aa r X i v : . [ m a t h . K T ] O c t K -theory of valuation rings Shane Kelly and Matthew Morrow
Abstract
We prove several results showing that the algebraic K -theory of valuation rings behave as thoughsuch rings were regular Noetherian, in particular an analogue of the Geisser–Levine theorem. Wealso give some new proofs of known results concerning cdh descent of algebraic K -theory. Contents p -adic K -theory of valuation rings in characteristic p K -theory of valuation rings 11 Recall that a valuation ring is an integral domain O with the property that, given any elements f, g ∈ O ,either f ∈ g O or g ∈ f O . Despite rarely being Noetherian, valuation rings surprisingly often behave likeregular Noetherian rings. The theme of this article is to explore the extent to which this is reflected intheir algebraic K -theory. The main results are largely independent of one another.The primary new result is the following calculation of the p -adic K -theory of valuation rings ofcharacteristic p , analogous to Geisser–Levine’s theorem for regular Noetherian local rings [12]: Theorem 1.1.
Let O be a valuation ring containing F p , and n ≥ . Then K n ( O ) is p -torsion-free andthere is a natural isomorphism K n ( O ) /p r ≃ → W r Ω n O , log given by dlog[ · ] on symbols. Here W r Ω n O , log , often denoted by ν nr ( O ), is the subgroup of the de Rham–Witt group W r Ω n O generatedby dlog forms; see Remark 2.3 for further information.Let us sketch the ideas of the proof of Theorem 1.1. By a recent result of the second author joint withClausen–Mathew [7], the trace map tr : K n ( − ; Z /p r Z ) → TC n ( − ; Z /p r Z ) is naturally split injective onthe category of local F p -algebras. The proof of this injectivity reduces Theorem 1.1 to showing that thetopological Hochschild and cyclic homologies of O behave as though it were a regular Noetherian local F p -algebra, which in turn comes down to controlling the derived de Rham and derived de Rham–Wittcohomologies of O . This may be done by means of deep results of Gabber–Ramero and Gabber on thecotangent complex and de Rham cohomology of valuation rings.The second main result of the article is that the injectivity part of Gersten’s conjecture remains truefor valuation rings over a field; away from the characteristic this follows from the existence of alterations,while the torsion-freeness in Theorem 1.1 provides the missing ingredient at the characteristic:1hane Kelly and Matthew Morrow Theorem 1.2.
Let O be a valuation ring containing a field. Then K n ( O ) → K n (Frac O ) is injective forall n ≥ . Presheaves which satisfy this injectivity property are studied in [19] where they are called “torsionfree”.Another important feature of the K -theory of regular Noetherian rings is homotopy invariance. Weobserve that this extends to valuation rings: Theorem 1.3.
Let O be a valuation ring. Then(i) K n ( O ) → K n ( O [ T , . . . , T d ]) is an isomorphism for all d ≥ and n ∈ Z ;(ii) K n ( O ) → KH n ( O ) is an isomorphism for all n ∈ Z ;(iii) K n ( O ) = 0 for all n < . In fact, the “proof” of Theorem 1.3 is simply the fact, observed by Gersten and Weibel in the1980s, that stable coherence and finite global dimension are enough for the usual proof (from the regularNoetherian case) of the assertions to go through. Since these homological properties of valuation ringshave been known since the 1970s, Theorem 1.3 has in principle been available for many years. But weare not aware of it having been previously noticed. We refer the interested reader to Remark 3.4 forhistorical comments and relations to properties of algebraic K -theory in the cdh topology.Remaining on the theme of cdh topology, we also give a new proof that K [ p ] satisfies cdh descent incharacteristic p in the following sense: Theorem 1.4.
Let Y ′ (cid:15) (cid:15) / / X ′ (cid:15) (cid:15) Y / / X be an abstract blow-up square of schemes in which p is nilpotent; assume that X is quasi-compact quasi-separated and that the morphisms X ′ → X and Y ֒ → X are of finite presentation. Then the resultingsquare K ( X )[ p ] (cid:15) (cid:15) / / K ( X ′ )[ p ] (cid:15) (cid:15) K ( Y )[ p ] / / K ( Y ′ )[ p ] is homotopy cartesian. Theorem 1.4 is only conceivably new in the case that X is not Noetherian (and one can probablyeven reduce to that case via Noetherian approximation), but the method of proof is completely differentto what has appeared previously: we replace all the schemes by their perfections (having reduced to thecase of F p -schemes) and then use results of Bhatt–Scholze [5] on the homological properties of perfectschemes to check that Tamme’s recent excision condition [37] is satisfied. See Remark 4.4 for somecomparisons to related results. Acknowledgements
We thank Georg Tamme for correspondence about Theorem 1.4 (in particular, he was already awarethat the results of Bhatt–Scholze could be used to apply [37] to blow-ups of perfect schemes) and BenAntieau for his comments. 2 -theory of valuation rings p -adic K -theory of valuation rings in characteristic p The goal of this section is to prove Theorem 1.1 from the introduction. More generally, we work with F p -algebras A satisfying the following smoothness criteria (Sm1–3), which we paraphrase by saying that A is Cartier smooth :(Sm1) Ω A is a flat A -module;(Sm2) H n ( L A/ F p ) = 0 for all n > C − : Ω nA → H n (Ω • A ) is an isomorphism for each n ≥ A is a smooth F p -algebra, or more generally if A is a regularNoetherian F p -algebra by N´eron–Popescu desingularisation (an alternative proof avoiding N´eron–Popescumay be found in the recent work of Bhatt–Lurie–Mathew [2, Thm. 9.5.1]), or even an ind-smooth F p -algebra. However, in the case in which A is ind-smooth, the forthcoming results trivially reduce to thesmooth case, where they are known.More interestingly, any valuation ring O of characteristic p is Cartier smooth: the first two criteriaare due to Gabber–Ramero [10, Thm. 6.5.8(ii) & Corol. 6.5.21], while the Cartier isomorphism (Sm3) isa recent unpublished result of Gabber obtained by refining his earlier work with Ramero. (Of course,resolution of singularities in characteristic p would imply that O is ind-smooth, but we wish to avoidassuming resolution of singularities.) Therefore Theorem 1.1 is a special case of the following moregeneral result: Theorem 2.1.
Let A be any Cartier smooth, local F p -algebra, and n ≥ . Then K n ( A ) is p -torsion-freeand there is a natural isomorphism K n ( A ) /p r ≃ → W r Ω nA, log given by dlog[ · ] on symbols. Remark 2.2.
The extra degree of generality afforded by the axiomatic set-up of Theorem 2.1 is notwithout interest. It also applies, for example, to essentially smooth, local S -algebras, where S is anyperfect ring of characteristic p . Remark 2.3.
We recall the definition of the group of dlog forms. Given a local F p -algebra A , we define W r Ω nA, log ⊆ W r Ω nA to be the subgroup generated by dlog forms d [ α ][ α ] ∧ · · ·∧ d [ α n ][ α n ] , where α , . . . , α n ∈ A × ;i.e., the image of dlog[ · ] : K Mn ( A ) → W r Ω nA . If A is not necessarily local, then W r Ω nA, log ⊆ W r Ω nA isdefined to be the subgroup generated Zariski locally by such forms; the fact that this coincides withthe more classical definition as the subgroup generated ´etale locally by dlog forms was proved in [29,Corol. 4.2(i)].In particular, under the hypotheses of Theorem 2.1, the canonical map K Mn ( A ) /p r → K n ( A ) /p r issurjective. We do not know whether it is an isomorphism (even under the usual assumption that A haslarge enough residue field), as we do not know of any way to control Milnor K -theory using derived deRham cohomology. We study in this subsection the behaviour of certain functors, such as de Rham–Witt groups, on Cartiersmooth F p -algebras. In essence, we must show that all familiar results about the de Rham–Witt complexextend from smooth algebras to Cartier smooth algebras, thereby justifying the well-known maxim thatit is the Cartier isomorphism which controls the structure of the de Rham–Witt complex.Given a functor F : F p -algs → Ab commuting with filtered colimits, let L F : F p -algs → D ≥ (Ab) bethe left Kan extension of its restriction to finitely generated free F p -algebras. More classically, followingQuillen’s theory of non-abelian derived functors, L F ( A ) is given by the simplicial abelian group q F ( P q ) (or more precisely the associated chain complex via Dold–Kan), where P • → A is any simplicialresolution of A by free F p -algebras.Let A ∈ F p -algs and F a functor as above. In general, the resulting canonial map H ( L F ( A )) → F ( A )need not be an isomorphism; when this is true we will say that F is right exact at A . When H i ( F ( A )) = 0 For any F p -algebra A , recall that the inverse Cartier map C − : Ω nA → H n (Ω • A ) is the linear map satisfying C − ( fdg ∧· · · ∧ dg n ) = f p g p − · · · g p − n dg ∧ · · · ∧ dg n . i >
0, we will say that F is supported in degree at A . Thus F is both right exact and supportedin degree 0 at a given A if and only if the canonical map L F ( A ) → F ( A )[0] is an equivalence, in whichcase we say that F is already derived at A .The following lemma is straightforward but will be used repeatedly: Lemma 2.4.
Let
F → G → H be a maps of functors F p -algs → Ab which commute with filtered colimits,such that → F ( R ) → G ( R ) → H ( R ) → is exact for any finitely generated free F p -algebra R . Fix A ∈ F p -algs .(i) Suppose that two of F , G , H are already derived at A . Then so is the third if and only if the sequence → F ( A ) → G ( A ) → H ( A ) → is exact.(ii) Assume that F and G are already derived at A , and that F ( A ) → G ( A ) is injective. Then H issupported in degree at A .(iii) Assume that F and H are supported in degree at A . Then G is supported in degree at G .Proof. The hypotheses provide us with a fibre sequence L F ( A ) → L G ( A ) → L H ( A ) mapping to F ( A ) →G ( A ) → H ( A ). The assertions are then all easy to check. Proposition 2.5.
The following functors are already derived at any Cartier smooth F p -algebra: Ω n − Z Ω n − B Ω n − W r Ω n − . Proof.
The case of Ω n − is classical: since the derived exterior power of an A -module may be computedin terms of any flat resolution, the functor Ω n − is already derived at any F p -algebra satsifying (Sm1–2).We now consider the complexes of functors(i) 0 −→ Ω n − C − −−−→ H n (Ω •− ) −→ −→ −→ Z Ω n − −→ Ω n − −→ Ω n − /Z Ω n − −→ −→ Ω n − /Z Ω n − d −→ Z Ω n +1 − −→ H n +1 (Ω •− ) −→ C − is an isomorphism) on any F p -algebra satisfying (Sm3), while thesecond two complexes are obviously exact on any F p -algebra. We now use Lemma 2.4(i) repeatedly.First it follows that H n (Ω •− ) is already derived at any Cartier smooth F p -algebra, whence the same istrue of Z Ω − = H (Ω •− ). By induction, assuming that Z Ω n − is already derived at any Cartier smooth F p -algebra, the second sequence shows that the same is true of Ω n − /Z Ω n − , and then the third sequenceshows that the same is true of Z Ω n +1 − . We have thus shown that Z Ω n − is already derived at any Cartiersmooth F p -algebra, for all n ≥ B Ω n − is also already derived at any Cartier smooth F p -algebra, for all n ≥ F p -algebra A , 0 =: B Ω nA ⊆ B Ω nA ⊆ · · · ⊆ Z Ω nA ⊆ Z Ω nA := Ω nA , inductively as follows, for r ≥ Z r Ω nA is the subgroup of Ω nA containing B Ω nA and satisfying Z r Ω nA /B Ω nA = C − ( Z r − Ω nA ).- B r Ω nA is the subgroup of Ω nA containing B Ω nA and satisfying B r Ω nA /B Ω nA = C − ( B r − Ω nA ).(Minor notational warning: B Ω nA = B Ω nA , but the inclusion Z Ω nA ⊆ Z Ω nA is strict if the inverse Cartiermap is not surjective.) Iterating the inverse Cartier map defines maps C − r : Ω nA −→ Z r Ω nA /B r Ω nA , which are (for purely formal reasons) isomorphisms for all r, n ≥ A satisfies (Sm3). By usingLemma 2.4(i) in the same way as above, a simple induction on r then shows that the functors Z r Ω n − and B r Ω n − are already derived at any Cartier smooth F p -algebra.4 -theory of valuation ringsWe now begin to analyse the de Rham–Witt groups, which are more subtle. We will use the followingcollection of functors F p -algs → Ab and maps between then:00 / / Z r − Ω n − − / / Ω n − − dV r − / / W r Ω n − /V r − Ω n − O O R / / W r − Ω n − / / W r Ω n − O O Ω n − V r − O O B r − Ω n − O O O O We break this diagram into sequences (not even necessarily complexes; for example, it is not clear if Z r − Ω n − A is killed by dV r − for arbitrary A ) of functors as follows:(i) 0 → Z r − Ω n − − → Ω n − − → Ker R → → Ker R → W r Ω n − /V r − Ω n − R −→ W r − Ω n − → → B r − Ω n − → Ω n − V r − −−−→ V r − Ω n − → → V r − Ω n − → W r Ω n − → W r Ω n − /V r − Ω n − → F p -algebra, by Illusie [20, (3.8.1) & (3.8.2)].We now claim inductively on r ≥ W r Ω nA is supported in degree 0 on any Cartier smooth F p -algebra. Note that we have already established stronger statements for Ω n − , Z r Ω n − , B r Ω nr for all r, n ≥
0. So Lemma 2.4(ii) and sequence (i) shows that Ker R is supported in degree 0 on any Cartiersmooth F p -algebra; the same argument using sequence (iii) shows that the same is true of V r − Ω n − . ThenLemma 2.4(iii) and sequence (ii) (and the inductive hypothesis) show that W r Ω n − /V r − Ω n − is supportedin degree 0 on any Cartier smooth F p -algebra; the same argument using sequence (iv) shows that thesame is true of W r Ω − , completing the inductive proof of the claim.To finish the proof we will show that W r Ω n − is right exact, i.e., H ( L W r Ω nA ) ≃ → W r Ω nA for all F p -algebras A . We may and do suppose that A is of finite type over F p . The key is to observe that thedata M n ≥ H ( L W r Ω nA ) r ≥ A . This follows from functoriality of left Kan extension and the fact that n = 0 part of the above sum is simply W r ( A ) (indeed, Lemma 2.4(i) and the short exact sequences0 → W r − → W r → W → W r are already derived at A ).Therefore universality of the de Rham–Witt complex yields natural maps λ A : W r Ω nA → H ( L W r Ω nA )such that the composition back to W r Ω nA is the identity. Picking a finitely generated free F p -algebra P and surjection P ։ A , the naturality provides us with a commutative diagram W r Ω nP (cid:15) (cid:15) λ P / / H ( L W r Ω nP ) (cid:15) (cid:15) ∼ = / / W r Ω nP (cid:15) (cid:15) W r Ω nA λ A / / H ( L W r Ω nA ) / / W r Ω nA P → A is surjective; the middle vertical arrow issurjective because H ( L W r Ω nA ) is a quotient of W r Ω nP (by definition of H , since we may as well supposethat P is the 0-simplices of a simplicial resolution of A ). The indicated isomorphism holds because P is free. Since both horizontal compositions are the identity, it follows from a diagram chase that all thehorizontal maps are isomorphisms. This completes the proof that W r Ω n − is right exact.We next study the usual filtrations on the Rham–Witt complex. Recall that if A is any F p -algebra,then the descending canonical , V - , and p - filtrations on W r Ω nA are defined for i ≥ i W r Ω nA := Ker( W r Ω nA R r − i −−−→ W i Ω nA ) , Fil iV W r Ω nA := V i W r − i Ω nA + dV i W r − i Ω n − A , Fil ip W r Ω nA := Ker( W r Ω nA p r − i −−−→ W r Ω nA ) . Standard de Rham–Witt identities show thatFil i W r Ω nA ⊇ Fil iV W r Ω nA ⊆ Fil ip W r Ω nA . It was proved by Illusie [20, Prop. I.3.2 & I.3.4] that these three filtrations coincide if A is a smooth F p -algebra. It was then shown by Hesselholt that the canonical and V -filtrations in fact coincide for any F p -algebra (the key observation is that for any fixed i ≥
1, the groups W r + i Ω nA / Fil iV W r + i Ω nA , for n ≥ r ≥
1, have an induced structure of a Witt complex over A ; a detailed proof in the generality of logstructures may be found in [18, Lem. 3.2.4]). For a general F p -algebra, the p - and canonical filtrationsneed not coincide, but we show that this holds for Cartier smooth algebras: Proposition 2.6.
Let A be a Cartier smooth F p -algebra, and i, n ≥ , r ≥ .(i) the functors Fil i W r Ω n − , Fil iV W r Ω n − , Fil ip W r Ω n − , p i W r Ω n − , W r Ω n − /p i W r Ω n − are already derivedat A ;(ii) Fil i W r Ω nA = Fil iV W r Ω nA = Fil ip W r Ω nA ;Proof. We begin by reducing all assertions to the inclusionFil i W r Ω nA ⊇ Fil ip W r Ω nA , ( † )so assume for a moment that this has been established.Note first (even without ( † )) from Proposition 2.5 and Lemma 2.4(i) that Fil i W r Ω n − = Ker( R r − i : W r Ω n − ։ W i Ω n − ) is already derived at A . Moreover, as explained before the proposition, one hasFil i W r Ω n − = Fil iV W r Ω n − ⊆ Fil ip W r Ω n − , with equality in the case of smooth F p -algebras, so left Kanextension and ( † ) give part (ii) and imply that Fil i W r Ω n − and Fil ip W r Ω n − are already derived at A . Thenapplying Lemma 2.4(i) to the functors Fil r − ip W r Ω n − → W r Ω n − p i −→ p i W r Ω n − (the first two terms of whichhave already been shown to be already derived at A ) shows that p i W r Ω n − is already derived at A , andthen applying the same argument to the sequence p i W r Ω n − → W r Ω n − → W r Ω n − /p i W r Ω n − shows that W r Ω n − /p i W r Ω n − is also already derived at A .It remains to prove ( † ). For any smooth F p -algebra R there is a natural map of short exact sequences0 / / Ω nR /B r − Ω nR V r − / / C − (cid:15) (cid:15) Fil r − W r Ω nR / / p (cid:15) (cid:15) β / / Ω n − R /Z r − Ω n − R / / C − (cid:15) (cid:15) / / Ω nR /B r Ω nR V r / / Fil r W r +1 Ω nR β / / Ω n − R /Z r Ω n − R / / p is characterised by the property that the compositionFil r − W r +1 Ω nA R −→ Fil r − W r Ω nA p −→ Fil r W r +1 Ω nA ⊆ Fil r − W r +1 Ω nA -theory of valuation ringsis multiplication by p . (The definition of β is unimportant but given in the top row by β ( V r − x + dV r − y ) = y , for x ∈ Ω nA and y ∈ Ω n − A ; for β in the bottom row replace r − r .)Upon left Kan extending and evaluating at A , we obtain the same map of short exact sequences butwith R replaced by A : here we use that Ω n − , B r Ω n − , Z r Ω n − , and Fil i W r Ω n − are already known to bederived at A . But the inverse Cartier maps C − : Ω nA /B r − Ω nA −→ Ω nA /B r Ω nA C − : Ω n − A /Z r − Ω n − A −→ Ω n − A /Z r Ω n − A are injective: this follows formally from the injectivity of C − : Ω nA → Ω nA /B Ω nA (similarly for n −
1) andthe surjectivity (by definition of the higher coboundaries and cocycles) of C − : B r − Ω nA → B r Ω nA /B Ω nA and C − : Z r − Ω n − A → Z r Ω n − A /B Ω n − A . Therefore the middle vertical arrow p : Fil r − W r Ω nA → Fil r W r +1 Ω nA is also injective and so, from the characterisation of p , we deduce thatKer(Fil r − W r +1 Ω nA p −→ Fil r − W r +1 Ω nA ) ⊆ Ker(Fil r − W r +1 Ω nA R −→ Fil r − W r Ω nA ) (= Fil r W r +1 Ω nA ) . It now follows easily by induction that Fil rp W r +1 Ω nA ⊆ Fil r W r +1 Ω nA . Indeed, this is vacuous if r = 0,while if ω ∈ Fil rp W r +1 Ω nA then Rω ∈ Fil r − p W r Ω nA ⊆ Fil r − W r Ω nA (the inclusion holds by the inductivehypothesis) and so ω belongs to Fil r − W r +1 Ω nA and is killed by p , whence the previous equation showsthat ω ∈ Fil r W r +1 Ω nA , as desired.Replacing r by r − r − ip W r Ω nA ⊆ Fil r − i W r Ω nA for i = 1.This easily extends to all i ≥ ω ∈ Fil r − ip W r Ω nA then p i − ∈ Fil r − p W r Ω nA , whence R ( p i − ω ) = 0 by the case i = 0; in other words, Rω ∈ Fil r − − ( i − p W r − Ω nA and so the inductive hypothesis implies Rω ∈ Fil r − − ( i − W r − Ω nA , i.e., R i ω = 0, as required. Corollary 2.7.
Let A be a Cartier smooth F p -algebra. The canonical maps [0 → W r Ω A /p d −→ · · · d −→ W r Ω n − A /p d −→ W r Ω nA /V W r − Ω nA d −→ −→ Ω ≤ nA and W r Ω • A /p → Ω • A are quasi-isomorphisms.Proof. The sequence of functors W r Ω n − V −→ W r +1 Ω n − F r −−→ Z r Ω n − −→ F p -algebra [20, (3.11.3)]. Since the three functors are alreadyderived at A by Proposition 2.5, left Kan extension shows that the sequence is also exact when evaluatedat A . (Here we implicitly use right exactness of left Kan extension: if F → G is a map of functors F p -algs → Ab such that F ( R ) → G ( R ) is surjective for all free R , then H ( L F ( B )) → H ( L G ( B )) issurjective for all B ∈ F p -algs.)It follows that the functor W r Ω n − /V W r − Ω n − is already derived at A . Since the functors W r Ω i − /p and Ω i − are also already derived at A (by Proposition 2.6(i) and Proposition 2.5 respectively), the quasi-isomorphisms follow by left Kan extension from the case of finitely generated free F p -algebras, namely[20, Corol. 3.20]. Theorem 2.8.
Let A be a Cartier smooth F p -algebra. Then(i) W Ω nA := lim ←− r W r Ω nA is p -torsion-free.(ii) F i W Ω nA = d − ( p i W Ω nA ) for all i ≥ .Proof. Part (i) follows formally from the equality of the canonical and p -filtrations in Proposition 2.6(ii).For (ii), let us first fix a finite level r and show that an element x ∈ W r Ω nA is in the image of F : W r +1 Ω nA → W r Ω nA if and only if dx ∈ pW r Ω n +1 A . The implication “only if” following from theidentity dF = pF d , so we prove “if” by induction on r ; assume the result known at level r − dx ∈ pW r Ω n +1 A . Then dR r − e x = 0, whence e x ∈ F W Ω nA (by the r = 1 case of the right exactsequence from the start of proof of Corollary 2.7; this also proves the base case of the induction). Liftingto level r and recalling that Fil W r Ω nA = Fil V W Ω nA allows us to write x = F z + V z + dV z for some z ∈ W r +1 Ω nA , z ∈ W r − Ω nA , z ∈ W r − Ω n − A .Applying F d to the previous identity yields dz = F dV z = F dx − pF dz ∈ pW r − Ω n +1 A , whence theinductive hypothesis allows us to write z = F z for some z ∈ W r Ω nA . In conclusion, x = F z + V F z + dV z = F z + F V z + F dV z ∈ F W r +1 Ω nA , as required to complete the induction.We now prove (ii); it suffices to treat the case i = 1 as the general case then follows by inductionusing part (i). The identity V F = p shows that the kernel of F : W r +1 Ω nA → W r Ω nA is killed by p , henceis killed by R by Proposition 2.6(ii). So, using also the finite level case of the previous paragraphs, thesequence of pro abelian groups0 −→ { W r Ω nA } r F −→ { W r Ω nA } r d −→ { W r Ω n +1 A /pW r Ω n +1 A } r is exact, and taking the limit yields the short exact sequence0 −→ W Ω nA F −→ W Ω nA d −→ lim ←− r W r Ω n +1 A /pW r Ω n +1 A . ( † )But the sequence of pro abelian groups0 −→ { W r Ω n +1 A } r p −→ { W r Ω n +1 A } r −→ { W r Ω n +1 A /pW r Ω n +1 A } r is also exact (again since p -torsion is killed by R ), and so taking the limit shows that the final term in( † ) is W Ω n +1 A /pW Ω n +1 A , as required to complete the proof. Remark 2.9.
In terms of the functor η p [3, § ϕ : W Ω • A → η p W Ω • A is an isomorphism and hence provides a quasi-isomorphism ϕ : W Ω • A ∼ → Lη p W Ω • A . Recall from Remark 2.3 that, for any F p -algebra A , we have defined the subgroup of dlog forms W r Ω nA, log ⊆ W r Ω nA . Alternatively, standard de Rham–Witt identifies imply the existence of a uniquemap F : W r Ω nA → W r Ω nA /dV r − Ω n − A making the following diagram commute (in which π denotes thecanonical quotient map) W r +1 Ω nAR (cid:15) (cid:15) F / / W r Ω nAπ (cid:15) (cid:15) W r Ω nA F / / W r Ω nA /dV r − Ω n − A and it may then be shown that W r Ω nA, log = Ker( W r Ω nA F − π −−−→ W r Ω nA /dV r − Ω n − A ) . See [29, Corol. 4.2] for further details.The following argument is modelled on the proof in special case when r = 1 and A is smooth [7,Prop. 2.26]: Lemma 2.10.
Let A be a Cartier smooth F p -algebra. Then the square W Ω nA /p r F − / / (cid:15) (cid:15) W Ω nA /p r (cid:15) (cid:15) W r Ω nA F − π / / W r Ω nA /dV r − Ω nA is bicartesian. -theory of valuation rings Proof.
Since the vertical arrows are surjective, we must show that their kernels are isomorphic, i.e, that F − V r W Ω nA + dV r W Ω nA → W Ω nA /p r ) ≃ −→ Im( V r W Ω nA + dV r − Ω n → W Ω nA /p r ) . Surjectivity follows from surjectivity modulo p , which is a consequence of the identities in W Ω nA ( F − V ≡ − V mod p, ( F − d X i> V i = d. It remains to establish injectivity, so suppose that x ∈ W Ω nA is an element of the form x = V r y + dV r z such that ( F − x ∈ p r W Ω nA . Applying 1 + · · · + F r − we deduce that ( F r − x ∈ p r W Ω nA , whence dV r y = dx = p r F r dx − d ( F r − x ∈ p r W Ω nA and so dy = F r dV r y ∈ p r W Ω nA ; therefore y = F r y ′ forsome y ′ ∈ W Ω nA by Theorem 2.8(ii).We know now that x = p r y ′ + dV r z ; applying F r − F r − dV r z = d (1 − V r ) z isdivisible by p r , whence (1 − V r ) z ∈ F r W Ω nA by Theorem 2.8(ii) again. But 1 − V r is an automorphismof W Ω nA with inverse P i ≥ V ri (which commutes with F ), whence z = F r z ′ for some z ′ ∈ W Ω nA . Inconclusion x = p r y ′ + p r dz ′ , which completes the proof of injectivity.In the case of smooth F p -algebras, the following is an important result due to Illusie [20]: Theorem 2.11.
Let A be a Cartier smooth, local F p -algebra. Then the canonical map W s Ω nA, log /p r → W r Ω nA, log is an isomorphism for any s ≥ r ≥ .Proof. The sequence 0 −→ W Ω nA /p s − r p r −→ W Ω nA /p s −→ W Ω nA /p r −→ W Ω nA is p -torsion-free by Theorem 2.8. Taking F -fixed points and appealing toLemma 2.10 gives an exact sequence0 −→ W s − r Ω nA, log p r −→ W s Ω nA, log → W r Ω nA, log . But W s Ω nA, log → W r Ω nA, log is surjective since both sides are generated by dlog forms, and the analogoussurjectivity for r replaced by r − s shows that the image of the p r arrow in the previous line is p r W s Ω nA, log . This subsection is devoted to the proof of Theorem 2.1. Familiarity with topological cyclic homology isexpected (we use the “old” approach to topological cyclic homology in terms of the fixed points spectra TR r ( − ; p ) = THH ( − ) C pr − , though the argument could be reformulated in terms of Nikolaus–Scholze’sapproach [31] and the motivic filtrations of [4].)Given any smooth F p -algebra R , the homotopy groups of TR r ( R ; p ) have been calculated by Hesselholt[17, Thm. B]: TR rn ( R ; p ) ∼ = M i ≥ W r Ω n − iR . These isomorphisms are natural in R and compatible with r in the sense that the restriction map R r : TR r ( R ; p ) → TR r ( R ; p ) induces zero on W r Ω n − iR , for i >
0, and the usual de Rham–Wittrestriction map R r : W r Ω nR → W r Ω nR , for i = 0.Therefore left Kan extending shows that on any F p -algebra A there is a natural descending N -indexed filtration on TR r ( A ; p ) with graded pieces L i ≥ L W r Ω n − iA ; moreover, the restriction map R r : TR r ( A ; p ) → TR r ( A ; p ) is compatible with this filtration and behaves analogously to the previous smoothcase. Note that TR r ( − ; p ) : F p -algs → Sp is left Kan extended from finitely generated free F p -algebras. Indeed, this is truefor THH since tensor products and geometric realisations commute with sifted colimits, and then it is true by inductionfor TR r ( − ; p ) thanks to the isotropy separation sequences and the fact that homotopy orbits commute with all colimits.See also [4, Corol. 6.8] for the construction of this filtration in the case of THH . A be any Cartier smooth F p -algebra. Then L W r Ω n − iA ≃ W r Ω n − iA [0] by Proposition 2.5and so the previous paragraph implies that the description of TR rn ( − ; p ) from the smooth case remainsvalid for A . In particular we obtain natural isomorphisms of pro abelian groups { TR rn ( A ; p ) } r wrt R ∼ = { W r Ω nA } r wrt R , whence taking the limit calculates the homotopy groups of TR ( A ; p ) := holim r TR r ( A ; p )as TR n ( A ; p ) ∼ = W Ω nA . Writing TR ( A ; Z /p r Z ) = TR ( A ; p ) /p r , we apply Theorem 2.8(i) to obtain TR ( A ; Z /p r Z ) ∼ = W Ω nA /p r . The previous two isomorphisms are compatible with the Frobenius endormorphisms F of TR ( R ; Z p )and W Ω A (by left Kan extending the Frobenius compatibility of [17, Thm. B] from the smooth case),thereby proving that the homotopy groups of TC ( A ; Z /p r ) := hofib( F − TR ( A ; Z /p r Z ) → TR ( A ; Z /p r Z ))fit into a long exact homotopy sequence · · · −→ TC n ( A ; Z /p r Z ) −→ W Ω nA /p r F − −−−→ W Ω nA /p r −→ · · · In this long exact sequence we may replace each map F − F − π : W r Ω nA → W r Ω nA /dV r − Ω nA , byLemma 2.10, and so obtain a short exact sequence0 −→ Coker( W r Ω n +1 A F − π −−−→ W r Ω n +1 A /dV r − Ω nA ) −→ TC n ( A ; Z /p r Z ) −→ W r Ω nA, log −→ W r Ω nA, log = Ker( F − π ) from the start of § F p -algebra surjecting onto A and let R denote its Henselisation along the kernel; then R is a local ind-smooth F p -algebra and so the previous short exact sequence is also valid for R (since R is also Cartier smooth). Putting together the short exact sequences for A and R yields a commutativediagram with exact rows, in which we have included the trace maps from algebraic K -theory: K n ( R ; Z /p r Z ) tr , , (cid:15) (cid:15) / / Coker( W r Ω n +1 R F − π −−−→ W r Ω n +1 R /dV r − Ω nR ) (cid:15) (cid:15) / / TC n ( R ; Z /p r Z ) / / (cid:15) (cid:15) W r Ω nR, log (cid:15) (cid:15) / / K n ( A ; Z /p r Z ) tr , , / / Coker( W r Ω n +1 A F − π −−−→ W r Ω n +1 A /dV r − Ω nA ) / / TC n ( A ; Z /p r Z ) / / W r Ω nA, log / / c A : K n ( A ; Z /p r Z ) tr −→ TC n ( A ; Z /p r Z ) −→ W r Ω nA, log is an isomorphism. Since R is ind-smooth and local, the analogous composition c R for R in place of A is an isomorphism by Geisser–Levine [12] and Geisser–Hesselholt [11]; see [7, Thm. 2.48] for a reminderof the argument in the case r = 1.According to the main rigidity result of [7], the square of spectra formed by applying tr : K ( − ; Z /p r Z ) → TC ( − ; Z /p r Z ) to R → A is homotopy cartesian, yielding a long exact sequence of homotopy groups · · · −→ K n ( R ; Z /p r Z ) → K n ( A ; Z /p r Z ) ⊕ TC n ( R ; Z /p r Z ) −→ TC n ( A ; Z /p r Z ) −→ · · · . But the trace map tr : K n ( R ; Z /p r Z ) → TC n ( R ; Z /p r Z ) is injective (as c R is injective), whence this breaksinto short exact sequences and proves that the trace map square in the above diagram is bicartesian.Next, the vertical arrow between the cokernel terms is an isomorphism since R → A is Henselianalong its kernel: see the proof of [7, Prop. 6.12]. Therefore the right-most square in the diagram isbicartesian. Concatanating the two bicartesian squares shows that K n ( R ; Z /p r Z ) c R / / (cid:15) (cid:15) W r Ω nR, log (cid:15) (cid:15) K n ( A ; Z /p r Z ) c A / / W r Ω nA, log -theory of valuation ringsis bicartesian. But we have already explained that c R is an isomorphism, whence c A is also an isomor-phism. Finally, note that c A sends any symbol in K n ( A ; Z /p r Z ) to the associated dlog form in W r Ω nA, log ,by the trace map’s multiplicativity [11, §
6] and behaviour on units [11, Lem. 4.2.3]. The completes theproof of Theorem 2.1. K -theory of valuation rings Here we prove Theorem 1.2 from the introduction:
Theorem 3.1.
Let O be a valuation ring containing a field, and n ≥ . Then K n ( O ) → K n (Frac O ) isinjective.Proof. Let p ≥ O . It is enough to show that K n ( O )[ p ] → K n (Frac O )[ p ] is injective: indeed, if p = 1 (i.e., char Frac O = 0) then this is exactly the desiredassertion, while if p > p ] since K n ( O ) has been shown to be p -torsion-free(Theorem 1.1).By writing F = Frac O as the union of its finitely generated subfields, and intersecting O with each ofthese subfields, we may immediately reduce to the case that F is finitely generated over its prime subfield F . Fix a prime number ℓ = p and α ∈ Ker( K n ( O ) → K n ( F )). Pick a finitely generated F -subalgebra A ⊆ O such that Frac A = F and such that α comes from a class α A ∈ Ker( K n ( A ) → K n ( F )). Accordingto Gabber’s refinement of de Jong’s theory of alterations [21, Thm. 2.1 of Exp. X], there exist a regular,connected scheme X and a projective, generically finite morphism X → Spec A such that | K ( X ) : F | isnot divisible by ℓ .Let e O be the integral closure of O in K ( X ). The theory of valuation rings states that e O has onlyfinitely many maximal ideal m , . . . , m d and that each localisation e O m i is a valuation ring (indeed, theselocalisations classify the finitely many extensions of the valuation ring O to K ( X ) [10, 6.2.2]). By thevaluative criterion for properness, each commutative diagramSpec K ( X ) (cid:15) (cid:15) / / X (cid:15) (cid:15) Spec e O m i / / ∃ ρ i ✐✐✐✐✐✐✐✐✐✐✐ Spec e O / / Spec A may be filled in by a dashed arrow as indicated. Let x i ∈ X be the image of the maximal ideal of e O m i under ρ i .Since X is quasi-projective over F , the homogeneous prime avoidance lemma implies the existenceof an affine open of X containing the finitely many points x , . . . , x d . This is given by a regular sub- A -algebra B ⊆ K ( X ) contained inside T i e O m i = e O and such that Frac B = K ( X ). After replacing B byits semi-localisation at the radical ideal T i m i ∩ B , we may suppose in addition that B is semi-local. Thesituation may be summarised by the following diagram:Spec K ( X ) (cid:15) (cid:15) / / Spec B / / X (cid:15) (cid:15) Spec e O m i / / Spec e O / / O O Spec A The element α A vanishes in K n ( K ( X )), hence also in K n ( B ) since Gersten’s conjecture states that K n ( B ) → K n ( K ( X )) is injective [34, Thm. 7.5.11] (note that B is a regular, semi-local ring, essentiallyof finite type over a field). Therefore α vanishes in K n ( e O ). Since e O is an integral extension of O , itis the filtered union of its O -algebras which are finite O -modules; let O ′ ⊆ e O be such a sub- O -algebrasuch that α vanishes in K n ( O ′ ). Since finite torsion-free O -modules are finite free, there is a tracemap K n ( O ′ ) → K n ( O ) such that the composition K n ( O ) → K n ( O ′ ) → K n ( O ) is multiplication by11hane Kelly and Matthew Morrow m := | Frac O ′ : F | , which is not divisible by ℓ . In conclusion, we have shown that mα = 0 for someinteger m not divisible by ℓ .In other words, K n ( O ) ( ℓ ) → K n ( F ) ( ℓ ) is injective for all primes ℓ = p , which impies the injectivity of K n ( O )[ p ] → K n ( F )[ p ] and so completes the proof. Remark 3.2.
The following streamlined proof of Theorem 3.1 has been pointed out to us by Antieau.In case O contains Q , then resolution of singularities implies that O is a filtered colimit of essentiallysmooth local Q -algebras A , whence the desired injectivity immediately reduces to the Gersten conjecturefor each A . When O contains F p , we combine the p -torsion-freeness of Theorem 1.1 with the isomorphismof Remark 4.2 to reduce to the case that O is perfect; but then O is a filtered colimit of essentiallysmooth local F p -algebras by Temkin’s inseparable local uniformisation theorem [38, Thm. 1.3.2], whichas in characteristic 0 reduces the problem to the usual Gersten conjecture. Now we prove Theorem 1.3 from the introduction, entirely using classical methods; this is independentof Section 3.1. We remark that parts of the following theorem have also recently been noticed by Antieauand Mathew.
Theorem 3.3.
Let O be a valuation ring. Then(i) K n ( O ) → K n ( O [ T , . . . , T d ]) is an isomorphism for all d ≥ and n ∈ Z ;(ii) K n ( O ) → KH n ( O ) is an isomorphism for all n ∈ Z ;(iii) K n ( O ) = 0 for all n < .Proof. Since assertions (i)–(iii) are compatible with filtered colimits we may assume that Frac O is finitelygenerated over its prime subfield, and in particular that O is countable.The usual proofs of the assertions (i)—(iii), in the case of a regular Noetherian ring, continue to workfor O once it is known, for all d ≥
0, that(a) O [ T , . . . , T d ] is coherent, and(b) every finitely presented O [ T , . . . , T d ]-module has finite projective dimension.Indeed, see [39, Eg. 1.4] and [13] for a discussion of Quillen’s fundamental theorem and applications inthis degree of generality. See also [1, Thm. 3.33] for a proof of the vanishing of negative K -groups ofregular, stably coherent rings.It remains to verify that O has properties (a) and (b). For (a), the coherence of polynomial algebrasover valuation rings is relatively well-known: any finitely generated ideal I of O [ T , . . . , T d ] is a flat O -module (since torsion-free modules over a valuation ring are flat), hence is of finite presentation over O [ T , . . . , T d ] by Raynaud–Gruson [35, Thm. 3.6.4]. A textbook reference is [14, Thm. 7.3.3].Next we check (b). A classical result of Osofsky [32, Thm. A] [33] states that a valuation ring O hasfinite global dimension n + 1 if and only if ℵ n is the smallest cardinal such that every ideal of O can begenerated by ℵ n elements. In particular, O has finite global dimension ≤ O is ℵ -Noetherian ); but this is manifestly truein our case since we reduced to the case in which O is itself a countable set. In conclusion O has finiteglobal dimension, which is inherited by any polynomial algebra over it by Hilbert [40, Thm. 4.3.7], andthis is stronger than required in (b). Remark 3.4.
Let X be a Noetherian scheme of finite Krull dimension. A conservative family of pointsfor the cdh site X cdh is given by the spectra of Henselian valuation rings [9], and hence Theorem 3.3(iii)implies that K cdh n = 0 for n <
0, where K cdh n denotes the sheafification of the abelian presheaf K n ( − )on X cdh . In fact, this follows from the earlier result of Goodwillie–Lichtenbaum [15] that a conservativefamily of points for the rh site is given by the spectra of valuation rings.Since X cdh is known to have cohomological dimension = dim X [36], we immediately deduce from theresulting descent spectral sequence that K cdh n ( X ) = 0 for n < − dim X , where K cdh ( X ) := H ( X cdh , K )12 -theory of valuation ringsrefers to the cdh sheafification of the K -theory presheaf of spectra. Moreover, combining Theorem 3.3(ii)with Cisinski’s result [6] (proved around 2010) that KH satisfies cdh descent implies that K cdh ( X ) ≃ KH ( X ), whence one immediately gets the following consequences:(a) KH n ( X ) = 0 for n < − dim X ; this was first proved in 2016 by Kerz–Strunk [24] via their usageof Raynaud–Gruson flattening methods. (It is important to note that the proof of Theorem 3.3crucially uses Raynaud–Gruson to check that valuation rings are stably coherent.)(b) If p is nilpotent in X then K n ( X )[ p ] = 0 for n < − dim X (since the nilpotence of p implies KH ( X )[ p ] = K ( X )[ p ] and then we can apply (a); alternatively, once can avoid Cisinski’s result byappealing instead to Theorem 4.3 to see that K [ p ] satisfies cdh descent); this was first proved in2011 by the first author’s use of alterations and ℓ dh topologies [22]. Here we establish a surjectivity result (Corollary 3.6) which was not stated in the introduction but whichplays a role in the first author’s new approach to comparing cdh and ℓ dh topologies [23, Thm. 1]. Lemma 3.5.
Let k be a perfect field, O ⊇ k a valuation ring, and p ⊆ O an ideal along which O isHenselian.(i) If F : k -algs → Sets is a functor which commutes with filtered colimits and p is the maximal idealof O , then F ( O ) → F ( O / p ) is surjective.(ii) If F : k -algs → Spectra is a functor which commutes with filtered colimits, is nil-invariant, andsatisfies excision, then π n F ( O ) → π n F ( O / p ) is surjective for all n ∈ Z .Proof. (i). Assume p = m is the maximal ideal of O , and let α ∈ F ( O / m ). Since O / m is the filteredunion of A/ ( m ∩ A ), as A runs over all finitely generated k -subalgebras of O , we pick such an A suchthat α comes from some class α A/ m ∩ A ∈ F ( A/ ( m ∩ A )). Let q := m ∩ A and let A h q be the Henselisationof A q ⊆ O at q A q ; note that the inclusion A ֒ → O factors through A h q , by functoriality of Henselisation.The completion c A q admits a coefficient field containing k by Cohen [28, Thm. 60], whence the image of α A/ q in F ( A q / q A q ) may be lifted to K n ( b A q ); by Artin approximation (or N´eron–Popescu desingularision)it may therefore be lifted to α A h q ∈ F ( A h q ) (note here that A q is essentially of finite type over a field, inparticular it is excellent). The image of α A h q in F ( O ) is of course the desired lift of α .(ii): Replacing p by its radical (using invariance of F for locally nilpotent ideals), we may assumethat p is a prime ideal. Then standard theory of valuation rings states that the localisation O p is avaluation ring with maximal ideal p . Thus we may apply excision to the data p ⊆ O ⊆ O p and so obtaina long exact Mayer–Vietoris sequence . . . −→ π n F ( O ) −→ π n F ( O / p ) ⊕ π n F ( O p ) −→ π n F ( O p / p ) −→ . . . . But O p is Henselian along p (here we use the fact the Henselianness along an ideal depends only the idealas a non-commutative ring, not on the ambient ring [8, Corol. 1]), so part (i) implies that πF n ( O p ) → π n F ( O p / p ) is surjective for all n ∈ Z . Therefore the long exact sequence breaks into short exact sequencesand moreover each map π n F ( O ) → π n F ( O / p ) is surjective. Corollary 3.6.
Let O be a valuation ring containing a field, and p ⊆ O a prime ideal along which O isHenselian. Then K n ( O ) → K n ( O / p ) is surjective for all n ∈ Z .Proof. We apply Lemma 3.5(ii) to the functor KH , which satisfies all the hypotheses by Weibel [39], andthen use Theorem 3.3(ii) to identify K and KH of valuation rings. By “excision” we mean that whenever A → B is a homomorphism of k -algebras and I ⊆ A is an ideal which is sentisomorphically to an ideal of B , then the induced map of relative theories hofib( F ( A ) → F ( A/I )) → hofib( F ( B ) → F ( B/I ))is an equivalence.
In this section we give a new proof that K [ p ] satisfies cdh descent in characteristic p . In the followinglemma the notation A perf := lim −→ ϕ A denotes the colimit perfection of any F p -algebra A : Lemma 4.1.
Let A be an F p -algebra. Then the canonical map K ( A )[ p ] → K ( A perf )[ p ] is an equivalence.Proof. Since A perf is unchanged if we replace A by A red , and A → A red induces an equivalence on K [ p ](since K [ p ] is invariant under locally nilpotent ideals in characteristic p ), we may replace A by A red andso assume that the Frobenius ϕ is injective on A . It is then clearly enough to show that ϕ ( A ) → A induces an isomorphism on K [ p ]. Changing notation and taking a filtered colimit, this reduces to thefollowing claim:Let A ⊆ B be an extension of rings such that B = A [ b ] for some b ∈ A such that b p ∈ A .Then K ( A )[ p ] → K ( B )[ p ] is an equivalence.Let A ⊆ B be as in the claim and set a := b p ∈ A . The map A [ X ] /X p − a → B , X b is surjectiveand has nilpotent kernel (any element f in the kernel satisfies f p ∈ A , but A → B is injective), henceinduces an equivalence on K [ p ] by nil-invariance of K [ p ] in characteristic p . Thus we have reduced tothe special case B = A [ X ] /X p − a .To treat the special case we consider the base change square A [ Y ] /Y p − a (cid:31) (cid:127) g ′ / / B [ Y ] /Y p − aA ?(cid:31) f O O (cid:31) (cid:127) g / / B ?(cid:31) f ′ O O in which all maps are finite flat. The map f ′ is split by the B -algebra homomorphism B [ Y ] /Y p − a → B , Y X , whose kernel is nilpotent (similarly to the previous paragraph); therefore, again using nil-invariance of K [ p ], we have shown that f ′∗ is an equivalence on K [ p ]. But f ′∗ f ′∗ = p , so f ′∗ is also anequivalence on K [ p ]. The same argument, swapping the roles of X and Y , shows that g ′∗ and g ′∗ areequivalences on K [ p ].Base change for algebraic K -theory tells us that g ∗ f ∗ = f ′∗ g ′∗ : K ( A [ Y ] /Y p − a ) → K ( B ), whencewe deduce that g ∗ f ∗ is an equivalence on K [ p ]. But f ∗ is surjective on homotopy groups since f ∗ f ∗ = p ,whence it follows that both g ∗ and f ∗ are equivalences on K [ p ]. Remark 4.2.
The higher K -groups K n , n ≥
1, of any perfect F p -algebra are known to be uniquely p -divisible [26, Corol. 5.5]. Therefore the previous lemma implies that K n ( A )[ p ] ∼ = K n ( A perf ) for n ≥ K [ p ] in characteristic p (which is not really new; see Remark 4.4), namely Theorem 1.4from the introduction: Theorem 4.3.
Let Y ′ (cid:15) (cid:15) i / / X ′ (cid:15) (cid:15) Y / / X be an abstract blow-up square of schemes on which p is nilpotent; assume that X is quasi-compact quasi-separated and that the morphisms X ′ → X and Y ֒ → X are of finite presentation. Then the resulting -theory of valuation rings square K ( X )[ p ] (cid:15) (cid:15) / / K ( X ′ )[ p ] (cid:15) (cid:15) K ( Y )[ p ] / / K ( Y ′ )[ p ] is homotopy cartesian.Proof. Using once again nil-invariance of K [ p ] we may apply − ⊗ Z Z /p Z to the diagram and so reduceto the case of F p -schemes. By Lemma 4.1 it is then enough to prove the stronger statement that K ( X perf ) (cid:15) (cid:15) / / K ( X ′ perf ) (cid:15) (cid:15) K ( Y perf ) / / K ( Y ′ perf )is homotopy cartesian. To do this we appeal to Tamme’s excision condition [37, Thm. 18], which statesthat it is enough to check that the square of stable ∞ -categories D qc ( X perf ) (cid:15) (cid:15) / / D qc ( X ′ perf ) (cid:15) (cid:15) D qc ( Y perf ) / / D qc ( Y ′ perf )has the following two properties:(a) it is a pull-back of ∞ -categories;(b) the right adjoint Ri ∗ : D qc ( Y ′ perf ) → D qc ( X ′ perf ) is fully faithful.Both these properties about perfect schemes are due to Bhatt–Scholze. Firstly, the pull-back condition(a) is [5, Corol. 5.28]. Secondly, condition (b) is a consequence of Li ∗ Ri ∗ = id, which follows from thefact that if S → S/I is a surjection of perfect F p -algebras, then S/I ⊗ L S S/I ∼ → S/I [5, Lem. 3.16].
Remark 4.4.
We review some previous proofs of cdh descent properties of K -theory.(i) Firstly, if X were assumed to be of finite type over a perfect field of characteristic exponent p ≥ KH satisfies cdh descent, thereby giving the homotopy cartesian square for K [ p ] replaced by KH ; but then one can invert p and use that KH [ p ] = K [ p ] when p is nilpotent.Alternatively, under the same assumptions on X and the base field, trace methods were used bythe second author [30] to establish pro cdh descent, namely that the square of pro spectra { K ( Y ′ r ) } r (cid:15) (cid:15) / / K ( X ′ ) (cid:15) (cid:15) { K ( Y r ) } r / / K ( X )is homotopy cartesian, where Y r is the r th -infinitesimal thickening of Y in X (and similarly for Y ′ ).Since K [ p ] is invariant under nilpotent ideals when p is nilpotent, inverting p then yields the samehomotopy cartesian square of Theorem 1.4.(ii) Alternatively, assuming only that X is Noetherian and of finite Krull dimension, Cisinski showedthat KH satisfies cdh descent [6], again giving the desired homotopy cartesian square for KH ratherthan K [ p ]. 15hane Kelly and Matthew Morrow(iii) Kerz–Strunk–Tamme have proved pro cdh descent [25] for Noetherian schemes without any hy-potheses on resolution of singularities. As in (i), inverting p then gives the homotopy cartesiansquare of Theorem 1.4 assuming that X is Noetherian.(iv) Most recently, Land–Tamme have substantially clarified the theory of cdh descent for arbitrary“truncating invariants” [27]. References [1]
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Cambridge Studies in AdvancedMathematics . Cambridge University Press, Cambridge, 1994.Shane Kelly Matthew MorrowDepartment of Mathematics, CNRS & IMJ-PRG,Tokyo Institute of Technology, SU – 4 place Jussieu,2-12-1 Ookayama, Meguro-ku, Case 247,Tokyo 152-8551, Japan 75252 Paris [email protected] [email protected]@math.titech.ac.jp [email protected]