aa r X i v : . [ m a t h . K T ] A p r A K-theoretic approach to Artin maps
Dustin ClausenApril 4, 2017
Abstract
We define a functorial “Artin map” attached to any small Z -linear stable ∞ -category, which inthe case of perfect complexes over a global field F recovers the usual Artin map from the idele classgroup of F to the abelianized absolute Galois group of F . In particular, this gives a new proof ofthe Artin reciprocity law. Introduction
For a global field F with adele ring A F and abelianized absolute Galois group G abF , the Artin map attached to F is a certain homomorphism of abelian groups Art F : A × F /F × → G abF which plays a fundamental role in class field theory. There are also Artin maps associated to other kindsof number-theoretic fields: if F is a local field then we have Art F : F × → G abF , and if F is a finite field then we have Art F : Z → G abF . We will denote the left hand side of each of these maps by C F , irrespective of the kind of field F .The last map, where F is a finite field, is simple to produce: it sends ∈ Z to the Frobenius elementof G abF . Together with a certain functoriality in F , this uniquely determines the whole system of maps Art F , for all fields F as above. Nonetheless, the existence of such a system is a difficult theorem,equivalent to the Artin reciprocity law.The goal of this paper is to provide a new and more general approach to the existence of sucha system of maps Art F . We will recover Art F as π of a certain map between K-theoretic spectra,which we will also call the Artin map. One advantage of this approach is that it works for all F simultaneously, and the appropriate functoriality in F becomes transparent. Another advantage is thatthe extra flexibility and structure of K-theory lets us reduce the entire construction to the base case F = Z .In that base case the desired map will come from the homotopy-theoretic product formula of [C],which was already used there to give a new proof of the quadratic reciprocity law.1 emark 0.1. In fact, we only need a small piece of the product formula from [C], namely we need its K (1) -localization in the sense of stable homotopy theory. This is one of the mysterious aspects of thestory: most of the constructions make sense even without K (1) -localization, and it’s conceivable thatmany of the results also hold without K (1) -localization. However, the author is at a loss for how toprove this outside the case of imaginary quadratic fields. See Remark 4.8 for more. Our theory applies to more general objects than just the fields F as above. Basically, there is aversion of the theory for anything Z -linear in nature which one can take algebraic K-theory of. For us,the most convenient context is that of small idempotent-complete Z -linear stable ∞ -categories . Denotethe ∞ -category of these by PerfCat Z . Every ring R gives such a P ∈
PerfCat Z , namely P = Perf( R ) .We will attach two different spectra to any P , as well as an “Artin map” between these spectra.The source spectrum is the locally compact K-theory spectrum K(lc P ) , defined as follows: Definition 0.2.
Let
P ∈
PerfCat Z , and let lc P = Fun Z ( P , D b (LCA ℵ )) be the stable ∞ -category of “ P op -modules in locally compact abelian groups”. Here LCA ℵ is the exactcategory of second countable Hausdorff locally compact abelian groups.The locally compact K-theory of P is defined to be the K-theory spectrum K(lc P ) . When P = Perf( F ) for a field F as above, let us abbreviate lc F = lc Perf( F ) . Then there is a naturalhomomorphism C F → π K(lc F ) . It is an isomorphism if F is finite or global. If F is local it is not anisomorphism, but it would be one if we had properly accounted for the topology on F in the previousdefinition. In any case, to recover the classical Artin maps we only need the homomorphism. Remark 0.3.
To give an idea about how the “reciprocity” arises in this framework, let us describe themap C F → π K(lc F ) when F is a global field. The adele group A F can be viewed as an F -module in LCA ℵ , hence as an object in lc F . As such it has an action by A × F , which gives a map A × F → π K(lc F ) .We need to see that this map is trivial on F × . Consider the short exact sequence F → A F → A F /F in LCA ℵ . The first term is discrete and hence trivial in K-theory by an Eilenberg swindle with direct sums;the last term is compact and hence trivial in K-theory by an Eilenberg swindle with direct products.Thus the middle term is trivial as well. Since F × acts by automorphisms on this whole short exactsequence, we deduce that our map A × F → π K(lc F ) is trivial on F × .Note that this argument takes the same form as Tate’s observation ([Ta] Theorem 4.3.1) that thetheory of Haar measures formally implies the product formula for valuations of a global field. Indeed,the theory of Haar measures induces π K(lc Z ) ≃ R > . The target spectrum attached to P will be a certain kind of K-homology theory which we call SelmerK-homology . Morally, it is dual to etale sheafified algebraic K-theory; but since etale sheafified algebraicK-theory a priori is not defined for an arbitrary Z -linear stable ∞ -category and therefore does not havethe appropriate functoriality, we replace it by the following:2 efinition 0.4. Let
P ∈
PerfCat Z . Define the Selmer K-theory spectrum of P to be the homotopypullback K Sel ( P ) := L K( P ) × L TC( P ) TC( P ) . Here TC is topological cyclic homology with its cyclotomic trace map K → TC (see [DGM] for atextbook reference on the cyclotomic trace, and [BGT] Section 10 for its construction in this precisesetting) and L : Sp → Sp is the chromatic level ≤ localization functor on spectra, i.e. L is Bousfieldlocalization with respect to complex K-theory ([Bo] Section 4). There is a natural map K( P ) → K Sel ( P ) , and K Sel is designed to be the closest approximationto K-theory which satisfies etale descent and yet is still defined in the same generality with the samefunctoriality.
Remark 0.5.
This definition is motivated on the one hand by the work of Thomason and Gabber-Suslin([Th1]) on L K( X ) , and on the other hand by the work of Geisser-Hesselholt ([GH], [GH2]) on TC( X ) .Taken together, these imply that K Sel is indeed essentially just the etale hypersheafification of K onreasonable schemes. The fact that the etale hypersheafification of algebraic K-theory is still a functorof the category Perf( X ) (up to the natural periodization, see the footnote) is a miracle. Remark 0.6.
When X is global and mixed characteristic in nature, the definition of K Sel ( X ) bears asimilarity to the definition of Selmer goups. The theory L K plays the role of Galois cohomology, andgluing on TC plays the role of imposing integrality conditions over Spec( Z p ) . In Section 1 we will define certain duality functors d K (1) and d T C , both close relatives of Andersondualtiy, as well as a natural transformation d K (1) → d T C , which let us dualize the above definition:
Definition 0.7.
Let
P ∈
PerfCat Z . We define the Selmer K-homology of P to be the homotopypushout dK Sel ( P ) := d K (1) K( P ) ⊔ d K (1) TC( P ) d T C
TC( P ) . If F is a field as above, then there is a canonical isomorphism G abF ∼ → π dK Sel ( F ) , coming from anetale descent spectral sequence and Galois cohomological dimension estimates on F (Theorem 2.15). Remark 0.8.
As far as I know, the required cohomological dimension estimates in the number fieldcase can only be proved using the local-global principle for central simple algebras (as in [Se] II.6). Thisprinciple in turn is usually proved alongside the development of class field theory. Thus, at this pointour approach to Artin reciprocity is not exactly independent of the usual edifice.
Our main theorem is the following:
Theorem 0.9.
There is a canonical natural transformation
Art P : K(lc P ) → dK Sel ( P ) of functors PerfCat op Z → Sp such that for all finite, local, and global fields F , the composition C F → π K(lc F ) π Art −→ π dK Sel ( F ) ∼ ← G abF equals the usual Artin map. What is true is the following. Let X be a noetherian algebraic space of virtual finite etale cohomological dimension andsuch that ( X ⊗ F p ) red is regular. Then: K Sel ( − ) is a hypersheaf on X et ; the map K → K Sel is an equivalence rationally;and on p -completion it is an equivalence on etale stalks at points of characteristic p , whereas at points of characteristic = p the former is the connective cover of the latter and the latter is the 2-periodization of the former. emark 0.10. If we take P = Perf( X ) for an algebraic space X essentially of finite type over Z , thenwe get a K-theoretic version of the Artin map from geometric class field theory, in a form for whichreciprocity laws become tautological, or rather reduce to simple mechanisms as in Remark 0.3. Futurework will be required to give a more detailed study of K(lc P ) and dK Sel ( P ) in this case, as well as toinvestigate the question of to what extent Art P is an equivalence (compare with [BlM] in the numberfield case at odd primes). Remark 0.11.
The definition of locally compact K-theory bears an analogy to the definition of K-homology for C*-algebras. In fact, it seems from several perspectives that X K(lc X ) plays therole of an algebraic K-homology theory. On the other hand X dK Sel ( X ) plays the role of anetale topological K-homology theory. Thus another way of thinking of our Artin maps is that they areRiemann-Roch-style natural transformations in the sense of [BFM], going from algebraic K-homologyto topological K-homology. Recall that the existence of such natural transformations on the levelof cohomology theories is essentially tautological, but its existence on the dual homology theories issurprising and encodes nontrivial information. Remark 0.12.
Every profinite abelian group is the product of its pro- p -completions over all primes p .Thus to produce the Artin map for the field F it is enough to produce, for every prime p , a p -primaryArtin map C F → (G abF ) b p . This is what we will do, and indeed our Selmer K-homology theory dK Sel will be p -complete for a prime p which will be fixed throughout the paper. To recover the statementsas written above, one can take the product over all primes p . Acknowledgements
I would like to thank Jacob Lurie for sharing so much relevant insight over the years in which I was hisPhD student, as well as for suggesting the “double Eilenberg swindle” approach to the K-theory of theexact category of locally compact abelian groups.I also thank Haynes Miller and John Tate for helpful comments on a previous version of this work,Clark Barwick and Steve Mitchell (through his papers) for teaching me many interesting and usefulfacts about algebraic K-theory, and Frank Calegari, Lars Hesselholt, Akhil Mathew, and Peter Scholzefor helpful exchanges.Finally, I thank the MIT math department, the National Science Foundation, the University ofCopenhagen, and Lars Hesselholt’s Niels Bohr Professorship for their support.
Contents p -adic dualizing module for TC( Z ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Locally compact K-theory 29
In this section we will recall a construction from [C]. Once the definitions are in place, this constructionwill supply the the “fundamental class” which completely specifies our system of Artin maps. We willalso define the dualizing objects which go into the definition of Selmer K-homology.For a ring R , let K( R ) denote the connective algebraic K-theory spectrum of R . For an E ∞ -ring R ,let Pic( R ) denote the connective Picard spectrum of R , obtained by delooping the space of invertible R -modules using the smash product E ∞ -structure. For a prime p , let ( − ) b p : Sp → Sp denote the p -completion functor on spectra; thus S b p is the p -complete sphere. Recall (see [BhS] Appendix for an exposition) that if R is a commutative ring with associated Eilenberg-Maclane E ∞ -ring HR , then there is a canonical “determinant” map det : K( R ) → Pic( HR ) which sends the class of a f.g. projective R -module M to the line bundle Λ dim( M ) M placed in degree dim( M ) (suitably interpreted if Spec( R ) is disconnected). This map identifies Pic( H ( − )) with theZariski-sheafication of the degree ≤ Postnikov truncation of K( − ) .Fundamental to the construction of our Artin maps is the following theorem: Theorem 1.1.
Let p be a prime. The determinant map K( Z p ) → Pic( H Z p ) lifts along Pic( S b p ) → Pic( H Z p ) to a map J Z p : K( Z p ) → Pic( S b p ) whose restriction to K( Z ) factors through K( R ) to a map J R : K( R ) → Pic( S b p ) . This follows from combining Theorem 5.1 and Lemma 3.2 from [C], noting that the J Z p we meanhere is actually the negative of the J Z p of [C].We immediately see: Corollary 1.2.
1. The map J Z p sends ∈ π K( Z p ) to the class of S in π Pic( S b p ) .2. For u ∈ Z × p , the map J Z p sends [ u ] ∈ π K( Z p ) to the class of the equivalence · u : S b p ≃ S b p in π Pic( S b p ) . Remark 1.3.
The above theorem contains all the information we will need on J Z p and J R , but letus nonetheless recall that the J Z p and J R considered in [C] are certain specific maps with transparenttopological meaning. amely, if M Z p is a finite free Z p -module, then J Z p [ M Z p ] is the invertible p -adic spectrum whichcontrols duality on the classifying topos BM Z p (compare [Ba]). If we give M Z p an integral structure M Z , then this identifies with the analogous dualizing object for BM Z . The required factoring through K( R ) comes from the fact that BM Z can be modeled as the torus M R /M Z , where Atiyah duality showsthat the dualizing object for this torus only depends on the real vector space M R , and indeed is theone-point compactification of M R , which defines J R (compare [K] Theorem 10.1). Now fix the prime p and let L K (1) mean K (1) -localization at p . The J-homomorphisms of the previoussection have target Pic( S b p ) , which somewhat motivates the following definition. Definition 1.4.
Define ω K (1) = L K (1) Pic( S b p ) , and define a functor d K (1) : Sp op → Sp by d K (1) ( X ) = map( X, ω K (1) ) . Remark 1.5.
Note that d K (1) lands in K (1) -local spectra and factors through K (1) -localization, soone can restrict d K (1) to a functor L K (1) Sp op → L K (1) Sp without loss of generality. Remark 1.6.
There is an equivalence of spectra ω K (1) ≃ Σ L K (1) S (see below). It follows from thisand [HM] Theorem 8.8 that ω K (1) is equivalent to a twist of the p -adic Anderson dual of L K (1) S , thetwist being given by a certain “exotic” K (1) -local invertible spectrum W (equivalent to S if p > ,but not if p = 2 ). Thus on K (1) -local spectra d K (1) is a W -twist of p -adic Anderson duality, and inparticular there are short exact sequences → Ext Z p ( π n − ( W ∧ M ) , Z p ) → π − n d K (1) M → Hom Z p ( π n ( W ∧ M ) , Z p ) → for all n ∈ Z and M ∈ L K (1) Sp . Remark 1.7.
When p = 2 , then despite the appearance of the strange W , in some respects d K (1) is better behaved than p -adic Anderson duality. For example KO b p is d K (1) self-dual, and the dualitypairing is canonically induced by J R . Its p -adic Anderson dual is Σ KO b p , and that duality pairing, likeAnderson duality itself, is apparently only canonical in the homotopy category. There is a canonical identification π − L K (1) S = Hom( Z × p , Z p ) coming from the KU -based Adamsspectral sequence, interpreted as a continuous homotopy fixed point spectral sequence ([DH]). Todescribe it concretely, recall that Z × p acts on the K (1) -local ring spectrum KU b p by Adams operations ψ x , x ∈ Z × p , and that π − KU b p = 0 . Given a class in π − L K (1) S represented by a point c ∈ Σ L K (1) S ,choose a null-homotopy κ : c ∼ → ∗ in Σ KU b p . Then for x ∈ Z × p , the value c ( x ) ∈ Z p via the aboveidentification is the homotopy class of the element ∗ κ − → c ≃ ψ x ( c ) ψ x κ → ψ x ( ∗ ) ≃ ∗ in π ΩΣ KU b p = π KU b p = Z p .Thus, if X is a K (1) -local spectrum and ϕ ∈ Hom( Z × p , Z p ) , then ϕ gives a functorial degree − operation on π ∗ X : α ∈ π n X α · ϕ ∈ π n − X. Our main result in this section is the following: 6 roposition 1.8.
1. The Z p -module π ω K (1) is free on [ S ] , the image of the class of S in Pic( S b p ) .2. The homomorphism Z × p = π Pic( S b p ) → π ω K (1) , which we will denote x [ x ] , has kernel µ p − and cokernel isomorhic to Z p / Z p .3. For ϕ ∈ Hom( Z × p , Z p ) = π − L K (1) S and x ∈ Z × p , there is the relation [ x ] · ϕ = 2 ϕ ( x ) · [ S ] in π ω K (1) . To prove this, we will study ω K (1) using Rezk’s logarithm ([Re]): Definition 1.9.
Let log K (1) : ω K (1) ∼ → Σ L K (1) S denote the result of applying the K (1) -local Bousfield-Kuhn functor to the pointed map Ω ∞ Ω Pic( S b p ) = GL ( S b p ) → Ω ∞ S b p given by x x − . The reason log K (1) is an equivalence is that x x − is an equivalence after looping once.Note that π − L K (1) S = Hom( Z × p , Z p ) is a free Z p -module of rank one. Recall also that π L K (1) S = Z p ⊕ ( Z p / Z p ) , with the first factor generated by and the second factor generated by η · ℓ for anygenerator ℓ of π − L K (1) S .The following is proved using Rezk’s formula ([Re] Theorem 1.9), or more precisely the special caseof Rezk’s formula which says that log K(1) : Pic( KU b p ) → Σ KU b p on π is the map x p log( x p − ) : Z × p → Z p . Proposition 1.10.
Consider the composition
Pic( S b p ) → ω K (1) log K (1) −→ Σ L K (1) S .1. On π , it sends the class of S to the generator ℓ of Hom( Z × p , Z p ) = π − L K (1) S defined by ℓ ( x ) = 12 p log( x p − ) .
2. On π , it sends the class of u : S b p ≃ S b p in Z × p to the element of π L K (1) S given by p log( u p − ) + u − · η · ℓ. Proof.
This is explained in [C] Proposition 4.2 for p > ; some slight additions allow us to handle p = 2 .For statement 1, consider the map Pic( S b p ) → Pic( KU b p ) Z × p induced by functoriality. The Bott periodicity class β ∈ π KU gives a nullhomotopy S ∧ KU b p ≃ KU b p of the image of S in Pic( KU b p ) , and ψ x ( β ) = x · β for x ∈ Z × p . Thus [ S ] ∈ π Pic( S b p ) is detectedin H ( B Z × p ; π Pic( KU b p )) = Hom( Z × p , Z × p ) by the identity map x x . By Rezk’s formula forthe logarithm on KU b p and naturality it follows that ( π log K (1) )([ S ]) ∈ π − L K (1) S is detected in7 ( B Z × p , π KU b p ) = Hom( Z × p , Z p ) by the map which sends x to p log( x p − ) . This forces the claimsince Hom( Z × p , Z p ) is torsion-free.For statement 2, the fact that π (log K (1) )([ u ]) projects to p log( u p − ) on the Z p -factor of π L K (1) S follows immediately from Rezk’s formula on KU b p . For p > we are finished; for p = 2 we deduce that π (log K (1) )([ u ]) − p log( u p − ) = ǫ ( u ) · η · ℓ for some homomorphism ǫ : Z × → Z / Z , and we need tosee that ǫ ( u ) is trivial if and only if u is (mod ).Since p log( u p − ) is always divisible by and hence is killed by η whereas η · ℓ = 0 , multiplying by η shows that ǫ is also characterized by π (log K (1) )( η · [ u ]) = ǫ ( u ) · η · ℓ. Since every order 2 character on Z × factors through ( Z / Z ) × , it suffices to see that ǫ ( − isnontrivial and ǫ (5) is trivial. The non-triviality of ǫ ( − follows from the identity [ −
1] = η · [ S ] andstatement 1. For the triviality of ǫ (5) , we claim in fact that η · [ p ] = 0 in Pic( S [1 /p ]) for any prime p which is (mod ). For this, note that for any prime p there is a map of spectra Σ K( F p ) → Pic( S [1 /p ]) which carries ∈ π K( F p ) to [ p ] ∈ π Pic( S [1 /p ]) (namely the map J F p of [C] Section 3.1, perhapsfirst considered by Quillen and/or Tornehave, see [Q1]). When − is a square (mod p ), we have that η = [ − ∈ π K( F p ) vanishes on any map to an order 2 group. Since π Pic( S [1 /p ]) = π S is of order2, we deduce the claim.Statement 3 is immediate from statement 1.Translating this knowledge back to ω K (1) using the log equivalence ω K (1) ≃ Σ L K (1) S , we deduceProposition 1.8 above. p -adic dualizing module for TC( Z ) Remark 1.11.
In this section, we will use known calculations of Galois cohomology with Tate-twistcoefficients for the field Q p (as in [Se] II.5). Such calculations are of course closely tied up with localclass field theory. It is certainly possible, and possibly preferable, to avoid the use of Galois cohomology bygiving purely TC-theoretic proofs of the results in this section. But we decided to use Galois cohomologyfor brevity.In any case, the proof of Artin reciprocity we eventually give can be easily modified so as not torely on the results of this section, and in particular doesn’t require such detailed knowledge of Galoiscohomology, or of TC-theory. For more on this, see Remark 4.7. Consider the diagram of maps of E ∞ -rings TC( Z ) → TC( Z p ) tr ←− K( Z p ) → K( Q p ) . The first map is an equivalence on p -completion, the second map is an equivalence on p -completion indegrees ≥ , and the third map is an equivalence on p -completion in degrees ≥ . Thus all the mapsare equivalences on L K (1) . Let us denote by R the K (1) -localization of any of the above spectra, with the understanding that one always uses theabove maps to pass between different choices of which spectrum has been localized. Recalling the map J Z p : K( Z p ) → Pic( S b p ) from Theorem 1.1, let us also denote L K (1) J Z p by j Z p : R → ω K (1) . j Z p as a point of the R -module d K (1) R .Here are some relevant classes in π ∗ R in degrees ∗ = − , , . Definition 1.12.
Define the following classes in π ∗ R :1. For x ∈ Q × p = π K( Q p ) , let [ x ] ∈ π R denote its image in R .2. (a) For ϕ ∈ Hom( Z × p , Z p ) = π − L K (1) S , let [ ϕ ] ∈ π − R denote its image in R .(b) Consider the generator f ∈ π − TC( Z ) b p ∼ → π − TC( F p ) b p , detected in the homotopy fixedpoint spectral sequence for TC( F p ) b p = TC( F p ) Frob Z b p = ( H Z p ) Frob Z by the homomorphism Frob Z → Z p = π TC( F p ) b p sending Frob to . Denote also the image of f in π − R by f .3. Consider the unit ∈ π R , as well as the element ǫ := f · [ p ] ∈ π R . Now we define another duality functor.
Definition 1.13.
Define a
TC( Z ) -module ω T C by the cofiber sequence
TC( Z ) b p → R → ω T C , and define a functor d T C : Mod op TC( Z ) → Sp by d T C ( M ) = map TC( Z ) ( M, ω
T C ) . Remark 1.14.
Since ω T C is p -complete, d T C factors through p -completion and lands in p -completespectra. The main results in this section are as follows.
Theorem 1.15.
The ring π R is Z p [ ǫ ] /ǫ . Elements of [ R, ω K (1) ] are uniquely characterized by theireffect on π , and for j Z p we have ( π j Z p )(1) = [ S ] , ( π j Z p )( ǫ ) = 2[ S ] . Theorem 1.16.
The map R · j Z p −→ d K (1) R is an equivalence. Corollary 1.17.
We can define a natural transformation n : d K (1) → d T C on TC( Z ) -modules as thecomposition map( − , ω K (1) ) ≃ map TC( Z ) ( − , d K (1) TC( Z )) → map TC( Z ) ( − , ω T C ) , where the last map is given by composing with d K (1) TC( Z ) = d K (1) R ∼ ← R → ω T C . Here the wrong-way equivalence is · j Z p and the last map is the tautological one from the cofiber sequencedefining ω T C . heorem 1.18. The
TC( Z ) b p -module ω T C is equivalent to the Z p -Anderson dual d Z p TC( Z ) b p , via anequivalence (unique up to homotopy) which carries ǫ ∈ π ω T C to the canonical class in π d Z p TC( Z ) b p .Thus on p -complete modules, d T C identifies with Z p -Anderson duality. Remark 1.19.
These results should be compared with the very similar K-theoretic local duality resultin [BlM] Theorem 1.2.
Remark 1.20.
The proof of Theorem 1.18 will also show that n is an equivalence on K (1) -local TC( Z ) -modules. Thus, if p > , then by combining Theorem 1.18 and Remark 1.6 we obtain two identificationsof d K (1) with d Z p on R -modules. These identifications necessarily differ by a non-trivial unit in π R .This is the main source of subtlety in our definition of Selmer K-homology: even when p > it is notequivalent to the Anderson dual of the Selmer K-theory defined the introduction. We start by studying some relations between the classes of Definition 1.12. For this, take theinterpretation R = L K (1) K( Q p ) , and use Thomason’s descent spectral sequence ([Th1]), or in otherwords the spectral sequence associated to the etale-sheafified Postnikov filtration of L K (1) K( − ) . Thistakes the form E i,j = H j − i (BG Q p ; Z p ( j/ ⇒ π i L K (1) K( Q p ) , where as usual Z p ( j/
2) = 0 for j odd. Since the cohomological dimension of BG Q p is two, there areno differentials and the result is as follows: Lemma 1.21.
For n = 2 j − odd, there is a canonical identification π n R = H (BG Q p ; Z p ( j )) . For n = 2 j even, there is a canonical short exact sequence → H (BG Q p ; Z p ( j + 1)) → π n R → H (BG Q p ; Z p ( j )) → . It is easy to see what the classes of Definition 1.12 look like in these terms:
Lemma 1.22.
1. For x ∈ Q × p , the class [ x ] ∈ π R = H (BG Q p ; Z p (1)) corresponds to the Kummertorsor of p -power roots of x . In other words, if we choose a collection ( x /p n ) n of compatible p n -roots of x in an algebraic closure, then [ x ] is represented by the 1-cocycle σ ( z n ) n where σ ( x /p n ) = z n · x /p n .
2. (a) For ϕ ∈ Hom( Z × p , Z p ) , the class [ ϕ ] ∈ π − R = Hom cont (G Q p , Z p ) corresponds to thecomposition G Q p → Gal( Q p ( ζ p ∞ ) / Q p ) = Z × p ϕ −→ Z p . (b) The class f ∈ π − R = Hom cont (G Q p , Z p ) corresponds to the composition G Q p → Gal( Q unrp / Q p ) b p ≃ −→ Z p , where the isomorphism sends Frob to .3. (a) The class ∈ π R projects to ∈ H (BG Q p ; Z p ) = Z p in the above short exact sequence. b) The class ǫ ∈ π R lives in the H (BG Q p ; Z p (1)) -subgroup, and there equals ∈ Z p underthe identification H (BG Q p ; Z p (1)) ∼ −→ T p ( Q / Z ) = Z p coming from the inv identification H (BG Q p ; Q / Z (1)) ≃ Q / Z .Proof. Claim 1 is standard and straightforward.For claim 2(a), note that Snaith’s presentation of KU ([Sn]) realizes KU b p as the p -completed Bott-localization of (Σ ∞ + Bµ p ∞ ) b p . Further, since the Adams operations are characterized by their effect on π , it follows that the automorphisms of the group µ p ∞ induce the Adams operations on KU b p . Usingthis one makes a map of E ∞ -ring spectra KU b p → L K (1) K( Q p ( ζ p ∞ )) which is Z × p -equivariant for the Adams action on the left and the Galois action on the right. Then claim1(b) follows by functoriality.Claim 2(b) is also immediate from functoriality.Claim 3 follows from the multiplicativity of Thomason’s descent spectral sequence, and standardcup product calculations in the Galois cohomology of Q p .Comparing with known calculations in the Galois cohomology of Q p , we deduce: Proposition 1.23.
1. We have π R = Z × p /µ p − ⊕ Z p , with the first factor given by the [ x ] for x ∈ Z × p and the second factor generated by [ p ] .2. We have π − R = Hom( Z × p , Z p ) ⊕ Z p , with the first factor given by the [ ϕ ] for ϕ ∈ Hom( Z × p , Z p ) and the second factor generated by f .3. We have π R = Z p ⊕ Z p , with the first factor generated by and the second by ǫ . In these terms, the filtration of π R induced by Thomason’s spectral sequence REF is given by the subgroup ⊕ Z p . (Concretely, ⊕ Z p is the kernel of π R → π L K (1) K( Q p ) = Z p .)Furthermore, the following multiplicative relations hold:1. ǫ = 0 ;2. For x ∈ Z × p and ϕ ∈ Hom( Z × p , Z p ) , [ x ] · [ ϕ ] = ϕ ( x ) · ǫ. This gives in particular π R = Z p [ ǫ ] /ǫ , which is part of Theorem 1.15. We can also deduce anotherpart: Corollary 1.24.
Consider again j Z p as a map R → ω K (1) . We have ( π j Z p )(1) = [ S ] , ( π j Z p )( ǫ ) = 2[ S ] . roof. If we K (1) -localize Corollary 1.2 we see that π j Z p sends to [ S ] and π j Z p sends [ u ] to [ u ] for u ∈ Z × p . But combining Proposition 1.8 and Proposition 1.23 Relation 2 shows that this secondstatement implies that π j Z p sends ǫ to · [ S ] .Next we turn to Theorem 1.16, for which we need to analyze d K (1) K( Q p ) . Following [Mi], thiscan also be done using a Thomason-style descent spectral sequence. Namely, as we will see in moredetail later, we can consider a version of the Brown-Comenentz dual of d K (1) K (Section 2.3) and itsassociated descent spectral sequence (Corollary 2.30, plus Theorem 2.35 to identify the E -page). If ( − ) denotes Pontryagin duality, then this takes the form E i,j = H j − i (BG Q p ; Q p / Z p ( j/ ⇒ (cid:0) π − i d K (1) K( Q p ) (cid:1) . Dualizing back again, we deduce:
Lemma 1.25.
For n = 2 j + 1 odd, there is a canonical identification π n d K (1) R = H (BG Q p ; Q p / Z p ( − j )) . For n = 2 j even, there is a canonical short exact sequence → H (BG Q p ; Q p / Z p ( − j )) → π n d K (1) R → H (BG Q p ; Q p / Z p ( − j + 1)) → . By Corollary 1.24, we know that j Z p sends ǫ to · [ S ] . Thus to prove Theorem 1.16, it suffices toshow: Proposition 1.26.
Let c ∈ [ R, ω K (1) ] = π d K (1) R . The following are equivalent:1. ( π c )( ǫ ) = 2 · [ S ] .2. c projects to the usual generator inv ∈ H (BG Q p ; Q p / Z p (1)) in the short exact sequence ofLemma 1.25.Under these conditions, the multiplication by c map R → d K (1) R is an equivalence.Proof. We use that the the Brown-Comenentz dual of d K (1) K( − ) is a module over L K (1) K( − ) , sothat the descent spectral sequence for the former is a module over the descent spectral sequence for thelatter. We claim that on the E -page, this module structure is induced by the cup product. To verifythis, we need to see that the induced pairing on homotopy group sheaves is the tautological productstructure. This follows from Corollary 2.18, which gives that the homotopy groups of the stalks of theformer identify with those of the latter tensored with Q p / Z p , compatibly with the module structure andwith all cospecialization maps. Thus by Lemma 2.33 (which is actually trivial in the case of a field), weget the same identification on homotopy group sheaves, giving the claim.We deduce that the multiplication by c map π n R → π n d K (1) R respects the filtrations of Lemmas 1.21 and 1.25, and on the associated graded pieces is given by themap H i (BG Q p ; Z p ( j )) → H − i ( BG Q p ; Q p / Z p (1 − j )) of capping with the image of c in H (BG Q p ; Q p / Z p (1)) .12f 2 holds, then local duality implies that this capping with the image of c is an isomorphism,hence multiplication by c is an equivalence R → d K (1) R . To finish, we need to show 1 ⇔
2. Fixa c ′ satisfying 2. Given the isomorphism · c ′ : π R → π d K (1) R we just proved, it suffices to showthat c ′ sends ǫ to · [ S ] (assuming this, write an arbitrary c as ( a + bǫ ) c ′ ; then c projects to a · inv in H (BG Q p ; Q p / Z p (1)) and sends ǫ to a · [ S ] , proving the claim). Note by multiplicativity andthe description of ǫ in the descent spectral sequence (Theorem 1.23) that c ′ · ǫ is the generator ofthe subgroup Z p = H (BG Q p ; Q p / Z p ) ⊂ π d K (1) R . On the other hand, by the construction of thespectral sequence and the identification of the stalks in Corollary 2.18, this generator corresponds tothe map K( Q p ) → ω K (1) given as the composition K( Q p ) → K( Q p ) → ω K (1) where the second map sends to · [ S ] . The claim follows.This proves Theorem 1.16. We can also finish the proof Theorem 1.15: what’s missing is to showthat an element of [ R, ω K (1) ] is determined by its effect on π . But this is obvious from the fact that j Z p and j Z p · ǫ form a basis, and the description of their effect on π (Corollary 1.24 plus ǫ = 0 ).Finally, we turn to Theorem 1.18. First, note that since π − R ≃ Z p ⊕ Z p is a free Z p -module, Extsout of it vanish, and thus a class in the Z p -Anderson dual c ∈ π d Z p R = [ R, I Z p ] is uniquely determined by its effect on π , which will be a homomorphism π R → Z p . One can analyze d Z p R exactly as we analyzed d K (1) R above, and the analog of Proposition 1.26 impliesthe following, which in fact was proved in [BlM] Theorem 1.2: Proposition 1.27.
Let c ∈ d Z p R correspond to the homomorphism π R → Z p sending to and ǫ to . Then the multiplication by c map R → d Z p R is an equivalence. To analyze ω T C , we need the following.
Lemma 1.28.
The map
TC( Z ) b p → L K (1) TC( Z ) = R has the following properties on π ∗ :1. It is an isomorphism in degrees ∗ ≥ .2. In degree ∗ = 1 , it is injective with image Z × p /µ p − ⊕ (1.23).3. In degree ∗ = 0 , it is injective with image Z p ⊕ .4. In degree ∗ = − , it is injective with image ⊕ Z p .5. In degrees ∗ ≤ − , the source is . roof. By McCarthy’s theorem (2.5) and p -adic continuity (2.6), it follows that K( Z p ) b p → TC( Z p ) b p is an isomorphism in degrees ≥ and ≤ − , and in degree − we have that π − TC( Z p ) b p is a free Z p -module on f . Then all the claims except claim 1 follow from Proposition 1.23.For claim 1, one can either refer to the calculation of the homotopy type of the spectrum TC( Z ) b p given in [BoM], [Ts] for p > and [Ro] for p = 2 to directly verify the claim, or else one can transferthe claim to K( Q p ) b p and use the norm residue isomorphism theorem plus the fact that Q p has Galoiscohomological dimension (cf [W] VI.4).It follows that π ω T C is a free Z p -module on the image of [ p ] , that π ω T C is a free Z p -moduleon the image of ǫ , and that π − ω T C identifies with the free Z p -module of rank one Hom( Z × p , Z p ) . Inparticular there is a unique class class in d Z p ω T C = [ ω T C , I Z p ] sending ǫ to . We can formally extendthis class to a unique up to homotopy TC( Z ) b p -module map α : ω T C → d Z p TC( Z ) b p sending ǫ to the tautological generator of π d Z p TC( Z ) b p = Hom( Z p , Z p ) given by the identity. Byconstruction one sees that there is a commutative square R · c / / (cid:15) (cid:15) d Z p R (cid:15) (cid:15) ω T C α / / d Z p TC( Z ) b p Comparing Lemma 1.28 parts 1 and 5 with Proposition 1.27, we deduce that α is an isomorphism indegrees ≥ and ≤ − . It is an isomorphism in degree by construction. In degree it is an isomorphismby the identity [ p ] · f = ǫ . In degree − it is an isomorphism by the identity [ ϕ ] · [ u ] = ϕ ( u ) · ǫ (Proposition1.23). This proves Theorem 1.18. Again in this section we will fix a prime p . Using the duality functors d K (1) and d T C and the natural transformation n from the previous section,we can define our chosen p -adic dual to etale K-theory. Definition 2.1.
Let
P ∈
PerfCat Z be a small idempotent-complete Z -linear stable ∞ -category. Definethe ( p -adic) Selmer K-homology spectrum of P , denoted dK Sel ( P ) , to be the pushout of d K (1) K( P ) d K (1) tr ←− d K (1) TC( P ) n −→ d T C
TC( P ) . Here tr : K( P ) → TC( P ) is the cyclotomic trace map between the localizing invariants K and TC (asin [BGT] Section 10).For a ring (or DGA) A we set dK Sel ( A ) = dK Sel (Perf( A )) , and similarly for a qcqs algebraic space (or derived algebraic space) X over Z we set dK Sel ( X ) = dK Sel (Perf( X )) . emark 2.2. The version of algebraic K-theory which satisfies localization is the nonconnective algebraicK-theory. On the other hand we have a convention that K( A ) for a ring (or connective DGA) A denotes the connective algebraic K-theory of A . Thus, according to our conventions K( A ) is theconnective cover of K(Perf( A )) . Note, however, that K-theory only contributes to dK Sel through its K (1) -localization, so this distinction doesn’t affect the above definition. With TC there is no ambiguity: TC( A ) = TC(Perf( A )) for any DGA A . Even when A is connective TC( A ) is not necessarily connective,but it’s close: π n (TC( A ) b p ) = 0 for n < − . Remark 2.3.
By construction, dK Sel is a functor
PerfCat op Z → Sp which satisfies localization, i.e. it sends fiber-cofiber sequences to fiber-cofiber sequences, i.e. if M → N → P is a Verdier quotient sequence up to idempotent completion, then dK Sel ( P ) → dK Sel ( N ) → dK Sel ( M ) is a fiber sequence of spectra. In particular, by the standard Thomason-Trobaugh argument ([Ta] Section2, [CMNN] Appendix A), X dK Sel ( X ) is a Nisnevich co-sheaf on qcqs derived algebraic spaces over Z . But one of the main points is that it is even an etale co-sheaf, see Section 2.3. Remark 2.4.
For concreteness, let us posit that our derived algebraic spaces are locally modeled on
Spec of a connective E ∞ -algebra over Z . Most of the statements hold in the weaker context wherethe local models are connective E -algebras, and some hold with local models just quasi-commutative E -algebras (or DGAs). However, none of our examples of interest will be derived, so the reader canignore these kinds of technicalities. There is a strong dichotomy between the behavior of Selmer K-homology at p and away from p :away from p only the term d K (1) K( P ) contributes, while at p only the term d T C
TC( P ) does. Beforesaying this more precisely, we need to recall some facts about K-theory and TC-theory, starting with thefundamental result of McCarthy ([Mc]): Theorem 2.5.
Define the functor F as the fiber of tr : K → TC . Let A → B be a morphism ofconnective DGAs over Z such that π A → π B is surjective with nilpotent kernel. Then F( A ) ∼ → F( B ) . Next we give a slight amplification of some results from [GH2] on p -adic continuity in K-theory andTC-theory: Theorem 2.6.
Let A be a connective DGA over Z , and for all n ∈ N let A ⊗ Z Z /p n Z denote thederived tensor product of A with Z /p n Z .1. For all ∗ ∈ Z , the map π ∗ (TC( A ) /p ) → “ lim ←− n ” π ∗ (TC( A ⊗ Z Z /p n Z ) /p ) is an isomorphism in the Pro-category of abelian groups. (This is even true “uniformly in A ”,meaning it is an isomorphism in the Pro-category of functors from connective DGAs to abeliangroups.) . Let R = π A , and suppose the following:(a) A is quasi-commutative, i.e. the action of R on π ∗ A is commutative (ensuring a reasonabletheory of Zariski localization on A , see [L1] 7.2.3);(b) ( R, pR ) is a henselian pair;(c) R has bounded p -torsion ( ∃ N ∈ N such that R [ p N +1 ] = R [ p N ] ; this holds if R is noetherian,or R lives over either Z [1 /p ] or Z /p n Z for some n , or R ( p ) is flat over Z ( p ) ).Then if π Zar ∗ denotes the functor of taking homotopy groups and then Zariski sheafifying over Spec( A ) , the map π Zar ∗ (K( − ) /p ) → “ lim ←− n ” π Zar ∗ (K( − ⊗ Z Z /p n Z ) /p ) is an isomorphism in the Pro-category of abelian group sheaves on Spec( A ) Zar for each ∗ ∈ Z .(This is also true uniformly in A provided we fix N .)Proof. For 1, by the standard devissage (cf [GH2] Section 3) it suffices to show that A → “ lim ←− n ” A ⊗ Z Z /p n Z , is a pro-equivalence in Mod( H Z ) after applying ( − ) /p = − ⊗ Z Z /p Z . But the fiber of this map is thepro-system represented by ( . . . A p → A p → A ) . Tensoring with Z /p Z makes all the maps nullhomotopic,hence the system is pro-zero.For 2, [GH2] Remark 1.3.2 (based on the argument of [Sus] Section 3) gives π Zar ∗ K( − ) /p ∼ −→ “ lim ←− n ” π Zar ∗ K( − / ( p n )) /p, where here the dash runs over Zariski localizations of R and by − / ( p n ) we mean the ordinary moddingout by the ideal generated by p n . On the other hand, since R [ p N +1 ] = R [ p N ] (and therefore also thesame for any Zariski localization of R ), an argument similar to that of claim 1 gives π ∗ TC( − ) /p ∼ −→ “ lim ←− n ” π ∗ TC( − / ( p n )) /p where again ( − ) runs over Zariski localizations of R . Then looking at the compatible comparison maps A → R and A ⊗ Z Z /p n Z → R/ ( p n ) , McCarthy’s theorem 2.5 lets us reduce claim 2 to claim 1. Remark 2.7.
Using the Milnor sequence, it’s easy to see that 1 implies
TC( A ) b p ∼ −→ lim ←− n TC( A ⊗ Z Z /p n Z ) b p . Similarly, since K( − ) satisfies Zariski descent, the conclusion of 2 implies that K( A ) b p ∼ −→ lim ←− n K( A ⊗ Z Z /p n Z ) b p if we add the hypothesis that the topological space Spec( A ) is e.g. noetherianof finite Krull dimension, so that the Zariski descent spectral sequences converge to the correct answer. The following is a version of [GH2] Theorem B. [GH2] requires that R be p -torsionfree, but the proof only requires information of the deep enough p -power congruencesubgroups, and these are the same for R and the p -torsion-free ring R/R [ p N ] . (Thanks to Akhil Mathew for this remark.) heorem 2.8. Let A be a connective DGA over Z which is quasi-commutative (i.e. the action of π A on π ∗ A is commutative; this ensures a reasonable theory of etale A -algebras, [L1] 7.5.1), and let ( − ) h denote the functor of henselization at p . Then:1. TC( A ) /p → TC( A h ) /p is an equivalence.2. Consider the cyclotomic trace map tr : K( A h ) → TC( A h ) . Assume the following:(a) π ( A ⊗ Z F p ) is a nilpotent thickening of a regular ring;(b) π A has bounded p -torsion.Then if π et ∗ denotes the functor of taking homotopy groups and then sheafifying over Spec( A ) et ,the map π et ∗ (tr /p ) : π et ∗ K( A h ) /p → π et ∗ TC( A h ) /p is an isomorphism.Proof. For the first claim, note A ⊗ Z F p ∼ → A h ⊗ Z F p , so TC( A ) /p ∼ → TC( A h ) /p by p -adic continuity(Theorem 2.6).For the second claim, consider the fiber F( A h ) /p of tr /p , and let x ∈ π ∗ (F( A h ) /p ) . It suffices toshow that x vanishes locally on Spec( A ) et . Since even ( − ) h vanishes over the open Spec( A [1 /p ]) , itsuffices to see that x vanishes etale-locally around every characteristic p point of Spec( A ) .First assume the claim for A ′ := π ( A ⊗ Z F p ) red replacing A . Then there is an etale cover { A ′ i } of A ′ such that x vanishes in each F( A ′ i ) /p . Lift each A ′ i arbitrarily to an etale A -algebra A i ; theneach characteristic p point is covered by some A i . By p -adic continuity for TC and K we get p -adiccontinuity for F, so the homotopy groups of the fiber of F( A hi ) /p → lim ←− n F( A i ⊗ Z Z /p n Z ) /p vanishZariski-locally on ( A i ) h , and therefore vanish etale-locally at each characteristic p point of A i . Thusit suffices to see that the image of x in lim ←− n F( A i ⊗ Z Z /p n Z ) /p vanishes. But McCarthy’s theoremimplies that this limit diagram is constant and even further has value F( A ′ i ) /p , in which x vanishes byassumption, whence the claim.Thus we have reduced to the case where A is a regular F p -algebra, and we want to see that tr( A ) /p : K( A ) /p → TC( A ) /p is a π et ∗ -isomorphism. Since F p is perfect, A is necessarily regular over F p . If furthermore A is smooth over F p , then this is proved in [GH] Theorem 4.2.2. By Popescu’stheorem every regular F p -algebra is a filtered colimit of smooth ones, so it suffices to see that both K( − ) /p and TC( − ) /p commute with filtered colimits of smooth F p -algebras. For K( − ) /p this isautomatic because K( − ) commutes with all filtered colimits. For TC( − ) /p , recall that TC( − ) is theinverse limit of a tower TC n ( − ) where each TC n ( − ) commutes with filtered colimits. But [H] TheoremB implies that on each homotopy group, the limit diagram TC( − ) /p → lim ←− n TC n ( − ) /p is pro-constantin the category of functors SmAlg F p → Ab . Thus it remains a limit diagram after passing to filteredcolimits, proving the claim. Remark 2.9.
We crucially used the theorem from [GH] that tr : K( A ) /p → TC( A ) /p is an isomorphismon π et ∗ for a smooth F p -algebra A . This is not an easy theorem — it relies on a number of nontrivialprecursors in both K-theory and in TC-theory. In any case, if this theorem is true for an arbitrary F p -algebra A , then the regularity hypothesis on π ( − ⊗ Z F p ) can be dropped here and everywhere elseit appears in this paper. emma 2.10. Recall that F denotes the fiber of tr : K → TC . Let P ∈
PerfCat Z .1. Suppose that TC( P ) b p = 0 . Then d K (1) K( P ) ∼ −→ dK Sel ( P ) .2. Suppose that L K (1) F( P ) = 0 . Then d T C
TC( P ) ∼ −→ dK Sel ( P ) .Proof. Both claims are immediate from the pushout square defining dK Sel . Proposition 2.11.
Let
P ∈
PerfCat Z .1. We have TC( P ) b p = 0 , and hence d K (1) K( P ) ∼ −→ dK Sel ( P ) , whenever P is Z [1 /p ] -linear.2. We have L K (1) F( P ) = 0 , and hence d T C
TC( P ) ∼ −→ dK Sel ( P ) , whenever one of the followingholds:(a) P admits an F p -linear structure;(b) P = Perf( A ) for a connective DGA A over Z /p n Z , or P = Perf( X ) for a qcqs derivedalgebraic space X over Z /p n Z .(c) P = Perf( A ) for a quasi-commutative connective DGA A over Z such that ( R, pR ) ishenselian, R has bounded p -torsion, and either R is local or Spec( R ) is noetherian of finiteKrull dimension ( R = π A ).Proof. For 1, if P is Z [1 /p ] -linear, then TC( P ) b p is a module over TC( Z [1 /p ]) b p = TC( Z [1 /p ] b p ) b p = 0 ,hence is zero.For 2(a), If P is F p -linear, then L K (1) F ( P ) b p is a module over L K (1) K( F p ) = L K (1) H Z p = 0 , henceis zero.For 2(b), if P is Z /p n Z -linear and the conclusion of McCarthy’s theorem holds for P → P ⊗ Z /p n Z F p ,then we deduce L K (1) F ( P ) = 0 from the F p -linear case. By McCarthy’s theorem this holds if P =Perf( A ) for a connective DGA A . By localization it then also holds if P = Perf( X ) for a qcqs derivedalgebraic space X .2(c) follows from the p -adic continuity (Theorem 2.6) and claim 2(b). Remark 2.12.
The upshot is that away from p , the theory dK Sel is controlled by L K (1) K , whereasformally completed at p (or even henselian at p ), the theory dK Sel is controlled by TC b p . Remark 2.13.
In contrast, when P is genuinely global and mixed characteristic, then all three of theterms contribute to the pushout defining dK Sel ( P ) , in a manner reminiscent of the definition of Selmergroups. Remark 2.14.
When P = Perf( X ) for an algebraic space X of finite type over Z with ( X ⊗ Z F p ) red regular, one can think that the term d K (1) K( X ) is associated to the etale theory of the algebraic space X Z [1 /p ] , that the term d K (1) TC( X ) is associated to the the etale theory of the rigid-analytic space X an Q p , and that the term d T C
TC( X ) is associated to the de Rham theory of the formal algebraic space X b p . The gluing together of these terms has something to do with p -adic Hodge theory. For the purposes of this paper, the main thing we will want to prove about dK Sel is the following,which will be proved in Section 2.5. 18 heorem 2.15.
Let X be a locally noetherian derived algebraic space over Z , and suppose ( X ⊗ Z F p ) red is regular.Then there is a natural map e X : H ( X et ; Z p ) → π dK Sel ( X ) , where H ( X et ; Z p ) stands for the pro- p -abelian completion of the etale fudamental group of X , orequivalently the Pontryagin dual of H ( X et ; Q p / Z p ) .The map e X is an isomorphism if one of the following conditions holds:1. X has (mod p ) etale cohomological dimension ≤ and X ⊗ F p has (mod p ) etale cohomologicaldimension ≤ .2. X = Spec( F ) for a field F of virtual (mod p ) Galois cohomological dimension ≤ . Remark 2.16.
More specifically, this map e X will be an edge map in a “co-descent” spectral sequencefor dK Sel . Especially when X lives over Z [1 /p ] , it behaves similarly to the comparison map H ( X ; Z ) → K ( X ) between singular homology and K-homology in topology, which is an edge map in the Atiyah-Hirzebruch spectral sequence. The above theorem will follow from two others: an etale co-descent result for dK Sel , and a (partial)calculation of the co-stalks of dK Sel . We start with the latter.
To study the “etale co-stalks of dK Sel ” means to study the values dK Sel ( A ) when A is a strictly henselianlocal ring (or, in the derived case, a quasi-commutative connective DGA over Z whose π is a strictlyhenselian local ring; we will also call such an object a strictly henselian local ring). We start with thecase of residue characteristic = p . The following is a standard consequence of Gabber-Suslin rigidity([G], [Sus]). Proposition 2.17.
Let A be a strictly henselian local ring with residue characteristic = p . Then:1. Let Z p (1) = T p ( A × ) denote the Tate module of p -power roots of unity in A (or π A ). There arenatural isomorphisms Z p (1) ≃ π K( A ) b p ∼ → π L K (1) K( A ) , and π ∗ K( A ) b p (resp. π ∗ L K (1) K( A ) ) is a polynomial algebra (resp. Laurent polynomial algebra)over Z p on the invertible Z p -module Z p (1) placed in degree 2.2. The above identifications are functorial under all ring homomorphisms between such A ’s (not justthe local ones).Proof. Since A is local with residue characteristic = p , it follows that p is a unit in A . We deduce K( A ) b p ∼ → K( π A ) b p , either by comparison either with TC using McCarthy’s theorem (Theorem 2.5) orby comparison with the cohomology of BGL( − ) using the group-completion theorem. Hence we canassume A is an ordinary commutative ring. (This also explains why π ∗ K ( A ) b p should have a productstructure even when A is only a DGA.)Then this is a standard consequence of Gabber-Suslin rigidity ([G], [Sus]), which gives K( A ) b p ≃ ku b p ,the equivalence being induced by a zig-zag of commutative ring homomorphisms between A and C . Letus just recall that the identification Z p (1) = π K( A ) b p arises from the short exact sequence calculatingthe homotopy group of a p -completion ([Bo] Proposition 2.5) and the identification A × = π K( A ) .19 orollary 2.18. With A as in the previous proposition, there is a unique class j A ∈ π dK Sel ( A ) = π d K (1) K( A ) = [K( A ) , ω K (1) ] such that ( π j A )(1) = 2 · [ S ] ∈ π ω K (1) . This class j A is functorial under all ring homomorphisms, and multiplication by j A gives an equivalence L K (1) K( A ) ∼ → dK Sel ( A ) . In particular there are canonical functorial isomorphisms π j dK Sel ( A ) ≃ Hom( Z p ( − j ) , Z p ) and π j +1 dK Sel ( A ) = 0 for all j ∈ Z .Proof. If we choose an equivalence K( A ) b p ≃ ku b p of ring spectra and use the equivalence log : ω K (1) ≃ Σ L K (1) S , we get [K( A ) , ω K (1) ] ≃ [ KU, Σ L K (1) S ] . Then the existence and uniqueness of a class j A ∈ [K( A ) , ω K (1) ] such that ( π j A )(1) = 2 · [ S ] , as well as the property · j A : L K (1) K( A ) ≃ d K (1) K( A ) ,follow from an easy calculation in the K (1) -local category using the KU -based Adams spectral sequence(cf [HM] Lemma 8.16), and the functoriality follows from the uniqueness. Remark 2.19.
Just as with the j Z p and j R from earlier (Remark 1.3), when A is an ordinary commutativering this homotopy class of maps j A ∈ [ L K (1) K( A ) , ω K (1) ] can be identified with the K (1) -localizationof a certain canonical map of spectra J etA : K( A ) → Pic( S b p ) . This is the etale J-homomorphism of [Q2]and [F], which sends the class of a finite free A -module M to the stable etale p -adic homotopy type ofthe cofiber M / ( M − Spec( A ) ) , where M = Spec(Sym A ( M ∨ )) is the vector bundle corresponding to M . One sees that even this unlocalized J etA is well-defined and functorial in A using standard facts inetale homotopy theory. Furthermore J et C defined in this way is naturally homotopic to the composition K( C ) → K( R ) J R −→ Pic( S b p ) , by the comparison between etale homotopy and classical homotopy over C .In these terms it is geometrically clear why j A should send to · [ S ] : when M is the unit A -module, M / ( M − Spec( A ) ) = A / ( A − is a -sphere in etale homotopy theory. The characteristic p counterpart to Proposition 2.17 is the following: Proposition 2.20.
Let A be a strictly henselian local ring of residue characteristic p . Suppose fur-thermore that π ( A ⊗ Z F p ) is a nilpotent thickening of a regular ring and π A has bounded p -torsion.Then:1. The trace map K( A ) b p → TC( A ) b p is an equivalence.2. We have π TC( A ) b p = Z p and π ∗ TC( A ) b p = 0 for ∗ < .Proof. Since k has characteristic p and A is henselian, the pair ( π A, pπ A ) is henselian. Thus Theorem2.8 shows that K( A ) → TC( A ) is an isomorphism on (mod p ) homotopy after sheafification over Spec( A ) et . But A is strictly henselian, so this means it is already an isomorphism on (mod p ) homotopy,hence on p -completion, giving claim 1. Claim 2 is immediate from claim 1. Corollary 2.21.
Let A be as in the previous proposition. Then there is a unique class c A ∈ π dK Sel ( A ) = π d T C
TC( A ) = [TC( A ) , ω T C ] TC( Z ) such that ( π c A )(1) = ǫ ∈ π ω T C . Furthermore this class c A is functorial in A for all ring homomorphisms, π dK Sel ( A ) is a free Z p -moduleon c A , and π n dK Sel ( A ) = 0 for all n > . roof. From Theorem 1.18 we have that ω T C identifies with the p -adic Anderson dual of TC( Z ) b p . Thusfor a TC( Z ) b p -module M and n ∈ Z , there is a short exact sequence → Ext Z p ( π n − M, π ω T C ) → π − n d T C M → Hom Z p ( π n M, π ω T C ) → where the quotient map records the effect of a map Σ − n M → ω T C on π . Furthermore, π ω T C is afree Z p -module on ǫ ∈ π ω T C . The claim follows.
Remark 2.22.
The π part of the corollary also holds for A = Z p . Indeed, π TC( Z p ) b p = Z p , whereas π − TC( Z p ) b p is free of rank over Z p and hence Ext’s out of it vanish. Thus the same argument works. Now we put the two cases (residue characteristic p vs. = p ) together to gain more global informationabout π dK Sel . For this the key is the following, which is already implicit in the proof of Proposition1.26:
Lemma 2.23.
Let Q p denote an algebraic closure of Q p . The composition R = L K (1) K( Z p ) → L K (1) K( Q p ) j Q p −→ ω K (1) is homotopic to j Z p · ǫ .Proof. By Theorem 1.15, we can check this by seeing that the two maps in question have the sameimage on , ǫ ∈ π R . Again by 1.15, j Z p · ǫ kills ǫ and sends to · [ S ] . The above composition kills ǫ because π L K (1) K( Q p ) ≃ π KU b p = 0 , and it sends to · [ S ] by the construction of j Q p (2.18). Theorem 2.24.
Let A denote a strictly henselian local ring. If the residue characteristic equals p ,assume that π A has bounded p -torsion and that π ( A ⊗ F p ) is a nilpotent thickening of a regular ring.1. The isomorphism π dK Sel ( A ) ≃ Z p , given by the class j A in residue characteristic = p and c A in residue characteristic p , is functorialunder all ring homomorphisms between such A ’s, where Z p means the constant functor with value Z p . Furthermore, π dK Sel ( A ) = 0 .
2. If we require A to have residue characteristic = p , then functorially π n dK Sel ( A ) ≃ Hom Z p ( Z p ( − n ) , Z p ) , and π n +1 dK Sel ( A ) = 0 . Proof.
All of the claims are clear from Corollaries 2.18 and 2.21 except that the isomorphism π dK Sel ( A ) ≃ Z p is functorial under homomorphisms A → B when A and B have different residue characteristics.In this case A has residue characteristic p and B has residue characteristic = p . But since we knowfunctoriality in residue characteristic p and = p separately, and the functor π dK Sel ( − ) lands in isomor-phisms in both cases, we can use zig-zags of maps to reduce to where A = W ( F p ) and B = Frac( A ) .By Remark 2.22 we can even take A = Z p instead, so B = Q p . In this case, we find after unwindingthe definitions that the claim is equivalent to saying that the composition R → L K (1) K( Q p ) j Q p −→ ω K (1) is homotopic to ǫ · j Z p up to a Z p -multiple of j Z p . But Lemma 2.23 says that it is exactly homotopic to ǫ · j Z p . 21 .3 Etale co-descent for Selmer K-homology It is a bit awkward to talk about co-sheaves and co-descent, so following [Mi] we will use a trick withduality to reduce to sheaves instead. We start with the observation that dK Sel canonically takes valuesin a more refined ∞ -category than just Sp : Definition 2.25.
Let Sp π ⊂ Sp denote the thick subcategory consisting of those spectra all of whosehomotopy groups are finite, and let Pro(Sp π ) denote its pro-category. Since Sp π is stable, it is co-tensored over finite spectra; therefore Pro(Sp π ) is co-tensored over Ind of finite spectra, which is allspectra. So map( X, P ) ∈ Pro(Sp π ) naturally for X ∈ Sp and P ∈ Pro(Sp π ) .If ω is a p -complete spectrum such that ω/p ∈ Sp π , then we can canonically lift ω to an object of Pro(Sp π ) , namely ω = “ lim ←− n ” ω/p n , and thereby also map( X, ω ) ∈ Pro(Sp π ) functorially in X ∈ Sp . Explicitly, if we write X = lim −→ i X i with X i finite, then map( X, ω ) = “ lim ←− i,n ” map( X i , ω/p n ) . A similar definition works if ω is module over a ring spectrum, e.g. ω T C or R over TC( Z ) . Sincethe homotopy of both ω K (1) /p and ω T C /p is finite in all degrees, we can thus canonically view d K (1) and d T C , and therefore dK Sel , as taking values in
Pro(Sp π ) . From now on we do this implicitly. Remark 2.26.
The standard t -structure on Sp restricts to a t -structure on Sp π whose heart is theabelian category of finite abelian groups. It follows that Pro(Sp π ) gets a t -structure whose heart isPro of the category of finite abelian groups, i.e. the category of profinite abelian groups. In particulareach homotopy group π ∗ dK Sel ( P ) is canonically a profinite abelian group. The isomorphisms of theprevious section all promote to isomorphisms of profinite abelian groups (necessarily, since the structureof profinite group on Z p is unique). Let ( − ) denote the Pontryagin duality Hom c ( − ; Q / Z ) , implementing an anti-equivalence betweentorsion abelian groups and profinite abelian groups. Recall that ( − ) admits a lift to spectra: there isa contravariant involutive self-equivalence X X on Sp π , called Brown-Comenentz duality, such that there are canonical isomorphisms π n ( X ) ≃ ( π − n X ) for all n ∈ Z . More precisely, we have X := map( X, I Q / Z ) where I Q / Z is a certain spectrum with π I Q / Z = Q / Z , and the canonical isomorphism π n ( X ) ≃ ( π − n X ) comes from the induced pairing π n X ⊗ π − n X → π I Q / Z . If we apply Brown-Comenentz duality termwise to an object X ∈ Pro(Sp π ) , we get an object of Ind(Sp π ) . But we may as well take the colimit of that Ind-system in spectra to get a spectrum whichwe also denote X , whose homotopy groups are canonically π n ( X ) = ( π − n X ) . X as a plain abelian group is torsion, and it determines andis uniquely determined by the homotopy of X ∈ Pro(Sp π ) as a profinite abelian group, by Pontryaginduality.The trick from [Mi] it to consider the presheaf X dK Sel ( X ) of honest spectra instead of theco-presheaf X dK Sel ( X ) of Pro- π ∗ -finite spectra; by the above discussion, they carry the sameinformation on homotopy groups anyway. Remark 2.27.
The functor d K (1) ( − ) from spectra to spectra preserves colimits. Therefore we havea canonical equivalence d K (1) ( X ) ≃ X ∧ ( d K (1) S ) . Thus d K (1) ( − ) is equivalent to smashing with a certain p -power torsion L -local spectrum, namely ( d K (1) S ) . (The spectrum is also W ∧ M Q p / Z p , where W is the K (1) -local “fake S ” of Remark 1.6and [HM] Lemma 7.3).Similarly, since d T C is equivalent to p -adic Anderson duality (Theorem 1.18), the functor d T C ( − ) is equivalent to smashing with a Moore spectrum M Q p / Z p . With all this as background, we have:
Theorem 2.28.
The contravariant functor X dK Sel ( X ) from qcqs derived algebraic spaces to spectra is an etale sheaf.If we restrict to X which are virtually of finite (mod p ) etale cohomological dimension and whichhave bounded p -torsion in π of their structure sheaf, then it is even an etale Postnikov sheaf (it is theinverse limit of its etale-sheafified Postnikov tower).Proof. It suffices to prove the claims separately for ( d T C
TC) , ( d K (1) TC) , and ( d K (1) K) . Forthe last two, since they satisfy localization and are valued in L -local spectra, [CMNN] Theorem A.14implies that they are automatically etale sheaves, and in [CM] it will be seen that under the bounded p -torsion and finite virtual dimension hypotheses they are even automatically etale Postnikov sheaves.(Both works use some of Thomason’s methods from [Th1]; one can also deduce the claim in a majorityof cases of interest from Thomason’s work itself.) Thus it suffices to consider d T C TC .Since ( d T C ) is equivalent to smashing with a Moore spectrum M Q p / Z p (Remark 2.27), it has t -amplitude ≤ . It therefore suffices to see that TC( − ) is an etale Postnikov sheaf on qcqs derivedalgebraic spaces. This is basically [GH] Corollary 3.2.2 in a slightly different language; a proof in thecurrent language, following the same idea as [GH], will be recorded in [CM]. Remark 2.29.
There is also a more general result: if P is tensored over Perf( X ) , then ( U et −→ X ) dK Sel ( P ⊗
Perf( X ) Perf( U )) is an etale sheaf or etale Postnikov sheaf under the same hypotheses, withthe same proof. Corollary 2.30.
In the situation above when F = dK Sel ( − ) is an etale Postnikov sheaf, we get aconditionally convergent “descent” spectral sequence E i,j = H j − i ( X et ; π etj F ) ⇒ π i ( F ( X )) . Here π etj ( − ) denotes the etale sheafification of π j ( − ) . We have indexed this spectral sequence so that d goes from E i,j to E i − ,j +1 and d from E i,j to E i − ,j +2 and so on. .4 Sheafified homotopy groups and the edge map We would like to dualize Theorem 2.24 to obtain information on the stalks of π j (dK Sel ( − ) ) =( π − j dK Sel ( − )) . But for this we need to be sure that dK Sel ( − ) commutes with the appropriatecolimits, so that its stalks are indeed the same as its values on strictly henselian local rings. Here is thelemma which assures this: Lemma 2.31.
Let X be a derived algebraic space such that classical algebraic space underlying ( X ⊗ Z F p ) is locally a nilpotent thickening of something regular, and X locally has bounded p -torsion in itsstructure sheaf on π . If Y = lim ←− i ∈ I U i is a filtered limit of affine schemes etale over X defining a strict henselization Y of X , then (dK Sel ( − )) sends this limit diagram to a colimit diagram.Proof. Both functors ( d K (1) ) and ( d T C ) are equivalent to smashing with some p -primary torsionspectrum (Remark 2.27), so it suffices to see that K( − ) /p and TC( − ) /p commute with the filteredcolimit in question. For K( − ) /p this is automatic. For TC( − ) /p , recall the natural transformation K(( − ) h ) /p → TC( − ) /p from Theorem 2.8. The functor K(( − ) h ) preserves filtered colimits, so it suffices to see that the fiber G of this natural transformation preserves the filtered colimit in question. But Theorem 2.8 implies boththat the filtered colimit of the G ( U i ) is zero, and that G ( Y ) is zero.Combining with Theorem 2.24, we get: Proposition 2.32.
For X a derived algebraic space such that the classical algebraic space underlying ( X ⊗ Z F p ) is locally a nilpotent thickening of something regular and X locally has bounded p -torsion in π of its structure sheaf, we have the following information on the stalks of π ∗ (dK Sel ( − ) ) over X et :1. The stalks of π (dK Sel ( − ) ) are canonically Q p / Z p , compatibly with all co-specialization maps.2. The stalks of π n (dK Sel ( − ) ) at points of characteristic = p are canonically Q p / Z p ( n ) , compat-ibly with all co-specialization maps between characteristc = p points, for all n ∈ Z ;3. The stalks of π n +1 (dK Sel ( − ) ) at points of characteristic = p are zero, for all n ∈ Z .4. The stalks of π j (dK Sel ( − ) ) at points of characteristic p are zero for j < . To go from knowledge of the stalks to knowledge of the sheaves themselves, we use the followinglemma.
Lemma 2.33.
Let X be an algebraic space whose underlying topological space is “locally path con-nected” in the sense that it has a basis of open subsets whose category of points is connected. (Notethat locally noetherian implies locally path connected). Denote by p ∗ : Sh( X ) → PSh(pt X ) the functor from etale sheaves (of sets) on X to presheaves on the category of points of the etale toposof X , given by taking stalks. uppose F ∈
Sh( X ) is such that p ∗ F sends every morphism in pt X to an isomorphism (that is,every co-specialization map for F is an isomorphism). Then for any G ∈
Sh( X ) , the map Hom ( G , F ) → Hom ( p ∗ G , p ∗ F ) is a bijection.Proof. The functor p ∗ has a right adjoint p ∗ , which can be determined as follows: let F ∈ PSh(pt X ) ,and let U → X be be etale. Then we need Hom ( U, p ∗ F ) = Hom ( p ∗ U, F ) , which gives us the formula ( p ∗ F )( U ) = lim ←− pt U F. By adjunction, the claim holds if and only if the unit map
F → p ∗ p ∗ F is an isomorphism. Since X et has enough points, the functor p ∗ detects isomorphisms. Therefore by adjunction identities it sufficesto see that if F ∈ PSh(pt X ) sends every morphism in pt X to an isomorphism, then the counit p ∗ p ∗ F → F is a monomorphism. In other words, given a point x of X et and an etale neighborhood U of x , weneed to see that every element of lim ←− pt U F is determined by its value at x , potentially after shrinking U . But since F sends all morphisms to isomorphisms, the map lim ←− pt U F → F ( x ) will itself be injectiveprovided that every two objects in pt U are connected by a zig-zag of maps. This can be arranged byour local path connectivity hypothesis.The following is an immediate corollary: Corollary 2.34.
Notation as in the lemma, if F , G ∈
Sh( X ) both have the property that all theirco-specialization maps are isomorphisms, then to give an isomorphism F ≃ G is equivalent to givingisomorphisms F x ≃ G x for all x ∈ pt X compatible with all co-specialization maps. Combining this corollary with the description of the stalks of π j dK Sel ( − ) (Proposition 2.32), weget: Theorem 2.35.
Let X be a locally noetherian derived algebraic space such that ( X ⊗ F p ) red is regular.The etale sheafified homotopy group π etj (dK Sel ( − ) ) ∈ Sh( X et ; Ab) satisfies:1. When j = 0 , it is canonically isomorphic to the constant sheaf Q p / Z p .2. When j = − , it is zero.3. If X lives over Z [1 /p ] , then for j = 2 n it is canonically isomorphic to ( Q p / Z p )( n ) and for j = 2 n + 1 it is zero.
25n particular, the vanishing of this sheaf when j = − and its identification with Q p / Z p when j = 0 implies: Corollary 2.36.
Let X be as in the previous theorem. There is an edge map in degree − for thedescent spectral spectral sequence of dK Sel ( − ) over X et which goes π − dK Sel ( X ) → H ( X et ; Q p / Z p ) . Note that one doesn’t even need convergence of the spectral sequence, i.e. that dK Sel ( − ) is anetale Postnikov sheaf, for the edge map to be defined. Definition 2.37.
Let X be as in the previous theorem. Define the map e X : H ( X et ; Z p ) → π dK Sel ( X ) to be the negative of the Pontryagin dual of the edge map of the previous corollary. The definition of e X proves the first part of Theorem 2.15. Before proving the rest (giving conditionson when e X is an isomorphism), we record the following unwinding of the description of e X when X = Spec( F ) for a field F of characteristic = p . There is a similar description (using c F instead of j F )when F has characteristic p , but we won’t need that. Proposition 2.38.
Let F be a field of characteristic = p , let b g ∈ H (Spec( F ) et ; Z p ) = (G F ) ab b p , and fixa separable closure F of F . Denote by i : K ( F ) → K ( F ) the map induced by the inclusion. Recall thehomotopy class j F ∈ [K( F ) , ω K (1) ] from Corollary 2.18. Make the following choices:1. Choose a representative K( F ) → ω K (1) for j F , and denote it just by j .2. Choose a representative g ∈ Gal(
F /F ) of b g .3. Choose a homotopy κ : j ≃ j ◦ g .Then e Spec( F ) ( b g ) ∈ π dK Sel ( F ) = π Map(K( F ) , ω K (1) ) identifies with the class of the followingself-homotopy of the map j ◦ i : K( F ) → ω K (1) : j ◦ i κ ≃ ( j ◦ g ) ◦ i ≃ j ◦ ( g ◦ i ) = j ◦ i. Corollary 2.39.
Let F be a field of characteristic = p . Then the composite G F → H (Spec( F ) et ; Z p ) e F −→ π dK Sel ( F ) = [Σ K( F ) , ω K (1) ] ev ∈ π F ) −→ π ω K (1) sends g ∈ G F to [ χ ( g )] ∈ π ω K (1) , where χ : G F → Z × p is the p -cyclotomic character.Proof. If we choose a ring spectrum equivalence K( F ) b p ≃ ku b p using Gabber-Suslin, then by looking on π we deduce that the action of g ∈ G F on K( F ) b p identifies with the Adams operation ψ χ ( g ) on ku b p .Thus we need to see that the Toda bracket of S → ku b p − ψ u −→ ku b p → ω K (1) [ u ] ∈ π ω K (1) for u ∈ Z × p , where the first map is the unit and the last map is the unique (up tohomotopy) map which sends to · [ S ] ∈ π ω K (1) . Note that this Toda bracket is independent of anychoices, because [ S, Ω ku b p ] = [ ku b p , Ω ω K (1) ] = 0 .As usual, this is a simple calculation in K (1) -local homotopy theory using the log equivalence ω K (1) ≃ Σ L K (1) S (1.9). (The proof of the Adams conjecture also gives a less calculational proof – seethe following remark.) Remark 2.40.
If we use the representative J etF : K( F ) → Pic( S b p ) of j F given by the etale J-homomorphism (Remark 2.19), then the previous proposition and corollary become much more vividand canonical.Indeed, in that case (following [Sul]) we have a canonical choice of homotopy J etF ◦ g ≃ J etF comingfrom the action of g on the pair of schemes ( M , M − , so we get a canonical map of spectra E ( g ) :K( F ) → Σ − Pic( S b p ) = Aut( S b p ) associated to g ∈ Gal(
F /F ) which refines e ( b g ) ∈ [Σ K( F ) , ω K (1) ] .Explicitly, Ω ∞ E ( g ) sends the class of the finite free F -module M to the automorphism of the sphere M F / ( M F − induced by the action of g ∈ G F . Then Corollary 2.39 results from the fact that when M is the unit module, the action of g on the 2-sphere M F / ( M F −
0) = A / G m is through the cyclotomiccharacter on H . In the previous section we dualized the edge map in the descent spectral sequence for dK Sel ( − ) toproduce the comparison map e X : H ( X et ; Z p ) → π dK Sel ( X ) whenever X is a locally noetherian derived algebraic space with ( X ⊗ F p ) red regular. Now we proveTheorem 2.15, which gives conditions under which e X is an isomorphism. Theorem 2.41.
Suppose X as above has (mod p ) etale cohomological dimension ≤ , and that X ⊗ F p has (mod p ) etale cohomological dimension ≤ . Then e X is an isomoprhism.Proof. By Theorem 2.28, the descent spectral sequence E i,j = H j − i ( X et ; π etj F ) ⇒ π i F ( X ) for F = dK Sel ( − ) converges. But:1. All terms of cohomological degree > are zero, by the cohomological dimension hypothesis on X .2. For j odd all terms of cohomological degrees > are zero, because π etj F is then supported on X ⊗ F p by Theorem 2.35.3. The sheaf π et − F is by Theorem 2.35.Thus the only term contributing to π − dK Sel ( X ) is H ( X et ; Q p / Z p ) , and all differentials to and from H ( X et ; Q p / Z p ) are zero. Hence the edge map in that degree is an isomorphism, as desired. In otherwords, the claim is true for degree reasons. 27o finish the proof of Theorem 2.15, we need to see that e X is also an isomorphism if X = Spec( F ) for a field F of virtual (mod p ) Galois cohomological dimension ≤ . By the cases already handledabove, we can assume that F does not have honest (mod p ) Galois cohomological dimension ≤ . Thisimplies that p = 2 and F has characteristic zero. (This follows by combining Artin-Schreier theory withresults in the cohomology of profinite groups, as is nicely explained in [Sc].)The difficulty with this case is that there are non-zero differentials in the spectral sequence, andthey need to behave just right to ensure the result. This is somewhat subtle: if we had for example used Z p -Anderson duality in place of d K (1) to define dK Sel ( − ) , then the E -terms of the spectral sequencewould be the same, but the differentials would behave differently and the result would fail, e.g. when F = R or F = Q .More specifically, a quick look at the spectral sequence shows that it suffices to prove the following: Proposition 2.42.
Take p = 2 . Let F a field of virtual (mod ) Galois cohomological dimension ≤ d forsome d ∈ Z ≥ . For brevity, let F denote the Postnikov sheaf dK Sel ( − ) on the topos Spec( F ) et = BG F .Then in the region j − i > d of the descent spectral sequence E i,j = H j − i (BG F ; π etj F ) ⇒ π i ( F ) , the group E i,j = E i,j is zero except when j is even and i − j is (mod ), and the d -differential E i,j → E i − ,j +2 is an isomorphism when i − j is (mod ). Indeed, if we take d = 2 then it follows from this proposition that the d differential E − , → E − , must be zero (because the following d : E − , → E − , is an isomorphism), and then all of the higherdifferentials on E − , will also be zero because their targets will be zero from the E page onwards.Thus the full group E − , = H (BG F ; Q / Z ) contributes to π − F ( F ) . But on the other hand nothingelse contributes, because all other terms of total degree − in the spectral sequence will be zero from E onwards.Thus, it suffices to prove Proposition 2.42. For this, we first reduce to the case where F is real-closed.Recall the following theorem from [Sc]: Theorem 2.43.
Let F be a field of virtual (mod 2) cohomological dimension ≤ d . There exists aprofinite space X F and a map of toposes f : BC × X F → BG F such that:1. The points of X F are the orderings on F ;2. For such a point x ∈ X F , the corresponding map f x : BC → BG F is induced by BC =BG F x → BG F where F x denotes the real closure of F with respect to the ordering x .3. For any 2-torsion sheaf of abelian groups M on BG F , the map on cohomology H i (BG F ; M ) → H i ( BC × X F ; f ∗ M ) is an isomorphism for i > d . It follows from this that for an etale Postnikov sheaf of spectra M on BG F , the comparison mapof descent spectral sequences for M and for f ∗ M is an isomorphism on E in the region j − i > d .Because our sheaf of interest M = dK Sel ( − ) has d = 0 for degree reasons, it follows we need onlyprove the analog of Proposition 2.42 for f ∗ M . On the other hand, since X F is profinite and hence of28ohomological dimension 0, for M as in 3 the assignment U H i ( BC × U ; f ∗ M ) on opens U ⊂ X F is a sheaf (namely, it is R i π ∗ f ∗ M for π : BC × X F → X F ), and indeed the descent spectral sequencefor f ∗ M is the global sections of a spectral sequence of sheaves over X F . The claims in Proposition2.42 can be checked on the stalks, which are the descent spectral sequences for f ∗ x M by a simplebase-change comparison. This gives the reduction to the case where F = F x is real-closed.Thus, to prove Proposition 2.42 and hence the last remaining case of Theorem 2.15 it suffices toshow: Theorem 2.44.
Let F be a real-closed field. Consider the homotopy fixed point spectral sequence E i,j = H j − i ( BC ; π j N ) ⇒ π i ( N C ) for C = Gal( F ( √− /F ) acting on N := d K (1) K( F ( √− . In the range j − i > , the group E i,j is zero except when j is even and i − j is (mod ), and the d -differential E i,j → E i − ,j +2 is anisomorphism when i − j is (mod ).Proof. By a zig-zag of maps of real-closed fields we can connect F to R . Thus Suslin rigidity ([Sus])implies that the homotopy type of the C -spectrum K( F ( √− b is the same for F as for R , so wecan assume F = R . There K( R ( √− b = ku b p again by Suslin, so N = ( d K (1) KU ) with C actingon KU by complex conjugation.Since we already know π n N = Q / Z ( n ) and π n +1 N = 0 (Proposition 2.32), the claim about the E i,j groups is an easy calculation in group cohomology. Furthermore, we see that when these groups arenonzero (in the region j − i > ) they are cyclic of order two. It remains to calculate the d differentials.Recall that all the differentials in the homotopy fixed point spectral sequence for the complexconjugation C -action on KU E i,j = H j − i ( BC ; π j KU ) ⇒ π i ( KO ) are well-known, namely the E = E page as a CDGA is Z [ t ± , e ] / e with e ∈ E , and t ∈ E , , withdifferential d determined by d e = e · t − and d t = e . Furthermore E = E ∞ .Now we use that N = ( d K (1) KU ) is a C -equivariant module over KU , so that the homotopyfixed point spectral sequence for N is a module over the homotopy fixed point spectral sequence for KU .Moreover, from Proposition 2.18 it follows that π ∗ N with its C action identifies with ( π ∗ KU ) ⊗ Q / Z ,compatibly with the module structure. Thus on the E -page the module structure is given by the usualcup product. One sees that the whole region j − i > on the E -page is generated as a moduleover Z [ t ± , e ] / e by the non-trivial class c in E , = H ( C ; Q / Z (1)) = µ . On the other hand,one calculates as usual that π − d K (1) KO = [ KO, Σ L K (1) S ] = 0 , so that π N C = 0 . This forcesthat d is nontrivial on c , which in turn determines all the differentials using the module structure over Z [ t ± , e ] / e , in particular verifying the claim. Remark 2.45.
The proof we just gave of Proposition 2.42, and hence of the second claim in Theorem2.15, doesn’t quite need the hypothesis that X is Spec of a field: it only needs that the real spectrumof X is profinite (see [Sc]). Definition 3.1.
Let
LCA ℵ denote the category of second-countable locally compact Hausdorff abeliangroups and continuous homomorphisms. Certainly LCA ℵ is additive, so we can consider it as an exact ategory with respect to its maximal exact structure, which amounts to saying that a null-composite A → B → C sequence is short exact provided A → B identifies A with a closed subgroup of B and B → C identifies C with the quotient by this closed subgroup. Let lc Z denote the Z -linear stable ∞ -category given as lc Z = D b (LCA ℵ ) , the bounded derived ∞ -category of the exact category LCA ℵ (the natural enhancement of the usualbounded derived category). Definition 3.2.
Let
P ∈
PerfCat Z . Define lc P = Fun Z ( P , lc Z ) , the Z -linear stable ∞ -category of Z -linear functors from P to lc Z .If R is a ring (or DGA), we set lc R = lc Perf( R ) , and if X is a qcqs derived algebraic space we set lc X = lc Perf( X ) . Remark 3.3.
Since
LCA ℵ is idempotent-complete and essentially small, so is D b (LCA ℵ ) ([BaS] Theo-rem 2.8, [L1] Lemma 1.2.4.6); thus so is lc P for any P ∈
PerfCat Z , and hence P 7→ lc P can be viewedas a functor PerfCat op Z → PerfCat Z . We have two goals in this section. First, we would like to produce the map C F → π K(lc F ) claimedin the introduction, for F a finite, local, or global field. We won’t do the work to fully understand lc F in these cases, though it is possible.Our other goal, crucial for producing the Artin maps, is to fully understand K(lc Z ) , or more generallyany localizing invariant of lc Z . The result, suggested by Lurie, is as follows: Theorem 3.4.
Let A : PerfCat Z → A be any functor to a stable ∞ -category which satisfies localization,i.e. sends fiber-cofiber sequences to (fiber-)cofiber sequences. Then there is a canonical cofiber sequence A (Perf( Z )) → A (Perf( R )) → A (lc Z ) . Remark 3.5.
A description of the data defining this cofiber sequence will fall out from the proof.Namely, the first map is induced by − ⊗ Z R , the second map is induced by the inclusion of finite-dimensional real vector spaces inside LCA ℵ , and the nullhomotopy of the composite of these two mapsis gotten as follows: consider the short exact sequence Z → R → T in LCA ℵ . The first term is discrete, and hence canonically trivialized by A using an Eilenberg swindlewith direct sums; similarly the last term is compact and hence canonically trivialized by A using anEilenberg swindle with products. Thus the middle term R is canonically trivialized by A as well. Wecan tensor this short exact sequence with an arbitrary element of Perf( Z ) to deduce a trivialization ofthe composite Perf( Z ) → Perf( R ) → lc Z after applying A , as desired. lc Z (or rather, its variant without the second-countability restriction — but that is justa technicality) has been studied comprehensively in [HS]. From that study it is quite easy to “calculate” lc Z : Theorem 3.6.
Fix a countable index set I , and for later clarity also denote I by J . As a stable ∞ -category, lc Z is generated by the objects ⊕ I Z and Q I T of LCA ℵ sitting in degree zero, and thehom-complexes between these generators are as follows:1. The maps hom( ⊕ I Z , ⊕ J Z ) → Q I hom( Z , ⊕ J Z ) ← Q I ⊕ J hom( Z , Z ) are equivalences, and hom( Z , Z ) is Z concentrated in degree zero, generated by the identity map.2. The map hom( ⊕ I Z , Q J T ) → Q I × J hom( Z , T ) is an equivalence, and hom( Z , T ) is the group R / Z concentrated in degree zero.3. The map hom( Q I T , ⊕ J Z ) ← ⊕ I × J hom( T , Z ) is an equivalence, and hom( T , Z ) is Z concen-trated in degree − , with the generator correpsonding to the extension class of Z → R → T .4. The maps hom( Q I T , Q J T ) → Q J hom( Q I T , T ) ← Q J ⊕ I hom( T , T ) are equivalences, and hom( T , T ) is Z concentrated in degree zero, generated by the identity map.Proof. Certainly lc Z is generated by the objects of LCA ℵ sitting in degree zero. By the structure theoryof locally compact abelian groups, every object of LCA ℵ R n . Thus lc Z is generated by the compact groups, the discrete groups, and R . Thanks to the short exact sequence Z → R → T , we actually only need the compact groups andthe discrete groups. Every countable discrete abelian group has a two-step resolution by free countablediscrete abelian groups, so the discrete ones are generated by ⊕ I Z . By Pontryagin duality, the compactones are also generated by Q I T . Thus in total lc Z is generated by ⊕ I Z and Q I T .Calculations 1 and 2 are easy, because Z and ⊕ I Z are projective objects of LCA ℵ . Similarly 4 iseasy because by duality T and Q I T are injective. For 3, one can calculate hom( T , Z ) by the projectiveresolution Z → R of T , getting the second claim in 3. What remains is the first claim in 3; that is thetrickiest, since Q I T has no projective resolution and ⊕ J Z has no injective resolution. We refer to [HS]Example 4.10 for the proof.To prove Theorem 3.4, we will use this calculation to recognize lc Z as a Verdier quotient D ′ / C ,where C is equivalent to Perf( Z ) and D ′ is equivalent to Perf( R ) “up to Eilenberg swindles”. In moredetail, we will define a “cone” construction, which takes as input a functor F : C → D of stable ∞ -categories and outputs a new stable ∞ -category cone( F ) . To build cone( F ) , we enlarge D along Eilenberg swindles in such a way that F becomes fully faithful, and then we take the usualVerdier quotient. Theorem 3.4 follows from an identification of lc Z with the cone of the base-changefunctor Perf( Z ) → Perf( R ) . We will also interpret the desired map C F → π lc F (for F a finite, local,or global field) in terms of the cone construction. We first define minimalist versions of the standard Ind and Pro categories.31 efinition 3.7.
Let C be a stable ∞ -category. We define ind( C ) to be the smallest stable full subcat-egory of Ind( C ) which contains the constant countable coproduct ⊕ N X for all X ∈ C , and we define pro( C ) to be the dual construction, pro( C ) = ind( C op ) op ⊂ Ind( C op ) op = Pro( C ) . Remark 3.8.
Note that
C ⊂ ind( C ) , since any X ∈ C is the cofiber of the “shift by one” map ⊕ N X → ⊕ N X . Remark 3.9.
The Milnor telescope construction shows that ind( C ) is closed under colimits along dia-grams of the type X f → X f → X f → . . . . In particular ind( C ) is idempotent-complete. Remark 3.10. If C is essentially small, then so is ind( C ) . Remark 3.11.
As in [L2] Proposition 5.3.5.10, one sees that ind( C ) satisfies the following universalproperty: it is closed under constant countable coproducts, and if E is any stable ∞ -category which isclosed under constant countable coproducts, then giving an exact functor ind( C ) → E which commuteswith constant countable coproducts is equivalent (via restriction) to giving an exact functor C → E . Remark 3.12. If C has a R -linear structure for some E -ring R , i.e. it is a module over Perf( R ) , then ind( C ) acquires a canonical such structure as well, and in fact a unique one which preserves countableconstant coproducts. This can be seen by applying the universal property of the previous remark. Fromthis one deduces that all the material of this section is valid without change in the R -linear context aswell. We will apply this with R = Z . Now, let F : C → D be an exact functor between stable ∞ -categories. Our category cone( F ) will bedefined as a Verdier quotient of another category precone( F ) , which in turn is an “extension” of ind( C ) by D by pro( C ) . Before giving the definition of precone( F ) , we note that there is an obvious notion ofa map from an object of Ind( C ) to an object of C , from an object of C to an object of Pro( C ) , and froman object of Ind( C ) to an object of Pro( C ) ; moreover, maps of the first two types can be composedto obtain a map of the third type. For example we can consider all of these as full subcategories of Ind(Pro( C )) or of Pro(Ind( C )) ; the mapping spaces we are interested in will be the same in either case. Definition 3.13.
Let F : C → D be an exact functor between stable ∞ -categories. We define a stable ∞ -category precone( F ) to consist of tuples ( I, V, P, α, β ) , where:1. I ∈ ind( C ) , V ∈ D , and P ∈ pro( C ) ;2. α is a map I → P ;3. β is a factorization of F ( α ) through V . Calculations in precone( F ) are easy right from the definition. We express this as follows. Definition 3.14.
Let F : C → D be an exact functor between stable ∞ -categories. Define functors δ : ind( C ) → precone( F ) , ν : D → precone( F ) , κ : pro( C ) → precone( F ) by δ ( I ) = (Σ − I, , , , , ν ( V ) = (0 , V, , , , and κ ( P ) = (0 , , Σ P, , . Proposition 3.15.
1. The functors δ , ν , and κ are fully faithful.2. Their essential images semi-orthogonally decompose precone( F ) as h ind( C ) , D , pro( C ) i . (Thus precone ( F ) is generated by these three full subcategories, and maps X → Y vanish if X ∈ pro( C ) and Y ∈ ind( C ) ∪ D or X ∈ D and Y ∈ ind( C ) .)3. The remaining mapping spectra in precone( F ) are as follows. For I ∈ ind( C ) , V ∈ D , and P ∈ pro( C ) , we have: • map( δ ( I ) , ν ( V )) = map( F ( I ) , V ) ; • map( ν ( V ) , κ ( P )) = map( V, F ( P )) ; • map( δ ( I ) , κ ( P )) = cofib(map( I, P ) → map( F ( I ) , F ( P ))) .Moreover, the composition maps are the obvious ones. Corollary 3.16.
There is a canonical null-composite sequence δ ( X ) → ν ( F ( X )) → κ ( X ) , functorial in X ∈ C . This null-composite sequence is the subject of the following universal property of precone( F ) : Proposition 3.17.
Let F : C → D be an exact functor between stable ∞ -categories, let precone( F ) be as in Definition 3.13, and suppose given another stable ∞ -category E .Let Fun precone ( F ; E ) denote the ∞ -category whose objects are tuples ( d, v, k, c ) where d, k : C → E and v : D → E are exact functors and c is a null-composite sequence d → v ◦ F → k, such that the constant countable coproduct ⊕ N d ( X ) and the constant countable product Q N k ( X ) existin E for each X ∈ C , and let Fun precone (precone( F ) , E ) denote the full subcategory of Fun(precone( F ) , E ) consisting of those functors which are exact and preserve the countable constant coproducts of δ ( X ) ’sand constant countable products of κ ( X ) ’s.Then the functor Fun precone (precone( F ) , E ) → Fun precone ( F ; E ) , given by composition with δ , ν and κ , is an equivalence.Proof. Let us simply describe how to produce produce the functor in the other direction, which is all wewill use in the end; checking that it is indeed an inverse is anyway straightforward. Thus suppose giventhe data ( d, v, k, c ) ∈ Fun precone ( F ; E ) , and let us produce the required functor f : precone( F ) → E .Take ( I, V, P, α, β ) ∈ precone( F ) . The objects I, V, and P give us the objects d ( I ) , v ( V ) , and k ( P ) in E . Then α and β and the null-composite sequence d → v ◦ F → k combine to give a null-composite sequence d ( I ) → v ( V ) → k ( P ) in E . We define f ( I, V, P, α, β ) to be the exactness defectof this null-composite sequence. (The exactness defect is equivalently the cofiber of d ( I ) mapping tothe fiber of v ( V ) → k ( P ) , or the fiber of the cofiber of d ( I ) → v ( V ) mapping to k ( P ) .)33ow, the way we will pass from precone( F ) to cone( F ) is to enforce that this null-compositesequence δ → ν ◦ F → κ should be a fiber-cofiber sequence. The exactness defect of that sequenceidentifies with the functor ι : C → precone( F ) defined by ι ( X ) = ( X, F ( X ) , X, id , id) . It’s easy to check that this functor is fully faithful, and so ι ( C ) is a full stable subcategory of precone( F ) .Thus we can form the Verdier quotient precone( F ) /ι ( C ) . Definition 3.18.
Let F : C → D be an exact functor between stable ∞ -categories. We define cone( F ) to be the idempotent-completed Verdier quotient precone( F ) /ι ( C ) , where precone( F ) is as in Definition3.13 and ι is as above. Calculations in cone( F ) are hardly more difficult than in precone( F ) . Indeed, one can explicitlywrite down an ι ( C ) -Ind-injective resolution for any object X = ( I, V, P, α, β ) of precone( F ) . Namely,there is an obvious map ι ( I ) → X in Ind(precone( F )) , and its cofiber is the ι ( C ) -injective resolution of X . Dually, we can also describethe ι ( C ) -projective resolution of X as the fiber of X → ι ( P ) in Pro(precone( F )) .One upshot of this is the following: Lemma 3.19. If X and Y are objects of precone( F ) which each lie in one of the generating full sub-categories ind( C ) , D , or pro( C ) , then Map(
X, Y ) is the same in cone( F ) as in precone( F ) (Proposition3.15), with one exception: if X ∈ pro( C ) and Y ∈ ind( C ) , then Map cone ( κ ( X ) , δ ( Y )) = Map( X, Σ Y ) , whereas it is in precone( F ) . Here
Map( X, Σ Y ) carries the only formal meaning it can have; it is the same if calculated in either Ind(Pro( C )) or in Pro(Ind( C )) .There is also a universal property for cone( F ) analogous to that of precone( F ) ; we just have toreplace “null-composite sequence” with “fiber-cofiber sequence”: Proposition 3.20.
Let F : C → D be a functor between stable ∞ -categories, let cone( F ) be as inDefinition 3.18, and suppose given an idempotent-complete stable ∞ -category E .Let Fun cone ( F ; E ) denote the ∞ -category whose objects are tuples ( d, v, k, c ) where d, k : C → E and v : D → E are exact functors and c is a fiber-cofiber sequence d → v ◦ F → k, such that the constant countable coproduct ⊕ N d ( X ) and the constant countable product Q N k ( X ) exist in E for each X ∈ C , and let Fun cone (cone( F ) , E ) denote the full subcategory of Fun(cone( F ) , E ) consisting of those functors which are exact and preserve the countable constant coproducts of δ ( X ) ’sand constant countable products of κ ( X ) ’s.Then the functor Fun cone (cone( F ) , E ) → Fun cone ( F ; E ) , given by composition with δ , ν and κ , isan equivalence. roof. Note that a functor cone( F ) → E preserves the relevant coproducts and products if and onlyif its restriction to precone( F ) does; this follows from the fact that precone( F ) → cone( F ) preservesthose coproducts and products, which is a consequence of the above discussion of mapping spaces in cone( F ) . Thus the statement to be proved follows by combining the universal property of precone (Proposition 3.17) with the universal property of the Verdier quotient.Also, the value of cone( F ) on localizing invariants is easy to describe. Namely, let PerfCat denotethe ∞ -category of idempotent-complete small stable ∞ -categories and exact functors between them.By a localizing invariant of PerfCat we will mean a functor A : PerfCat → A such that A is a stable ∞ -category and A preserves fiber-cofiber sequences. (The terminology is as in [BGT], but we don’trequire preservation of filtered colimits.) Proposition 3.21.
Let F : C → D be an exact functor between idempotent-complete small stable ∞ -categories. Then for any localizing invariant A : PerfCat → A , there is a canonical cofiber sequence A ( C ) F −→ A( D ) → A (cone( F )) . Proof.
Since A is localizing, by the definition of cone( F ) we have that A(cone( F )) identifies withthe cofiber of A ( C ) ι −→ A (precone( F )) . On the other hand, the functor F : C → D factors as thecomposition of ι with the projection precone( F ) → D . Thus it suffices to show that this projectioninduces an equivalence A (precone( F )) ∼ → A ( D ) . By the semi-orthogonal decomposition of Proposition3.15 and the fact that localizing invariants turn semi-orthogonal decompositions into direct sums, weneed only see that A (ind( C )) and A (pro( C )) are both trivial. But ind( C ) admits countable direct sumsand is thus trivial on any localizing invariant by an Eilenberg swindle, and dually for pro( C ) .We finish our discussion of the cone construction with some remarks and examples. Remark 3.22.
If we were to use
Ind( C ) and Pro( C ) instead of ind( C ) and pro( C ) in the above con-structions, then we would obtain a bigger variant of the precone and cone categories, which could bedenoted PreCone( F ) and Cone( F ) . It seems that in practice it doesn’t much matter whether one uses cone( F ) or Cone( F ) or something in between. We chose to focus on cone( F ) because it is essentiallysmall when C is, and because of its connection with D b (LCA ℵ ) . But for the remainder of these remarkswe’ll talk about Cone( F ) instead, because its formal properties are slightly more general and convenient. Remark 3.23.
There are two extreme examples of
Cone( F ) which are worth looking at. The first is when F is the zero functor C → , where C is arbitrary. The association C 7→
Cone(
C → is a derived analogof the lim ←→ , or “locally compact objects”, construction of [Be] A.3. Indeed, an object of PreCone(
C → can, by reindexing, be identified with a map I → Σ P where I ∈ Ind( C ) and P ∈ Pro( C ) . This map canbe thought of as classifying a formal extension of I by P ; the middle term of this imagined extension isthus “locally compact” if we think of Ind as signifying discrete and Pro as signifying compact. Passingfrom PreCone to Cone via the Verdier quotient has the effect of remembering only the middle term ofthe extension, forgetting how we “cut it in half” to get the extension itself. Note also that in this caseProposition 3.21 gives that A (Cone( C → ≃ cofib( A ( C ) → ≃ Σ A ( C ) for any localizing invariant A ; this can be compared the the result of [Sa] in the exact category context. Remark 3.24.
The other extreme is when the functor F : C → D is fully faithful. Then there is also theVerdier quotient D / C , and one can produce a canonical functor D / C →
Cone(
C → D ) as follows. Thefunctor F automatically has an Ind-right adjoint r : D →
Ind( C ) and a Pro-left adjoint l : D →
Pro( C ) . hen we can define a functor D →
PreCone(
C → D ) by d ( r ( d ) , d, l ( d ) , − , − ) where the last twobits of data are gotten from the appropriate units and counits for these adjunctions. This functor factorsthrough the quotient by C to define the desired functor D / C →
Cone(
C → D ) . Proposition 3.21 in thiscase says that this functor is an equivalence on localizing invariants. So we can think of Cone(
C → D ) as some “freed up” version of D / C , which exists even when C → D is not fully faithful.
Remark 3.25.
We can also connect the two examples. Namely, if
C → D is again fully faithful, we obtaina functor ∂ : D / C →
Cone(
C → by composing the functor of the previous remark with the projection D → . The reason for the notation is that, under the equivalence A (Cone( C → ≃ Σ A ( C ) , themap induced by ∂ on any localizing invariant A identifies with the boundary map in the localizationsequence A ( C ) → A ( D ) → A ( D / C ) . This is clear by functoriality. Basically, this cone business serves to realize certain operations on K-theoretic spectra at the moreprimitive level of categories and functors.
We start with the base case of lc Z = D b (LCA ℵ ) . Theorem 3.26.
There is a Z -linear equivalence α Z : cone(Perf( Z ) → Perf( R )) ∼ → lc Z , produced from the universal property of the cone (Proposition 3.20) via the data of the cofiber sequence Z → R → T inside lc Z and the action of the ring R on the middle term.Proof. More specifically, the object R ∈ lc Z carries an obvious R -action by multiplication, so it generatesa functor v : Perf( R ) → lc Z ; similarly, the cofiber sequence Z → R → T carries a Z -action and sogives the required cofiber sequence of functors needed to apply the universal property of the cone(Proposition 3.20). Note that the existence of the required coproducts and products follows from thefact that Z is discrete and T is compact, so the former admits a constant countable coproduct and thelatter a constant countable product. (Note also that the inclusion LCA ℵ ⊂ D b (LCA ℵ ) preserves thesecoproducts and products because Z is projective and T is injective; the same goes for any discrete orcompact group by the usual two-step resolution by free abelian groups and its Pontryagin dual.)To prove that the resulting functor α Z : cone(Perf( Z ) → Perf( R )) → lc Z is an equivalence, itsuffices to show that there is a full subcategory of the source whose image under α Z generates thetarget, and on which α Z is fully faithful. For this we can take the full subcategory on ⊕ N δ ( Z ) and Q N κ ( T ) . That the image under α Z generates follows from Theorem 3.6, and the full faithfulness of α Z on this full subcategory follows by comparing the calculations of mapping spaces in cones (Lemma3.19) with those in lc Z (Theorem 3.6). Actually, there are sign issues, but we can always make these work out by adjusting the identification cofib( X → ≃ Σ X used in identifying A (Cone( C → ≃ Σ A ( C ) , which is only canonical up to a sign. As for our sign conventionfor boundary maps, let us declare it to be such that the boundary map in the localization sequence for a DVR sends auniformizer in π K of the fraction field to the unit ∈ π K of the residue field, cf [W] Example 6.1.12. orollary 3.27. Theorem 3.4 holds: for any functor A : PerfCat Z → A to a stable ∞ -category whichpreserves fiber-cofiber sequences, there is a canonical cofiber sequence A (Perf( Z )) → A (Perf( R )) → A (lc Z ) . Proof.
This follows by combining the previous theorem with (the Z -linear analog of) Proposition 3.21.Now, for recovering the Artin reciprocity law we are interested in investigating lc F for F a finite,local, or global field. As an intermediary, to help with functoriality, we will also want to look at lc R for R the ring of integers in a non-archimedean local field. Proposition 3.28.
1. Let F be a global field, and A F its ring of adeles. There is a canonical F -linearcomparison functor α F : cone(Perf( F ) → Perf( A F )) → lc F determined by the cofiber sequence F → A F → A F /F of objects of lc Z , with its natural F -action and the extension of this action to an A F -action onthe middle term.2. Let F be a finite field. There is a canonical F -linear comparison functor α F : cone(Perf( F ) → → lc F determined by the cofiber sequence F → → Σ F in lc Z with its F -action.3. Let R be the ring of integers of a non-archimedean local field F , and let Perf m ( R ) ⊂ Perf( R ) denote the fiber of Perf( R ) → Perf( F ) . There is a canonical R -linear comparison functor α R : cone(Perf m ( R ) → → lc R , determined by the functorial cofiber sequence of R -modules in lc Z M → → Σ M for M ∈ Perf m ( R ) .4. Let F be a local field. There is a canonical F -linear comparison functor α F : Perf( F ) → lc F determined by the object F of lc Z with its F -action.Proof. For 1, 2, and 3, again this follows from the universal property of cone (Proposition 3.20) oncewe note that the left-hand term is discrete and the right-hand term is compact in each of our cofibersequences. For 4, this follows from the usual universal property of Perf.37ince K-theory is a localizing invariant, combining with Proposition 3.21 gives:
Corollary 3.29.
1. If F is a global field, K( α F ) gives a map of spectra cofib(K( F ) → K( A F )) → K(lc F ) , and therefore on π we get A × F /F × → π K(lc F ) .
2. If F is a finite field, K( α F ) gives a map of spectra Σ K( F ) → K(lc F ) , and therefore on π we get Z → π K(lc F ) .
3. If R is the ring of integers of a non-archimedean local field, then K( α R ) gives a map of spectra Σ K(Perf m ( R )) → K(lc R ) , and therefore on π we get Z → π K(lc R ) .
4. If F is a local field, K( α F ) gives a map of spectra K( F ) → K(lc F ) , and therefore on π we get F × → π K(lc F ) . Remark 3.30.
The map α (and therefore also K( α ) ) is an equivalence in cases 1 and 2, but not incases 3 and 4. But this failure in the latter cases is essentially a technicality which arises because ourdefinition of lc R did not require the topology on R to interact with the topology of the objects in LCA ℵ . Remark 3.31.
We can put the finite and global cases on a common footing by defining the adele ringof a finite field to be the zero ring . In fact, there is a common story for any commutative ring R essentially of finite type over Z . Namely, attached to any such R is an “adele ring” A R , which is notan ordinary ring but a co-connective E ∞ -ring, and also an object of lc Z . There is a cofiber sequence of R -modules in lc Z R → A R → ( ω R/ Z ) ∨ where ω R/ Z is the relative dualizing complex and ( − ) ∨ is derived Pontryagin duality hom( − ; T ) . Usingthis one gets a description of lc R for any such R in exactly the same terms as for a global field above.New language is required to set all of this up, but let us just indicate how things look in the case R = Z [ X ] . There A R can be represented by the homotopy pullback of Z (( X − )) → R (( X − )) ← R [ X ] . Note that the objects here do not themselves lie in
LCA ℵ , only the homotopy groups of the pullback(namely Z [ X ] and T [[ X − ]] ) do. This shows that our derived definitions are necessary to obtain the orrect theory in higher dimensions: more concretely, the adele ring of a higher-dimensional ring like Z [ X ] does not give a class in the K-theory of the exact category of R -modules in LCA ℵ , though it doesgive a class in K(lc R ) after things are set up properly.Note also that the ring Z (( X − )) is in some sense the “competion at ∞ ” of Spec( Z [ X ]) over Spec( Z ) , and the adele ring above is gotten by combining this “geometric completion at ∞ ” with the”arithmetic completion at ∞ ” R of Z . This is an instance of a general pattern with such adele rings. We will also need some functorial properties of these comparison functors α . Namely, for anyhomomorphism of rings R → R ′ there is a forgetful functor lc R ′ → lc R , and we would like to see thisreflected in the left-hand side of the comparison maps α of Proposition 3.28. We simply state theresults; the proofs are immediate from the definitions, especially if one uses the universal property ofcones (Proposition 3.20). Proposition 3.32.
1. Let F → L be a finite extension of local fields or finite fields. Then thefunctors α as in Proposition 3.28 intertwine the natural forgetful functor lc L → lc F with theforgetful functor Perf( L ) → Perf( F ) . (There is also a similar statement in the global field case,incorporating the forgetful map Perf( A L ) → Perf( A F ) .)In particular, on π K in the local field case, we get a commutative diagram L × / / (cid:15) (cid:15) π K(lc L ) (cid:15) (cid:15) F × / / π K(lc F ) where the left-hand map is the norm map, whereas in the finite field case we get a commutativediagram Z / / (cid:15) (cid:15) π K(lc L ) (cid:15) (cid:15) Z / / π K(lc F ) where the left-hand map is multiplication by the degree [ L : F ] .2. Let F → L be a homomorphism from a global field to one of its non-discrete completions L (which is then a local field). The functors α intertwine the natural forgetful functor lc L → lc F with the forgetful functor Perf( L ) → Perf( A F ) coming from the ring homomorphism A F → L of projection to the L -factor.In particular, on π K we get a commutative diagram L × / / (cid:15) (cid:15) π K(lc L ) (cid:15) (cid:15) A × F /F × / / π K(lc F ) where the left-hand map is induced by the map L × → A × F which is the identity in the L -componentof A F and is everywhere else. . Let F be a non-archimedean local field with ring of integers R and residue field k (which is thena finite field). The functors α intertwine the forgetful functor lc F → lc R with the functor ∂ :Perf( F ) → cone(Perf m ( R ) → induced by the localization sequence Perf m ( R ) → Perf( R ) → Perf( F ) as in Remark 3.25, and the functors α intertwine the forgetful functor lc k → lc R withthe forgetful functor Perf( k ) → Perf m ( R ) . In particular, on π K , we get a commutative diagram F × / / (cid:15) (cid:15) π K(lc F ) (cid:15) (cid:15) Z / / π K(lc R ) Z / / O O π K(lc k ) O O where the upper left map is the discrete valuation on F and the lower left map is the identity. We start by constructing the fundamental class j ∈ dK Sel (lc Z ) . Theorem 3.4 gives a fiber sequence dK Sel (lc Z ) → dK Sel ( R ) → dK Sel ( Z ) . Thus, to construct a point in dK Sel (lc Z ) , it is enough to construct a point in dK Sel ( R ) and a nullho-motopy of the image of that point in dK Sel ( Z ) .By the pushout defining dK Sel , this is equivalent to giving the following data:1. A point in d K (1) K( R ) .2. A point in d K (1) TC( Z ) .3. A homotopy between the images of these points in d K (1) K( Z ) .4. A nullhomotopy of the image of the second point in d T C
TC( Z ) .For data 1, we take the K (1) -localization of the map J R : K( R ) → Pic( S b p ) of Theorem 1.1. For data2, we take the K (1) -localization of J Z p , i.e. the map j Z p studied in Section 1.3. For data 3, we takethe homotopy gotten by K (1) -localizing Theorem 1.1. The existence of data 4 is tautological from theconstruction of the natural transformation n : d K (1) → d T C (Corollary 1.17). Thus we have specified apoint j ∈ dK Sel (lc Z ) . Remark 4.1.
In fact, one can calculate that π dK Sel (lc Z ) is a free Z p -module on the class of j . Forthis one needs to use some input from the etale cohomology of Spec( Z [1 /p ]) , notably a part of thestandard exact sequence for Brauer group of Q . Using this sort of calculation one can see the existenceof the class of j in π dK Sel (lc Z ) without referring to the explicit construction of [C], and thereby avoidthe use of [C] in this paper at the cost of borrowing more material from traditional class field theory.But we consider it to be interesting and possibly indicative of some fundamental structure that thereis an explicit topological construction which pins down j , not just up to homotopy. It also raises somefurther questions which may be related to explicit class field theory; see Remark 4.8. efinition 4.2. Let
P ∈
PerfCat Z . Since K-theory, TC-theory, and the trace map between themare multiplicative with respect to the tensor product in PerfCat Z ([BGT2]), the theory K Sel of theintroduction is also multiplicative. Moreover, the pushout diagram defining dK Sel , and hence dK Sel itself, is K Sel -linear. Thus the tautological pairing lc P ⊗ Z P → lc Z induces a map of spectra K Sel (lc P ) ⊗ dK Sel (lc Z ) → dK Sel ( P ) . Define the Artin map
Art P : K Sel (lc P ) → dK Sel ( P ) to be the evaluation of this pairing on the class j ∈ dK Sel (lc Z ) described above. Remark 4.3.
By construction,
Art P is a K Sel -linear natural transformation of functors
PerfCat op Z → Sp . Remark 4.4.
We will only be interested in the restriction of
Art P to a map K(lc P ) → dK Sel ( P ) . This restriction is perhaps more tangible from a concrete perspective: if a point in
K(lc P ) is representedby an object F ∈ lc P viewed as a Z -linear functor P → lc Z , then Art P ( F ) is simply the pullback of j by the functor F , using the functoriality of dK Sel . By Theorem 2.15 and standard Galois cohomological dimension estimates ([Se] II.5 and II.6), if F is a number field, local field, or finite field, then the edge map gives an isomorphism π dK Sel ( F ) ≃ (G abF ) b p , and similarly if R is the ring of integers of a non-archimedean local field with residue field F then π dK Sel ( R ) ≃ ( π et ( R ) ab ) b p = (G abF ) b p . On the other hand, Corollary 3.29 gives us comparison maps C F → π K(lc F ) . Composing with π Art F , we obtain homomorphisms a F : C F → (G abF ) b p for all such F . Moreover, the functoriality of Art together with Proposition 3.32 shows that these mapssatisfy the usual functoriality of Artin maps. The following lemma is then standard:
Lemma 4.5.
Suppose given such system of maps { a F } satisfying the functoriality properties of Propo-sition 3.29. If the map a Q p : Q × p → (cid:16) G ab Q p (cid:17) b p has the property that when we restrict it to Z × p andcompose it with the p -cyclotomic character (cid:16) G ab Q p (cid:17) b p → Z × p /µ p − we get the tautological map, then thesystem of maps { a F } identifies with the p -completion of the negative of the usual Artin maps. roof. First note that by the norm functoriality in field extensions (Proposition 3.32 part 1) and thefact that norm subgroups are open, the maps a F are automatically continuous in the local and globalcases. Now consider F = Q . Since Q ( ζ p ∞ ) is unramified outside p , the functoriality Proposition 3.32parts 2 and 3 implies that the effect of a Q on Q ( ζ p ∞ ) is trivial when restricted to Z × ℓ → C Q for ℓ = p .By continuity, it is then trivial on the product of these groups. But the quotient A × Q / ( Q × · Q ℓ = p Z × ℓ ) identifies with Z × p coming from the p -factor, so our hypothesis uniquely determines a Q on the p -cyclotomic extension. In particular, we see that a Q sends a uniformizer at ℓ to the inverse of the ℓ -Frobenius in Gal( Q ( ζ p ∞ ) / Q ) b p for ℓ = p . Therefore by Proposition 3.32 part 3 we find that a F ℓ sends to the inverse of Frobenius in the p -cyclotomic extension of F ℓ . Since ℓ has infinite order in Z × p /µ p − ,this implies that a F ℓ sends to the inverse of Frobenius in (G ab F ℓ ) b p . By the functoriality of Proposition3.32 part 1 we deduce the same for any finite field of characteristic = p . Now we turn to the case ofan arbitrary number field F . The uniformizers at primes outside a given finite subset generate a densesubgroup of A × F /F × , and every finite extension of F is unramified outside a finite subset, so from thecase of finite fields of charcteristic = p and functoriality we deduce that a F is also the negative of theusual Artin map in this case. Every local field of characteristic is the completion of a number field, sowe deduce from Proposition 3.32 part 2 that a F is also the usual Artin map in that case, and from thisand Proposition 3.32 part 3 we see that a F sends to Frobenius also for F finite of characteristic p .Then we can run the same argument again for global fields and local fields of characteristic p to deducethat a F is the usual Artin map in these remaining cases.To conclude our discussion, we need to show that a Q p has the indicated property. For this it sufficesto see: Lemma 4.6.
Consider the functor
Perf( Q p ) → lc Z classifying the object Q p ∈ lc Z with its Q p -action.The pullback of j ∈ dK Sel (lc Z ) along this functor, viewed as a point in dK Sel ( Q p ) = d K (1) K( Q p ) = d K (1) K( Z p ) , identifies with j Z p . Indeed, this lemma implies that
Art Q p : K( Q p ) → dK Sel ( Q p ) = d K (1) K( Q p ) is the K( Q p ) -linearmap classifying j Z p . But Corollary 1.2 implies that j Z p sends [ u ] to [ u ] , which by Corollary 2.39 translatesexactly into the desired claim.To prove Lemma 4.6, note that in terms of the identification cone(Perf( Z ) → Perf( R )) ≃ lc Z ofTheorem 3.26, this functor Perf( Q p ) → lc Z identifies with the composition of the “boundary functor” ∂ : Perf( Q p ) → cone(Perf { p } ( Z ) → associated to the localization sequence Perf { p } ( Z ) → Perf( Z p ) → Perf( Q p ) as in Remark 3.25, followed by the functor on cones induced by the inclusion Perf { p } ( Z ) → Perf( Z ) .Chasing through localization sequences in L K (1) K and TC b p , we get an identification dK Sel (cone(Perf { p } ( Z ) → ≃ fib( d K (1) TC( Z ) → d T C
TC( Z )) , such that the pullback from dK Sel (lc Z ) corresponds to remembering just the data 2 and 4 from thedescription of points of dK Sel (lc Z ) given above, whereas the pullback to dK Sel ( Q p ) = d K (1) K( Q p ) corresponds to the natural forgetful map to d K (1) TC( Z ) ≃ d K (1) K( Q p ) . This verifies the claim, bythe definition of the point j ∈ dK Sel (lc Z ) . 42 emark 4.7. Much of the complication of this paper, for example the material in section 1.3 and thefairly subtle definition of Selmer K-homology in Section 2, arose from a desire to have a completelyuniform framework for describing Artin maps, both in characteristic and characteristic p . If one iswilling to treat these cases separately, much less background is required. For characteristic we onlyneed d K (1) K , not anything to do with TC. And for characteristic p we only need d T C TC — and wecan define this duality in an easier way since everything is linear over TC( F p ) b p = H Z p ⊕ Ω H Z p insteadof the more complicated TC( Z ) b p . Remark 4.8.
Let us see what our description of the Artin maps says in more concrete terms, when F has characteristic = p . Suppose given an element c ∈ C F , and an element g ∈ Gal(
F /F ) ab . Thenfrom both c and g we can produce a homotopy class of maps of spectra K( F ) → Ω Pic( S b p ) . For c , if we make a similar construction to that of j but using the unlocalized J ’s of Theorem 1.1we can make a map of spectra K(lc Z [1 /p ] ) → Pic( S b p ) . Then the class c ∈ π K(lc F ) and the pairing K( F ) ⊗ K(lc F ) → K(lc Z [1 /p ] ) give the desired map K( F ) → Ω Pic( S b p ) .On the other hand, for g we can use the construction of Remark 2.40, meaning we look at the actionof g on the ℓ -adic etale one-point compactifications of F -vector spaces base-changed to F .What we have proved in this paper is that after K (1) -localization, the classes associated to c and g become equal if and only if c and g match up under the Artin map. This raises a natural question: isthe same statement true without the K (1) -localization? I would conjecture that it is. For finite fields F , the map K( F ) → Ω Pic( S b p ) associated to ∈ C F identifies with the map J F of [C] Section 3.1 asa consequence of the product formula proved in [C], and thus this conjecture reduces to Conjecture 4.5of [C]. Already that case seems hard. The only case for which it seems currently plausible to attack thisconjecture is that of imaginary quadratic fields F , where one can work with elliptic curves with complexmultiplication to mediate between the J ’s of Theorem 1.1 and the J et ’s of Remark 2.19. References [Ba] Tilman Bauer, p-compact groups as framed manifolds , Topology, 2004.[BFM] Paul Baum, William Fulton, Robert Macpherson,
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