Sup´er. (4) 52 , 6 (2019), 1537–1601.[38]
Nakamura, J.
On the Milnor K -groups of complete discrete valuation fields. Doc. Math. 5 (2000),151–200.[39]
Nakamura, J.
On the structure of the Milnor K -groups of complete discrete valuation fields.In Invitation to higher local fields (M¨unster, 1999) , vol. 3 of
Geom. Topol. Monogr.
Geom. Topol.Publ., Coventry, 2000, pp. 123–135.[40]
Nesterenko, Y. P., and Suslin, A. A.
Homology of the general linear group over a local ring,and Milnor’s K -theory. Izv. Akad. Nauk SSSR Ser. Mat. 53 , 1 (1989), 121–146.[41]
Nisnevich, Y. A.
The completely decomposed topology on schemes and associated descent spectralsequences in algebraic K -theory. In Algebraic K -theory: connections with geometry and topology(Lake Louise, AB, 1987) , vol. 279 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.
Kluwer Acad.Publ., Dordrecht, 1989, pp. 241–342.[42]
Panin, I. A.
The equicharacteristic case of the Gersten conjecture.
Tr. Mat. Inst. Steklova 241 ,Teor. Chisel, Algebra i Algebr. Geom. (2003), 169–178.[43]
Quillen, D.
Higher algebraic K-theory. I. Algebr. K-Theory I, Proc. Conf. Battelle Inst. 1972,Lect. Notes Math. 341, 85-147 (1973)., 1973.[44]
Sato, K. p -adic ´etale Tate twists and arithmetic duality. Ann. Sci. ´Ecole Norm. Sup. (4) 40 , 4(2007), 519–588. With an appendix by Kei Hagihara.[45]
Schneider, P. p -adic points of motives. In Motives (Seattle, WA, 1991) , vol. 55 of
Proc. Sympos.Pure Math.
Amer. Math. Soc., Providence, RI, 1994, pp. 225–249.[46]
Suslin, A. A., and Yarosh, V. A.
Milnor’s K of a discrete valuation ring. In Algebraic K -theory ,vol. 4 of Adv. Soviet Math.
Amer. Math. Soc., Providence, RI, 1991, pp. 155–170.[47]
Tate, J.
Relations between K and Galois cohomology. Invent. Math. 36 (1976), 257–274.[48]
Totaro, B.
Milnor K -theory is the simplest part of algebraic K -theory. K -Theory 6 , 2 (1992),177–189.[49] van der Kallen, W. The K of rings with many units. Ann. Sci. ´Ecole Norm. Sup. (4) 10 , 4(1977), 473–515.[50] van der Kallen, W., Maazen, H., and Stienstra, J.
A presentation for some K ( n, R ). Bull.Amer. Math. Soc. 81 , 5 (1975), 934–936.[51]
Weibel, C. A.
The K -book , vol. 145 of Graduate Studies in Mathematics . American MathematicalSociety, Providence, RI, 2013. An introduction to algebraic K -theory.Morten L¨udersIMJ-PRG,SU – 4 place Jussieu,Case 247,75252 Paris [email protected] Matthew MorrowCNRS & IMJ-PRG,SU – 4 place Jussieu,Case 247,75252 Paris [email protected]@imj-prg.fr