A descent view on Mitchell's theorem
aa r X i v : . [ m a t h . K T ] A ug A DESCENT VIEW ON MITCHELL’S THEOREM
ELDEN ELMANTO, DENIS NARDIN, AND LUCY YANG
Abstract.
In this short note, we give a new proof of Mitchell’s theorem that L T( n ) K( Z ) ≃ n >
2. Instead of reducing the problem to delicate representation theory, we use recentlyestablished hyperdescent technology for chromatically-localized algebraic K-theory. Introduction and background
In this note, we give an alternate proof of the following result:
Theorem 1.0.1 (Mitchell) . For all primes p and n > , K( n ) ∗ K( Z ) = 0 . The proof Theorem 1.0.1 given in [Mit87] is relatively self-contained and depends on showingthat the unit map → K( Z ) factors through the “image of J.” This factoring relies on delicaterepresentation theory by way of the Barratt-Priddy-Quillen theorem. Since the latter spectrumis known to be acyclic for the Morava’s K( n ) for n >
2, the result follows. The value of thepresent note is that it locates the proof in its natural environment — Rognes’ redshift philosophyin algebraic K-theory.The starting point of Theorem 1.0.1 is Thomason’s seminal result that K(1)-localized alge-braic K-theory satisfies ´etale descent [Tho85]. Combined with the rigidity theorems of Suslin[Sus83] and Gabber [Gab92], one concludes that K(1)-local algebraic K-theory is, more or less,topological K-theory; we also refer the reader to [DFST82] for further elaboration on this pointof view.One can view Thomason’s theorem as a “Bott-asymptotic” version of a more refined state-ment — the Quillen-Lichtenbaum conjecture (now the Voevodsky-Rost theorem [Voe03, Voe11])which asserts that algebraic and ´etale K-theory agrees in high enough degrees. By analogy withthe Quillen-Lichtenbaum conjectures, Rognes has formulated the idea that taking algebraic K-theory increases “chromatic complexity” — demonstrating a “redshift”; we refer the reader to[Rog14] for a discussion.In line with this ideology, Thomason’s theorem is then viewed as saying that taking algebraicK-theory of a discrete commutative ring (which is K(1)-acyclic) yields an interesting answerK(1)-locally. At the next height, results of Ausoni-Rognes [Aus10, AR02] who confirmed that v acts invertibly on K(K( C ) ∧ p ) ⊗ V(1) where V(1) is a type 2 complex, hence is interestingK(2)-locally.We can view Mitchell’s result anachronistically as a demonstration of the strictness of redshift:while the 2-fold algebraic K-theory of a discrete ring is interesting K(2)-locally, the algebraicK-theory thereof itself is not.The value of our proof, if there is one, is that it is born in the same spirit as Thomason’sresults: we confirm Mitchell’s vanishing by way of ´etale hyperdescent. We believe that provingthe result this way places it within its proper context, at the cost of using more technology.1.1.
Acknowledgements.
We would like to thank Akhil Mathew and John Rognes for usefuldiscussions and Gabriel Angelini-Knoll, Ben Antieau, Markus Land, Lennart Meier and GeorgTamme for comments on an earlier draft. On Mitchell’s theorem A p -adic version of Mitchell’s theorem. As a warm-up, we first give a very sim-ple proof of the p -adic version of Mitchell’s theorem. So fix a prime p >
0; here our T( n )-localizations will be at this implicit prime. Theorem 2.1.1.
For n > , we have that L T( n ) K ( Z p ) ≃ . To begin, let C be the completion of the algebraic closure of Q p and O C be its ring of integerswhich is an integral perfectoid ring. Lemma 2.1.2.
For n > , we have that L T( n ) K ( O C ) ≃ .Proof. Consider the zig-zag of maps ku → K (C) j ∗ ←− K ( O C ) . The maps above are all p -adic equivalences:(1) for j ∗ this is [HN19, Lemma 1.3.7],(2) for the unlabeled arrow, we have Suslin rigidity [Sus84].Hence we conclude that L T( n ) K ( O C ) ≃ (cid:3) Remark 2.1.3.
Note that [HN19, Lemma 1.3.7] only uses very basic facts about perfectoidrings (that we can choose a pseudouniformizer π such that O C /π ≃ O C ♭ /π ♭ ) and the fact thatthe positive homotopy groups of the K-theory of local perfect F p -algebras are all p -divisible by[Hil81, Kra80]. Proof of Theorem 2.1.1.
Since K-theory is a finitary invariant , we can writeK ( O C ; Z p ) ≃ colim Q p ⊂ E ⊂ C K ( O E ; Z p ) , a colimit of E ∞ -rings, and the colimit ranges along finite extensions of Q p contained in C.Now, the source vanishes after applying L T( n ) , whence so is the target. Since L T( n ) commuteswith filtered colimits, we may find a finite extension E of Q p such that L T( n ) K ( O E ) ≃ E ∞ -ring; indeed, vanishing is equivalent to the unit beingnullhomotopic. Since the morphism of rings Z p → O E is finite flat, by the descent results of[CMNN20] we get that L T( n ) K( Z p ) ≃ Tot (cid:16) L T( n ) K (cid:16) O ⊗ Z p • +1E (cid:17)(cid:17) . We are now done, since all terms of the limit on the right hand side are modules over L T( n ) K ( O E ) =0. (cid:3) To conclude note that if A is T( n )-acyclic, then it is also K( n )-acyclic. Though we willnot need it, we can also reverse the implication if A is, furthermore, an E ∞ -ring spectrum by[LMT20, Lemma 2.3]. Corollary 2.1.4. If n > , we have that L K( n ) K ( Z p ) ≃ . In particular K( n ) ∗ K ( Z p ) = 0 . In more details: O C is p -adically isomorphic to the colimit of the O E ’s and K-theory preserves p -adicequivalences in this setting. This follows from, for example, the fact that given a morphism of rings A → B, fib(GL(A) → GL(B)) ≃ fib(M(A) → M(B)) and the formation of matrix rings evidently preserves localequivalences.
DESCENT VIEW ON MITCHELL’S THEOREM 3
Mitchell’s theorem.
We now give a proof of Mitchell’s theorem. We phrase this as.
Theorem 2.2.1.
For all primes p and n > , L T( n ) K( Z ) ≃ . Equivalently, L K( n ) K( Z ) ≃ and, in particular, K( n ) ∗ K( Z ) = 0 .Proof. The “equivalently” part follows from [LMT20, Lemma 2.3]. We first claim that that wecan work with rings which are ( p )-local. To see this, we claim that the mapL T( n ) K ( Z ) → L T( n ) K (cid:18) Z (cid:20) p (cid:21)(cid:19) , is an equivalence. Indeed, by localization and d´evissage [Qui10] we have a cofiber sequenceK ( F p ) → K ( Z ) → K (cid:18) Z (cid:20) p (cid:21)(cid:19) , and L T( n ) K( F p ) ≃ L T( n ) H Z p ≃ n >
2. Therefore, it suffices to show that L T( n ) K (cid:16) Z h p i(cid:17) ≃ (cid:3) Claim 2.2.2.
For any n > , the presheaf, L T( n ) K( − ) : Et op Z [ p ] → Spt is a hypercomplete sheaf of spectra.Proof.
While this is an immediate application of [CM19, Theorem 7.14], we will give a moredetailed proof here to highlight the ingredients. Since telescopic localization is invariant undertaking connective covers ([LMT20, Lemma 2.3(iii)]) we obtain a map of presheaves of E ∞ -rings:K cn ( − ; Z p ) → L T( n ) K cn ( − ; Z p ) ∼ = L T( n ) K ( − )Moreover L T( n ) K( − ) is an ´etale sheaf, thus this map factors through a canonical E ∞ -mapK cn ( − ; Z p ) ´et → L T( n ) K ( − )Since hypercompletion is smashing by [CM19, Corollary 4.39], it suffices to then prove thatK cn ( − ; Z p ) ´et is a hypersheaf on Et Z [ p ].As is proved by Thomason in [Tho85, Theorem 4.1] for odd primes (which relies on the Suslin-Merkerjuev theorem [MS82]) and Rosenschon and Østvær [ROsr05] for the prime 2 (which doesrely on the Milnor conjecture [Voe03]), L T(1)
K does satisfy ´etale hyperdescent. We consider thecanonical map K cn ( − ; Z p ) ´et → L T(1)
K and F be the fiber. The claim then follows if one canshow that the fiber F has ´etale hyperdescent.Let G denote the fiber of the map K cn ( − ; Z p ) → L T(1)
K. By the full strength of the Bloch-Kato conjectures [Voe11, Voe03] (see [CM19, Theorem 6.18], noting that Spec Z admits thedesired global bound by [Ser02, I.3.2]) we see that the fiber is truncated. Therefore the sheafi-fication, F ≃ G ´et is Postnikov complete, whence is indeed hypercomplete as desired. (cid:3) Claim 2.2.3.
For n > and for all strictly hensel local ring R with residue field κ of charac-teristic ℓ > and ( p, ℓ ) = 1 L T( n ) K(R) ≃ E. ELMANTO, D. NARDIN, AND L. YANG
Proof of Claim 2.2.3.
By Gabber rigidity [Gab92], the map K cn (R) → K cn ( κ ) is a p -adic equiv-alence, while by Suslin rigidity [Sus84] we have a further p -adic equivalence K cn ( κ ) ← ku. Sincetelescopic localization for n > T( n ) K (R) ≃ L T( n ) K cn (R) ≃ L T( n ) ku ≃ . (cid:3) Remark 2.2.4.
Because of the appeal to [CM19, Theorem 7.14], our proof is not “simple”,in contrast to the p -adic situation. This is because the Clausen-Mathew theorem dependson the resolution of the Quillen-Lichtenbaum conjectures. Specifically, the version of [CM19,Theorem 7.14] that we need, requires [CM19, Theorem 6.13] which ultimately proves thatthe restriction of K ´et to Et Z [1 /p ] is a ´etale hypersheaf. This latter result, in turn, relies onRost-Voevodsky’s resolution of the Bloch-Kato conjectures. Note that, in contrast to the p -adic situation, Theorem 2.2.1 asserts something global which prompts us to argue via stalks,whence an appeal to hyperdescent. Remark 2.2.5 (Personal communication by J. Rognes) . Morally speaking, our proof is acleaner packaging of the following method to prove Mitchell’s theorem using the Rost-Voevodskyresults. For simplicity let p be an odd prime and let V(1) := / ( p, v ) be a finite complex whichadmits a v -self map. Our goal is to prove that K( Z ) ⊗ V(1) is a bounded complex which sufficesto prove Mitchell’s theorem since inverting a positive degree self-map on a bounded complexannihilates it. Using the localization sequence, we reduce to proving the following assertions:(1) if ℓ = p , then v acts invertibly on K ∗ ( F ℓ ) /p for ∗ large enough,(2) v acts invertibly on K ∗ ( Q ) /p for ∗ large enough, and(3) if ℓ = p , then K ∗ ( F p ) /p is bounded above.The last statement follows from Quillen’s computation [Qui72] and the first statement fol-lows by a further application of Suslin rigidity [Sus83]. The second statements is where Rost-Voevodsky’s resolution of the Bloch-Kato conjectures is needed [Voe03, Voe11] (this is wherethe “overkill happens”), the motivic spectral sequence [FS02] and the fact that v is detectedby a periodicity operator on ´etale cohomology.As a consequence of the main theorem we can easily obtain two corollaries: Corollary 2.2.6.
The functor L T( n ) K | Cat perf Z is zero for n > . Corollary 2.2.7.
For n > , L T( n ) TC ( Z ) ≃ . Consequently, L T( n ) TC | Cat perf Z is zero for n > .Proof. Via the trace map, L T( n ) TC( Z ) is an L T( n ) K ( Z )- E ∞ -algebra, whence the result followsfrom Theorem 2.2.1. (cid:3) References [AR02] C. Ausoni and J. Rognes,
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