Unit L-functions for étale sheaves of modules over noncommutative rings
aa r X i v : . [ m a t h . N T ] S e p UNIT L -FUNCTIONS FOR ´ETALE SHEAVES OF MODULESOVER NONCOMMUTATIVE RINGS MALTE WITTE
Abstract.
Let s : X → Spec F be a separated scheme of finite type over afinite field F of characteristic p , let Λ be a not necessarily commutative Z p -algebra with finitely many elements, and let F • be a perfect complex of Λ-sheaves on the ´etale site of X . We show that the ratio L ( F • , T ) /L (R s ! F • , T ),which is a priori an element of K (Λ[[ T ]]), has a canonical preimage in K (Λ[ T ]).We use this to prove a version of the noncommmutative Iwasawa main conjec-ture for p -adic Lie coverings of X . Let p be a prime number and F the finite field with q = p v elements, and s : X → Spec F a separated finite type F -scheme. Let further Λ be an adic Z p -algebra, i. e. Λ is compact for the topology defined by the powers of its Jacobsonradical Jac(Λ). For any perfect complex of Λ-sheafs F • on the ´etale site of X wehave defined in [Wit09] an L -function L ( F • , T ) attached to F • . This is an elementin the first K -group K (Λ[[ T ]]) of the power series ring Λ[[ T ]] in the formal variable T commuting with the elements of Λ. The total higher direct image R f ! F • is againa perfect complex of Λ-sheaves and so we may form the L -function L (R f ! F • , T ),which is also an element of K (Λ[[ T ]]). Different from the situation where Λ is anadic Z ℓ -algebra with ℓ = p dicussed in [Wit09], the ratio L ( F • , T ) /L (R f ! F • , T )does not need to be 1 in K (Λ[[ T ]]).Let Λ h T i = lim ←− k Λ / Jac(Λ) k [ T ]denote the Jac(Λ)-adic completion of the polynomial ring Λ[ T ] and let b K (Λ h T i ) = lim ←− k K (Λ / Jac(Λ) k [ T ])denote its first completed K -group. If Λ is commutative, then b K (Λ h T i ) = K (Λ h T i ) = Λ h T i × is a subgroup of K (Λ[[ T ]]) = Λ[[ T ]] × and Emerton and Kisin [EK01] show that L ( F • , T ) /L (R f ! F • , T ) ∈ Λ h T i × . If Λ is not commutative, the canonical homomor-phism b K (Λ h T i ) → K (Λ[[ T ]])is no longer injective in general. Nevertheless, we shall prove: Theorem 0.1 (see Thm. 4.1) . There exists a unique way to associate to eachseparated F -scheme s : X → Spec F of finite type, each adic Z p -algebra Λ , and eachperfect complex of Λ -sheaves F • on X an element Q ( F • , T ) ∈ b K (Λ h T i ) such that (1) the image of Q ( F • , T ) in K (Λ[[ T ]]) is the ratio L ( F • , T ) /L (R s ! F • , T ) , (2) Q ( F • , T ) is multiplicative on exact sequences of perfect complexes and de-pends only on the quasi-isomorphism class of F • , (3) Q ( F • , T ) is compatible with changes of the ring Λ . Date : November 3, 2018.
Aside from the result of Emerton and Kisin, a central ingredient for the proofis the recent work of Chinburg, Pappas, and Taylor [CPT11], [CPT13] on the firstK-group of p -adic group rings. In fact, the main strategy of the proof is to reducethe assertion first to the case Λ = Z p [ G ] for a finite group G and then use theresults of Chinburg, Pappas, and Taylor to reduce it further to the case alreadytreated by Emerton and Kisin. In particular, we use almost exclusively methodsfrom representation theory, whereas the result of Emerton and Kisin itself may beconsidered as the geometric input.As an application, we deduce the following version of a noncommutative Iwasawamain conjecture for varieties over finite fields. Assume for the moment that X isgeometrically connected and let G be a factor group of the fundamental group of X such that G ∼ = H ⋊ Γ where H is a compact p -adic Lie group and Γ = Gal( F q p ∞ / F ) ∼ = Z p . We write Z p [[ G ]] = lim ←− Z p [ G/U ]for the Iwasawa algebra of G . Let S = { f ∈ Z p [[ G ]] : Z p [[ G ]] / Z p [[ G ]] f is finitely generated as Z p [[ H ]]-module } denote Venjakob’s canonical Ore set and write Z p [[ G ]] S for the localisation of Z p [[ G ]]at S . We turn Z p [[ G ]] into a smooth Z p [[ G ]]-sheaf M ( G ) on X by letting thefundamental group of X act contragrediently on Z p [[ G ]]. Let R Γ c ( X, F ) denotethe ´etale cohomology with proper support of a flat constructible Z p -sheaf F on X .For every continuous Z p -representation ρ of G , there exists a homomorphism ρ : K ( Z p [[ G ]] S ) → Q ( Z p [[ Γ ]]) × into the units Q ( Z p [[ Γ ]]) × of the field of fractions of Z p [[ Γ ]]. It is induced bysending g ∈ G to det([ g ] ρ ( g ) − ), with [ g ] denoting the image of g in Γ . On theother hand, ρ gives rise to a flat and smooth Z p -sheaf M ( ρ ) on X . Theorem 0.2. (1) R Γ c ( X, M ( G ) ⊗ Z p F ) is a perfect complex of Z p [[ G ]] -modules whose coho-mology groups are S -torsion. In particular, it gives rise to a class [R Γ c ( X, M ( G ) ⊗ Z p F )] − in the relative K -group K ( Z p [[ G ]] , Z p [[ G ]] S ) . (2) There exists an element e L G ( X/ F , F ) ∈ K ( Z p [[ G ]] S ) with the followingproperties: (a) (Characteristic element) The image of e L G ( X/ F , M ( ρ ) ⊗ Z p F ) underthe boundary homomorphism d : K ( Z p [[ G ]] S ) → K ( Z p [[ G ]] , Z p [[ G ]] S ) is [R Γ c ( X, M ( G ) ⊗ Z p F )] − . (b) (Interpolation with respect to all continuous representations) Assumethat ρ is a continuous Z p -representation of G . We let γ denote theimage of the geometric Frobenius in Γ . Then ρ ( e L G ( X/ F , F )) = L ( M ( ρ ) ⊗ Z p F , γ − ) in Q ( Z p [[ Γ ]]) × . This enhances the main result of [Bur11], which also asserts the existence of e L G ( X/ F , F ), but requires only that it satisfies the interpolation property withrespect to finite representations. In fact, we will prove in Section 5 an even moregeneral version of this theorem in the style of [Wit10], replacing Z p by arbitrary NIT L -FUNCTIONS 3 adic Z p -algebras and allowing schemes and coverings which are not necessarilyconnected.We refer to [Bur11] and [Wit13b] for applications of Theorem 0.2.1. Preliminaries on completed K -Theory For any topological ring R (associative and with unity) we let I R denote thelattice of all two-sided open ideals of R . For any n ≥ b K n ( R ) = lim ←− I ∈ I R K n ( R/I )the n -th completed K-group of R . The group b K n ( R ) becomes a topological groupby equipping each K n ( R/I ) with the discrete topology.Recall that we call R an adic ring if R is compact and the Jacobson radicalJac( R ) is open and finitely generated, or equivalently, R = lim ←− k R/ Jac( R ) k with R/ Jac( R ) k a finite ring. Fukaya and Kato showed that for adic rings thecanonical homomorphism K ( R ) → b K ( R ) is an isomorphism [FK06, Prop. 1.5.1].The same is true if R = A [ G ] with G a finite group and A a commutative, p -adicallycomplete, Noetherian integral domain with fraction field of characteristic 0 [CPT13,Thm. 1.2]. We can add the following rings to the list. Proposition 1.1.
Let R be a commutative topological ring such that (1) the topology of R is an ideal topology, i. e. I R is a basis of open neighbour-hoods of , (2) R = lim ←− I ∈ I R R/I, (3) the Jacobson radical
Jac( R ) is open.Then K ( R ) → b K ( R ) is an isomorphism.Proof. For any commutative ring A we have an exact sequence1 → A ) → K ( A ) → K ( A/ Jac( A )) → . Since R/ Jac( R ) carries the discrete topology we haveK ( R/ Jac( R )) = b K ( R/ Jac( R )) . Because of the completeness of R and because Jac( R ) is open we have1 + Jac( R ) = lim ←− I ∈ I R R ) / Jac( R ) ∩ I. The claim now follows from the snake lemma. (cid:3)
Let Λ be an adic ring. We are mainly interested in the topological ringsΛ h T i = lim ←− I ∈ I Λ Λ /I [ T ]where we give the polynomial ring Λ /I [ T ] the discrete topology. As every opentwo-sided ideal I of Λ is finitely generated as left or right ideal, we note that I Λ h T i = ker(Λ h T i → Λ /I [ T ]) and Jac(Λ h T i ) = Jac(Λ)Λ h T i . In particular, theopen ideals of Λ h T i are again finitely generated.We suspect that for all these rings K (Λ h T i ) and b K (Λ h T i ) agree, but we werenot able to prove this in general. By the above results they do agree if either Λis commutative or Λ = Z p [ G ] for a finite group G and this is all we need for ourpurposes. The following result is therefore only for the reader’s edification. MALTE WITTE
Proposition 1.2.
Let Λ be an adic ring. Then the homomorphisms Λ h T i × → K (Λ h T i ) → b K (Λ h T i ) are surjective.Proof. We note that Λ / Jac(Λ) is a finite product of finite-dimensional matrix ringsover finite fields. In particular, K n (Λ / Jac(Λ)[ T ]) = K n (Λ / Jac(Λ)) for all n by acelebrated result of Quillen. Since Λ / Jac(Λ) is semi-simple, the homomorphismΛ / Jac(Λ) × → K (Λ / Jac(Λ)) is surjective. Moreover, K (Λ / Jac(Λ)) = 0 since thisis true for all finite fields.By a result of Vaserstein (see [Oli88, Thm. 1.5]) we have for any ring R and anytwo-sided ideal I ⊂ Jac( R ) an exact sequence1 → V ( R, I ) → I → K ( R, I ) → V ( R, I ) the subgroup of 1 + I generated by the elements (1 + ri )(1 + ir ) − with r ∈ R and i ∈ I . Choosing R = Λ h T i and I = Jac(Λ)Λ h T i we conclude thatΛ h T i × → K (Λ h T i ) is surjective.Since the homomorphisms V (Λ h T i , Jac(Λ h T i )) → V (Λ /I [ T ] , Jac(Λ /I [ T ]))are surjective for any open two-sided ideal I ⊂ Jac(Λ) of Λ, we conclude thatR lim ←− k V (Λ / Jac(Λ) k [ T ] , Jac(Λ)Λ / Jac(Λ) k [ T ]) = 0 . Hence, K (Λ h T i , Jac(Λ h T i )) → lim ←− k K (Λ / Jac(Λ) k [ T ] , Jac(Λ)Λ / Jac(Λ) k [ T ])is surjective. Passing to the limit over the exact sequences1 → K (Λ / Jac(Λ) k [ T ] , Jac(Λ)Λ / Jac(Λ) k [ T ]) → K (Λ / Jac(Λ) k [ T ]) → K (Λ / Jac(Λ)[ T ]) → h T i we conclude thatK (Λ h T i ) → b K (Λ h T i )is surjective. (cid:3) If Λ is an adic ring and P a finitely generated, projective left Λ-module, we set P [ T ] = Λ[ T ] ⊗ Λ P, P h T i = Λ h T i ⊗ Λ P, P [[ T ]] = Λ[[ T ]] ⊗ Λ P Note that P h T i = lim ←− k P/ Jac(Λ) k P [ T ] , P [[ T ]] = lim ←− k P/ Jac(Λ) k P [[ T ]] , and that Λ[[ T ]] is again an adic ring.If Λ ′ is another adic ring acting on P from the right such that P becomes aΛ-Λ ′ -bimodule, then P/ Jac(Λ) P [ T ] is annihilated by some power Jac(Λ ′ ) m ( k ) ofthe Jacobson radical of Λ ′ . This shows that P h T i is a Λ h T i -Λ ′ h T i -bimodule andtherefore induces homomorphismsΨ P h T i : K n (Λ ′ h T i ) → K n (Λ h T i ) . At the same time, it shows that the system ( P/ Jac(Λ) k P [ T ]) k ≥ of Λ / Jac(Λ) k -Λ ′ / Jac(Λ ′ ) m ( k ) -bimodules induces homomorphismsΨ P h T i : b K n (Λ ′ h T i ) → b K n (Λ h T i ) , which are compatible with the above homomorphisms. The construction of Ψ P h T i extends in the obvious manner to complexes P • of Λ-Λ ′ -bimodules which are strictly NIT L -FUNCTIONS 5 perfect as complexes of Λ-modules. By a similar reasoning we also obtain change-of-ring homomorphismsΨ P [[ T ]] • : K n (Λ ′ [[ T ]]) → K n (Λ[[ T ]]) , as well as the corresponding versions for the completed K-theory.2. On K ( Z p [ G ] h T i )In this section, we use the results of Chinburg, Pappas, and Taylor [CPT11],[CPT13] to analyse K ( Z p [ G ] h T i ) for a finite group G .For a noetherian integral domain of finite Krull dimension R with field of frac-tions Q ( R ) of characteristic 0 we setSK ( R [ G ]) = ker(K ( R [ G ]) → K ( Q ( R )[ G ])) , Det( R [ G ] × ) = im( R [ G ] × → K ( O K [ G ] h T i )) / SK ( O K [ G ] h T i ) . Here, Q ( R ) denotes a fixed algebraic closure of Q ( R ). For any subgroup U of R [ G ] × we write Det( U ) for its image in Det( R [ G ] × ). We are mainly interested in the cases R = O K h T i or R = O K [[ T ]] with O K the valuation ring of a finite extension K/ Q p .Assume that L/K is a finite extension such that L is a splitting field for G , i. e. L [ G ] is a finite product of matrix algebras over L . Let M be a maximal Z p -orderinside L [ G ] containing O L [ G ]. Then M is a finite product of matrix algebras over O L such that K ( Mh T i ) → K ( Q ( O K h T i )[ G ])is injective. In particular, we haveSK ( O K [ G ] h T i ) = ker(K ( O K [ G ] h T i ) → K ( Mh T i )) . The same reasoning also applies to SK ( O K [ G ][[ T ]]). Moreover, note that thegroup K ( Mh T i ) injects into K ( M [[ T ]]), such that we also have an injectionDet( O K [ G ] h T i × ) ⊂ Det( O K [ G ][[ T ]] × ). Lemma 2.1.
For any finite group G and any finite field extension K/ Q p , theinclusion O K [ G ] → O K [ G ][[ T ]] induces an isomorphism SK ( O K [ G ]) ∼ = SK ( O K [ G ][[ T ]]) . In particular, K ( O K [ G ][[ T ]] , T O K [ G ][[ T ]]) = Det(1 + T O K [ G ][[ T ]]) . Proof.
The first equality is proved in [Wit09, Prop. 5.4]. Since T ∈ Jac( O K [ G ][[ T ]]),the map 1 + T O K [ G ][[ T ]] → K ( O K [ G ][[ T ]] , T O K [ G ][[ T ]])is surjective by the result of Vaserstein [Oli88, Thm. 1.5] that we already used inthe proof of Prop 1.2. The decompositionK ( O K [ G ][[ T ]]) = K ( O K [ G ]) × K ( O K [ G ][[ T ]] , T O K [ G ][[ T ]])induced by the inclusion O K [ G ] → O K [ G ][[ T ]] and the evaluation T ( O K [ G ][[ T ]]) ∩ K ( O K [ G ][[ T ]] , T O K [ G ][[ T ]]) = 1 . (cid:3) Lemma 2.2.
Let
L/K/ Q p be finite extensions such that L is a splitting field for G and let M be a maximal Z p -order inside L [ G ] containing O L [ G ] . Assume that f isin the intersection of K ( Mh T i , Jac( Mh T i )) and Det(1 + Jac( O K [ G ]) O K [ G ][[ T ]]) inside K ( M [[ T ]]) . Then there exists n ≥ such that f p n ∈ Det( O K [ G ] h T i ) . MALTE WITTE
Proof.
Let p k be the order of a p -Sylow subgroup of G . Since M /p k +2 M is a finitering, some power of Jac( M ) is contained in p k +2 M . Now p k M ⊂ O L [ G ] [Oli88,Thm. 1.4]. For large n we thence have f p n ∈ Det(1 + p O L [ G ] h T i ) ∩ Det(1 + p O K [ G ][[ T ]]) . By [CPT11, Prop. 2.4] the p -adic logarithm induces R -linear isomorphisms v : Det(1 + p R [ G ]) → p R [ C G ]where C G is the set of conjugacy classes of G and R is equal to either O K h T i or O K [[ T ]] for any K . In particular, v ( f p n ) ∈ p O L [ C G ] h T i ∩ p O K [[ T ]][ C G ] = p O K h T i [ C G ] . Hence, f p n ∈ Det(1 + p O K [ G ] h T i ). (cid:3) For any finite group G we let [ G, G ] denote the commutator subgroup of G andset G ab = G/ [ G, G ]. Lemma 2.3.
Let G be a finite p -group and K/ Q p a finite unramified extension.Let further A denote the kernel of O K [ G ] → O K [ G ab ] . The abelian group Det(1 + A O K [ G ][[ T ]]) / Det(1 + A O K [ G ] h T i ) is torsionfree.Proof. Let R be equal to either O K h T i or O K [[ T ]]. Write C G for set of the conju-gacy classes of G and let φ ( AR [ G ]) denote the kernel of the natural R -linear map R [ C G ] → R [ G ab ]. Note that φ ( AR [ G ]) is finitely generated and free as an R -module.For any choice of Frobenius lifts compatible with the inclusion O K [ G ] h T i ⊂ O K [[ T ]]we obtain a commutative diagramDet(1 + A O K [ G ] h T i ) / / ∼ = ν (cid:15) (cid:15) Det(1 + A O K [ G ][[ T ]]) ∼ = ν (cid:15) (cid:15) pφ ( A O K [ G ] h T i ) / / pφ ( A O K [ G ][[ T ]])with the horizontal arrows induced by the natural inclusion and the vertical iso-morphisms induced by the integral group logarithm [CPT11, Thm. 3.16]. Since O K [[ T ]] / O K h T i is a torsionfree abelian group, the same is true for the group pφ ( A O K [ G ][[ T ]]) /pφ ( A O K [ G ] h T i ). (cid:3) Recall that a semi-direct product G = Z /s Z ⋊ P with s prime to p is called p - Q p -elementary if P is a p -group and the image of t : P → ( Z /s Z ) × given by theaction of P on Z /s Z lies in Gal( Q p ( ζ s ) / Q p ) ⊂ ( Z /s Z ) × . For any divisor m of s ,and R = Z p , R = Z p h T i , or R = Z p [[ T ]] we set R [ m ] = Z [ ζ m ] ⊗ Z R and let R [ m ][ P ; t ] denote the twisted group ring for t , i. e. σr = t ( σ )( r ) σ for elements r ∈ R [ m ], σ ∈ P . Set H m = ker( t : P → Gal( Q p ( ζ m ) / Q p )) , B m = P/H m . We may write R [ G ] = Y m | s Z p [ m ][ P ; t ][Oli88, Prop. 11.6]. We then see that R [ G ] is a finitely generated, projective moduleover the subring Y m | s R [ m ][ H m ] ⊂ Y m | s R [ m ][ P ; t ] . NIT L -FUNCTIONS 7 We let r : Det( R [ G ] × ) → Y m | s Det( R [ m ][ H m ] × )denote the corresponding restriction map. We further set A = ker( Z p [ G ] → Y m | s Z p [ m ][ P/ [ H m , H m ]; t ]) ,A m = ker( Z p [ m ][ H m ] → Z p [ m ][ H ab m ]) , and let b m denote the order of B m . Lemma 2.4.
With the notation as above, (1) R [ m ][ P/ [ H m , H m ]; t ] is isomorphic to the ring of b m × b m matrices over itscentre R [ m ][ H ab m ] B m , (2) r induces an isomorphism r : Det(1 + AR [ G ]) → Y m | s Det(1 + A m R [ m ][ H m ]) B m . (3) Det(1 + A Z p [ G ][[ T ]]) / Det(1 + A Z p [ G ] h T i ) is a torsionfree abelian group.Proof. Assertion (1) is a theorem of Wall [Wal74, Thm. 8.3]. Assertion (2) followsfrom [CPT11, Thm 6.2 and Diagramm (6.7)]. Assertion (3) is then a consequenceof Lemma 2.3 and (2). (cid:3)
We now return to the case that G is an arbitrary finite group. For R = Z p , R = Z p h T i , and R = Z p [[ T ]] and any subgroup H ⊂ G we write Res GH for thechange-of-ring homomorphism K ( R [ G ]) → K ( R [ H ]) induced by the Z p [ H ]- Z p [ G ]-bimodule Z p [ G ]. Lemma 2.5.
Let f ∈ K ( Z p [ G ][[ T ]]) such that (1) for any p - Q p -elementary group H , Res GH f is in the image of the homomor-phism K ( Z p [ H ] h T i ) → K ( Z p [ H ][[ T ]]) , (2) f p n is in the image of K ( Z p [ G ] h T i ) → K ( Z p [ G ][[ T ]]) for some n ≥ .Then f is in the image of K ( Z p [ G ] h T i ) → K ( Z p [ G ][[ T ]]) .Proof. The homomorphisms K ( Z p [ G ] h T i ) → K ( Z p [ G ][[ T ]]) for each finite group G constitute a homomorphism of Green modules over the Green ring G G ( Z p [ G ])[CPT13, § f ℓ is in the image ofK ( Z p [ G ] h T i ) → K ( Z p [ G ][[ T ]]) for some integer ℓ prime to p . Because of (2) wemay choose ℓ = 1. (cid:3) Finally, we need the following vanishing result for SK , which is a variant of[FK06, Prop. 2.3.7]. Proposition 2.6.
Let K/ Q p be unramified and R = O K h T i or R = O K [[ T ′ ]] h T i forsome indeterminate T ′ . (More generally, R can be any ring satisfying the standinghypothesis of [CPT13] .) Let further G be a profinite group with cohomological p -dimension cd p G ≤ . For any open normal subgroup U ⊂ G there exists an opensubgroup V ⊂ U normal in G such that the natural homomorphism SK ( R [ G /V ]) → SK ( R [ G /U ]) is the zero map.Proof. For any finite group G , let G r denote the set of p -regular elements, i. e. thoseelements in G of order prime to p . The group G acts on G r via conjugation. Write MALTE WITTE Z p [ G r ] for the free Z p -module generated by G r . By [CPT13, Thm. 1.7] there existsa natural surjection R ⊗ Z p H ( G, Z p [ G r ]) → SK ( R [ G ]) . Since H ( G, Z p [ G r ]) is finite for all finite groups G , it suffices to show thatlim ←− U ⊂G H ( G /U, Z p [( G /U ) r ]) = 0 , where the limit extends over all open normal subgroups of G . After taking thePontryagin dual we deduce the latter fromlim −→ U ⊂G H ( G , Map(( G /U ) r , Q p / Z p )) = 0 . (cid:3) L -functions of perfect complexes of adic sheaves We will briefly recall some notation from [Wit09] (see also [Wit08] and [Wit10]).Let F be the finite field with q = p v elements and fix an algebraic closure F of F .For any scheme X in in the category Sch sep F of separated schemes of finite typeover F and any adic ring Λ we introduced a Waldhausen category PDG cont ( X, Λ)of perfect complexes of adic sheaves on X [Wit09, Def. 4.3]. The objects of thiscategory are certain inverse systems over the index set I Λ such that for I ∈ I Λ the I -th level is a perfect complex of ´etale sheaves of Λ /I -modules.Let us write K ( X, Λ) for the zeroth Waldhausen K-group of
PDG cont ( X, Λ),i. e. the abelian group generated by quasi-isomorphism classes of objects in thecategory
PDG cont ( X, Λ) modulo the relations[ G • ][ F • ] − [ H • ] − for any sequence 0 → F • → G • → H • → PDG cont ( X, Λ) which is exact (in each level I ∈ I Λ ).For any morphism f : X → Y in Sch sep F we have Waldhausen exact functors f ∗ : PDG cont ( Y, Λ) → PDG cont ( X, Λ) , R f ! : PDG cont ( X, Λ) → PDG cont ( Y, Λ)that correspond to the usual inverse image and the direct image with proper sup-port. As Waldhausen exact functors they induce homomorphisms f ∗ : K ( Y, Λ) → K ( X, Λ) , R f ! : K ( X, Λ) → K ( Y, Λ) . If Λ ′ is a second adic ring we let Λ op - SP (Λ ′ ) denote the Waldhausen category ofcomplexes of Λ ′ -Λ-bimodules which are strictly perfect as complexes of Λ ′ -modules.For each such complex P • we have a change-of-ring homomorphismΨ P • : K ( X, Λ) → K ( X, Λ ′ ) . The compositions of these homomorphisms behave as expected. In particular, Ψ P • commutes with f ∗ and R f ! , for f : X → Y , g : Y → Z we have ( g ◦ f ) ∗ = f ∗ ◦ g ∗ ,R( f ◦ g ) ! = R f ! ◦ R g ! , and R f ′ ! g ′∗ = g ∗ R f ! if W f ′ / / g ′ (cid:15) (cid:15) X g (cid:15) (cid:15) Y f / / Z is a cartesian square [Wit09, § NIT L -FUNCTIONS 9 For any A ∈ K ( X, Λ) we define the L -function L ( A, T ) ∈ K (Λ[[ T ]]) as follows.First, assume that X = Spec F ′ for a finite field extension F ′ / F of degree d , that Λis finite and that A is the class of a locally constant flat ´etale sheaf of Λ-moduleson X . This sheaf corresponds to a finitely generated, projective Λ-module P witha continuous action of the absolute Galois group of F ′ , which is topologically gen-erated by the geometric Frobenius automorphism F F ′ of F ′ . We then let L ( A, T ) bethe inverse of the class of the automorphism id − F F ′ T d of Λ[[ T ]] ⊗ Λ P in K (Λ[[ T ]]).If A is the class of any perfect complex of ´etale sheaves of Λ-modules on Spec F ′ ,we replace it by a quasi-isomorphic, strictly perfect complex P • and define L ( A, T )as the alternating product L ( A, T ) = Y k L ( P k , T ) ( − k . This then extends to a group homomorphism K (Spec F ′ , Λ) → K (Λ[[ T ]]). If Λ isan arbitrary adic ring and A ∈ K (Spec F ′ , Λ), then L ( A, T ) is given by the system( L (Ψ A/I ( A ) , T )) I ∈ I Λ in K (Λ[[ T ]]) = lim ←− I ∈ I Λ K (Λ /I [[ T ]]) . Finally, if X is any separated scheme over F , we let X denote the set of closedpoints of X and x : Spec k ( x ) → X the closed immersion corresponding to any x ∈ X . We set L ( A, T ) = Y x ∈ X L ( x ∗ ( A ) , T ) ∈ K (Λ[[ T ]]) . The product converges in the topology of K (Λ[[ T ]]) induced by the adic topologyof Λ[[ T ]] because T ∈ Jac(Λ[[ T ]]) and for any d there are only finitely many closedpoints of degree d in X . We have thus constructed a group homomorphism L : K ( X, Λ) → K (Λ[[ T ]]) , A L ( A, T ) , for any X in Sch sep F and any adic ring Λ.This construction agrees with [Wit09, Def. 6.4]. If Λ is commutative, such thatK (Λ[[ T ]]) ∼ = Λ[[ T ]] × via the determinant map, it is also seen to agree with theclassical definition used in [EK01]. Moreover, we note that for any pair of adicrings Λ and Λ ′ and any complex P • of bimodules in Λ op - SP (Λ ′ ), we have L (Ψ P • ( A ) , T ) = Ψ P [[ T ]] • ( L ( A, T ))for A ∈ K ( X, Λ).
Remark . Note that L ( A, T ) depends on the base field F . If F ′ ⊂ F is a subfield,then the L -function of A with respect to F ′ is L ( A, T [ F : F ′ ] ) [Wit09, Rem. 6.5].4. The construction of unit L -functions In this section, we prove the following theorem.
Theorem 4.1.
Let s : X → Spec F be a separated F -scheme of finite type. Thereexists a unique way to associate to each adic Z p -algebra Λ a homomorphism Q : K ( X, Λ) → b K (Λ h T i ) , A Q ( A, T ) , such that for A ∈ K ( X, Λ)(1) the image of Q ( A, T ) in K (Λ[[ T ]]) is the ratio L ( A, T ) /L (R s ! A, T ) , (2) if Λ ′ is a second adic ring and P • is in Λ ′ op - SP (Λ) , then Ψ P h T i • ( Q ( A, T )) = Q (Ψ P • ( A ) , T ) . Proof.
If we restrict to the class of adic rings Λ which are full matrix algebrasover commutative adic Z p -algebras, then the natural homomorphism b K (Λ h T i ) → K (Λ[[ T ]]) is injective and the existence of Q : K ( X, Λ) → b K (Λ h T i ) follows from[EK01, Cor. 1.8] and Morita invariance. In fact, we even know that the image of Q lies in the subgroup K (Λ h T i , Jac(Λ) T Λ h T i ). (The result of Emerton and Kisinis stated only for F = F p , but the result for general F follows from Remark 3.1and the simple observation that λ ( T ) ∈ Λ[[ T ]] is in Λ h T i if and only if for some n > λ ( T n ) ∈ Λ h T i .) We note further that it suffices to prove the assertion ofthe theorem for the class of finite Z p -algebras. Then general case then follows bytaking projective limits.We proceed by induction on dim X . So, we assume that the theorem has alreadybeen proved for all schemes in Sch sep F of dimension less than dim X . Note thatany open subscheme j : U → X with closed complement i : Z → X induces adecompositionK ( U, Λ) × K ( Z, Λ) ∼ = −→ K ( X, Λ) , ( A, B ) (R j ! A )(R i ! B )and if the theorem is true for U and Z , then it is also true for X . In particular, wemay reduce to the case that X is an integral scheme.If Λ is finite then each object F • in PDG cont ( X, Λ) is quasi-isomorphic to astrictly perfect complex P • of ´etale sheaves of Λ-modules on X . This means, P n isflat and constructible and for | n | sufficiently large, P n = 0. In particular, we maychoose j : U → X open and dense such that j ∗ P n is locally constant for each n .We may then find a finite connected Galois covering g : V → U with Galois group G and a complex P • in Z p [ G ] op - SP (Λ) such that j ∗ P • ∼ = Ψ P • ( g ! g ∗ Z p )[Wit09, Lemma 4.12]. Here, we consider g ! g ∗ Z p as an object in PDG cont ( U, Z p [ G ]).Since the function field of X is of characteristic p , the cohomological p -dimensionof its absolute Galois group 1. So, we may apply Prop. 2.6 and choose G (afterpossibly shrinking U ) large enough such thatΨ P h T i • : K ( Z p [ G ] h T i ) → K (Λ h T i )factors through Det( Z p [ G ] h T i × ).Let s : U → Spec F be the structure map. We will now show that α = L ( g ! g ∗ Z p , T ) /L (R s ! ( g ! g ∗ Z p ) , T ) ∈ K ( Z p [ G ][[ T ]])has a preimage in K ( Z p [ G ] h T i ). First, we note that by construction, α is inthe subgroup K ( Z p [ G ][[ T ]] , T Z p [ G ][[ T ]]). Further, by [Del77, Fonction L mod ℓ n , Theorem 2.2.(b)], we know that the image of α in K ( Z p [ G ] / Jac( Z p [ G ])[[ T ]])vanishes. Hence, using Lemma 2.1, we may view α as an element of the groupDet(1+Jac( Z p [ G ]) T Z p [ G ][[ T ]]). From [EK01, Cor. 1.8] and Lemma 2.2 we concludethat α p n is in Det( Z p [ G ] h T i × ) for some n ≥
0. By Lemma 2.5 it thence suffices toshow that for every Q p - p -elementary subgroup H ⊂ G , the elementRes HG α = L (Res HG g ! g ∗ Z p , T ) /L (R s ! Res HG g ! g ∗ Z p , T ) ∈ K ( Z p [ H ][[ T ]])has a preimage in K ( Z p [ H ] h T i ). Choosing the ideal A ⊂ Z p [ H ] as in Lemma 2.4and noting that by part (1) of this lemma, Z p [ H ] /A is a finite product of full matrixrings over commutative adic rings, we can again apply [EK01, Cor. 1.8] to see thatthe image of Res HG α inK ( Z p [ H ] /A [[ T ]]) / K ( Z p [ H ] /A h T i ) = Det( Z p [ H ] /A [[ T ]] × ) / Det( Z p [ H ] /A h T i × ) NIT L -FUNCTIONS 11 vanishes. From the exact sequenceDet(1+ A Z p [ H ][[ T ]]) / Det(1+ A Z p [ H ] h T i ) → Det( Z p [ H ][[ T ]] × ) / Det( Z p [ H ] h T i × ) → Det( Z p [ H ] /A [[ T ]] × ) / Det( Z p [ H ] /A h T i × ) → HG α does indeed have a preimagein K ( Z p [ H ] h T i ). Thus, the element α has a preimage e α in K ( Z p [ G ] h T i ).We return to the complex F • and define Q ( F • , T ) = Q ( i ∗ F • , T ) · Ψ P h T i • ( e α ) . We need to check that this definition does not depend on the choices of U , V , P • , and e α . For this, let j ′ : U ′ → U be an open dense subscheme with closedcomplement i ′ : Z ′ → U . Then g : U ′ × U V → U ′ is still a finite connected Galoiscovering with Galois group G . Let h : V ′ → U ′ × U V be a finite connected Galoiscovering with Galois group H and assume that g ′ = g ◦ h is Galois with Galoisgroup G ′ such that G ′ /H = G . Assume further that P ′• is a complex in Z p [ G ′ ] op - SP (Λ) such that Ψ P ′• ( g ′ ! g ′∗ Z p ) is quasi-isomorphic to ( j ′ ◦ j ) ∗ F • . By taking thestalks of Ψ P ′• ( g ′ ! g ′∗ Z p ) and Ψ P • ( g ! g ∗ Z p ) at any geometric point of U ′ , we see that P ′• ⊗ Z p [ G ′ ] Z p [ G ] is quasi-isomorphic to P • in Z p [ G ] op - SP (Λ). In particular,Ψ P ′ h T i • = Ψ P h T i • ◦ Ψ Z p [ G ] h T i : K ( Z p [ G ′ ] h T i ) → K (Λ h T i ) . As above, we construct a preimage e α ′ of L ( g ′ ! g ′∗ Z p , T ) /L (R( s ◦ j ′ ) ! ( g ′ ! g ′∗ Z p ) , T ) inK ( Z p [ G ′ ] h T i ). Since Det( Z p [ G ] h T i × ) → Det( Z p [ G ][[ T ]] × )is injective, we see that the image of e α/ Ψ Z p [ G ] h T i ( e α ′ ) in Det( Z p [ G ] h T i × ) must agreewith the unique preimage of L ( i ′∗ g ! g ∗ Z p , T ) /L (R( s ◦ i ′ ) ! i ′∗ g ! g ∗ Z p , T ), which in turnagrees with the image of Q ( i ′∗ g ! g ∗ Z p , T ) by the induction hypothesis. Hence,Ψ P h T i • ( e α ) = Ψ P h T i • ( Q ( i ′∗ g ! g ∗ Z p , T )Ψ Z p [ G ] h T i ( e α ′ )) = Q ( i ′∗ F • , T )Ψ P ′• ( e α ′ ) . Let i ′′ : Z ′′ → X be the closed complement of U ′ in X . By the uniqueness of Q for Z ′′ we conclude Q ( i ′′∗ F • , T ) = Q ( i ′∗ F • , T ) Q ( i ∗ F • , T )and so, the above definition of Q ( F • , T ) is independent of all choices.Assume now that Λ ′ is a second finite Z p -algebra and that W • is in Λ op - SP (Λ ′ ).Then Ψ W h T i • ( Q ( F • , T )) = Ψ W h T i • ( Q ( i ∗ F • , T )Ψ P h T i • ( e α ))= Q ( i ∗ Ψ W • ( F • ) , T )Ψ W ⊗ Λ P h T i • ( e α )= Q (Ψ W • ( F • ) , T ) . It is immediately clear from the definition that Q ( F • , T ) does only depend onthe quasi-isomorphism class of F • . Moreover, if Λ is finite, any exact sequence in PDG cont ( X, Λ) can be completed to a diagram0 / / P • / / (cid:15) (cid:15) P • / / (cid:15) (cid:15) P • / / (cid:15) (cid:15) / / F • / / F • / / F • / / j : U → X , i : Z → X , g : V → U , G as above and an exact sequence0 → P • → P • → P • → of complexes in Z p [ G ] op - SP (Λ) such that for k = 1 , , P • k ( g ! g ∗ Z p ) ∼ = j ∗ P • k . Choosing e α as above, we conclude Q ( F • , T ) = Q ( i ∗ F • , T ) Q ( i ∗ F • , T )Ψ P h T i • ( e α )Ψ P h T i • ( e α )= Q ( F • , T ) Q ( F • , T ) . Thus, Q is well-defined as homomorphism K ( X, Λ) → K (Λ h T i ) and satisfiesproperties (1) and (2) of the theorem.It remains to prove uniqueness. So let Q ′ be a second family of homomorphismsK ( X, Λ) → b K (Λ h T i ) ranging over all adic rings Λ such that (1) and (2) aresatisfied. If i : Z → X is any closed subscheme, thenK ( Z, Λ) → b K (Λ h T i ) , A Q ′ ( i ∗ A, T ) , still satisfies (1) and (2). Thus, by the induction hypothesis, we must have Q ′ ( i ∗ A, T ) = Q ( A, T ) . If Λ is finite and G is a finite group and P • a complex in Z p [ G ] op - SP (Λ) such thatΨ P h T i • factors through Det( Z p [ G ] h T i ), then by the injectivity ofDet( Z p [ G ] h T i ) → Det( Z p [ G ][[ T ]])and properties (1) and (2) we must have Q ′ (Ψ P h T i • ( B ) , T ) = Q (Ψ P h T i • ( B ) , T )for all B ∈ K ( X, Z p [ G ]). By the construction of Q we thus see that Q = Q ′ . Thiscompletes the proof of the theorem. (cid:3) We conclude with some ancillary observations. First, we note that Q depends inthe same way as the L -function on the choosen base field F : Proposition 4.2.
Let X be a separated scheme of finite type over F and F ′ ⊂ F bea subfield. Write Q ′ : K ( X, Λ) → b K (Λ h T i ) for the homomorphism resulting fromapplying Thm 4.1 to X considered as F ′ -scheme. Then Q ′ ( A, T ) = Q ( A, T [ F ′ : F ] ) for every A ∈ K ( X, Λ) .Proof. By Remark 3.1, the given equality holds for the images in K (Λ[[ T ]]). Nowone proceeds as in the proof of the uniqueness part of Thm 4.1 to show that it alsoholds in b K (Λ h T i ). (cid:3) As in [EK01], we can also define a version of Q relative to a morphism f : X → Y in Sch sep F by setting Q ( f ) : K ( X, Λ) → b K (Λ h T i ) , A Q ( f, A, T ) = Q ( A, T ) /Q (R f ! A, T ) , and extend [EK01, Lemma 2.4, 2.5] as follows. Proposition 4.3.
Let A be an object in K ( X, Λ) . (1) If f : X → Y , g : Y → Z are two morphisms in Sch sep F , then Q ( g ◦ f, A, T ) = Q ( g, R f ! A, T ) Q ( f, A, T ) . (2) If f : X → Y is quasi-finite, then Q ( f, A, T ) = 1 . NIT L -FUNCTIONS 13 Proof.
Assertion (1) follows immediately from the definition. For the second asser-tion, we note that for f : X → Y quasi-finite, L (R f ! A, T ) = Y y ∈ Y L ( y ∗ f ! A, T )= Y y ∈ Y Y x ∈ f − ( y ) L ( x ∗ A, T )= Y x ∈ X L ( x ∗ A, T ) = L ( A, T ) . In particular, the images of Q ( A, T ) and Q (R f ! A, T ) in K (Λ[[ T ]]) agree. By theuniqueness of Q , we must have Q ( A, T ) = Q (R f ! A, T )in b K (Λ h T i ). (cid:3) Finally, setting b K ( X, Λ) = lim ←− I ∈ I Λ K ( X, Λ /I ) , we observe that the constructions of the L -function and of Q extends to L : b K ( X, Λ) → K (Λ[[ T ]]) , Q : b K ( X, Λ) → b K (Λ h T i ) . We cannot say much about the canonical homomorphism K ( X, Λ) → b K ( X, Λ),but we suspect that it might be neither injective nor surjective in general.5.
A noncommutative main conjecture for separated schemes over F In this section, we will complete the ℓ = p case of the version of the noncommu-tative Iwasawa main conjecture for separated schemes X over F = F q considered in[Wit10]. We will briefly recall the terminology of the cited article.Recall from [Wit10, Def. 2.1] that a principal covering ( f : Y → X, G ) of X with G a profinite group is an inverse system ( f U : Y U → X ) U ∈ NS ( G ) of finiteprincipal G/U -coverings (not necessarily connected), where U runs through thelattice NS ( G ) of open normal subgroups of G . As a particular case, for k prime to p and Γ kp ∞ = Gal( F kp ∞ q / F q ), we have the cyclotomic Γ kp ∞ -covering( X kp ∞ = X × Spec F q Spec F q kp ∞ → X, Γ kp ∞ )[Wit10, Def. 2.5]. We will only consider principal coverings ( f : Y → X, G ) suchthat there exists a closed normal subgroub H ⊂ G which is a topologically finitelygenerated virtual pro- p -group and such that the quotient covering ( f H : Y H → X, G/H ) is the cyclotomic Γ p ∞ -covering. These coverings will be called admissiblecoverings for short [Wit10, Def. 2.6]. For such a group G = H ⋊ Γ p ∞ , if Λ is anyadic Z p -algebra, then the profinite group ring Λ[[ G ]] is again an adic Z p -algebra[Wit10, Prop. 3.2].For any admissible covering ( f : Y → X, G ) we constructed in [Wit10, Prop. 6.2]a Waldhausen exact functor f ! f ∗ : PDG cont ( X, Λ) → PDG cont ( X, Λ[[ G ]])by forming the inverse system over the intermediate finite coverings. As before, wewill denote the induced homomorphism f ! f ∗ : K ( X, Λ) → K ( X, Λ[[ G ]])by the same symbol.We also constructed a localisation sequence1 → K (Λ[[ G ]]) → K loc1 ( H, Λ[[ G ]]) d −→ K rel0 ( H, Λ[[ G ]]) → with K loc1 ( H, Λ[[ G ]]) = K ( w H PDG cont (Λ[[ G ]])) , K rel0 ( H, Λ[[ G ]]) = K ( PDG cont ,w H (Λ[[ G ]]))and certain Waldhausen categories w H PDG cont (Λ[[ G ]]) and PDG cont ,w H (Λ[[ G ]]),respectively [Wit10, § H ]] is noetherian, thereexists a left denominator set S such thatK loc1 ( H, Λ[[ G ]]) = K (Λ[[ G ]] S ) , K rel0 ( H, Λ[[ G ]]) = K (Λ[[ G ]] , Λ[[ G ]] S ) , where Λ[[ G ]] S is the localisation of Λ[[ G ]] at S and the K-groups on the right-handside are the usual groups appearing in the corresponding localisation sequence ofhigher K-theory [Wit13a, Thm. 2.18, Prop. 2.20, Rem 2.22]. In particular, if Λ iscommutative (hence, noetherian) and G = Γ kp ∞ , then the Ore set S is given by S = { λ ∈ Λ[[ Γ kp ∞ ]] : λ is a nonzerodivisor in Λ / Jac(Λ)[[ Γ kp ∞ ]] } and K loc1 ( H, Λ[[ G ]]) = K (Λ[[ G ]] S ) = Λ[[ Γ kp ∞ ]] × S . Let Λ ′ be a second adic ring and M • a complex in Λ[[ G ]] op − SP (Λ ′ ). Then f M • = Ψ M • ( f ! f ∗ Λ)is a perfect complex of smooth Λ ′ -adic sheaves on X with an additional right Λ-module structure. Using this complex we obtain a homomorphismΨ f M • : K ( X, Λ) → K ( X, Λ ′ ) . Furthermore, we can form the complex M [[ G ]] δ • = Λ ′ [[ G ]] ⊗ Λ M • in Λ[[ G ]] op - SP (Λ[[ G ]]) with the canonical left G -action and the diagonal right G -action.If G acts trivially on M • , then Ψ f M • is just the homomorphismΨ M • : K ( X, Λ) → K ( X, Λ ′ )that we have already used above. Moreover, we haveΨ M [[ G ]] δ • f ! f ∗ A = f ! f ∗ Ψ f M • ( A )[Wit10, Prop. 6.7]. Note that M [[ G ]] δ • also induces compatible homomorphismson K (Λ[[ G ]]), K loc1 ( H, Λ[[ G ]]) and K rel0 ( H, Λ[[ G ]]) [Wit10, Prop. 4.6].As a special case we may take Λ = Λ ′ = Z p and let ρ : G → GL n ( Z p ) be acontinuous left G -representation as in the introduction. We may then choose M tobe the Z p - Z p [[ G ]]-module obtained from ρ by letting G act contragrediently on Z np from the right. The sheaf f M is then just the smooth Λ-adic sheaf M ( ρ ) associatedto ρ and Ψ f M corresponds to taking the (completed) tensor product with this sheafover Z p .In [Wit10, Thm. 8.1] we have already shown that for each complex F • in theWaldhausen category PDG cont ( X, Λ) and each admissible covering ( f : Y → X, G )the complex of Λ-adic cohomology with proper supportR Γ c ( X, f ! f ∗ F • )is an object of PDG cont ,w H (Λ[[ G ]]). Hence, we obtain a homomorphismK ( X, Λ) → K rel0 ( H, Λ[[ G ]]) , A R Γ c ( X, f ! f ∗ A ) . NIT L -FUNCTIONS 15 We have also contructed an explicit homomorphismK ( X, Λ) → K loc1 ( H, Λ[[ G ]]) , A
7→ L G ( X/ F , A ) , such that d L G ( X/ F , A ) = R Γ c ( X, f ! f ∗ A ) − [Wit10, Def. 8.3].We let γ denote the image of the geometric Frobenius automorphism F F in Γ kp ∞ .If Ω is a commutative adic Z p -algebra, we write e S ⊂ Ω h T i for the denominator setconsisting of those elements which become a unit in Ω[[ T ]]. The proof of [Wit10,Lemma 8.5] shows that the evaluation T γ − extends to a ring homomorphismΩ h T i e S → Ω[[ Γ kp ∞ ]] S , ω ( T ) ω ( γ − ) . Note that Ω h T i e S is a semilocal ring, henceΩ h T i × e S = K (Ω h T i e S ) . Let s : X → Spec F be the structure map. Then the proof of [Wit10, Thm. 8.6]shows that for every M • in Λ[[ G ]] op - SP (Ω) and every A ∈ K ( X, Λ), we have L (cid:0) R s ! Ψ f M • ( f ! f ∗ A ) , T (cid:1) ∈ K (Ω h T i e S )and that Ψ Ω[[ Γ kp ∞ ]] Ψ M [[ G ]] δ • ( L G ( X/ F , A )) = L (cid:0) R s ! Ψ f M • ( f ! f ∗ A ) , γ − (cid:1) in K loc1 ( H, Ω[[ Γ kp ∞ ]]) = Ω[[ Γ kp ∞ ]] × S . So, L G ( X/ F , A ) satisfies the desired interpolation property with respect to thefunctions L (cid:0) R s ! Ψ f M • ( f ! f ∗ A ) , γ − (cid:1) , but not with respect to L (Ψ f M • ( f ! f ∗ A ) , γ − ).We will construct a modification of L G ( X/ F , A ) below.For any adic ring Λ, the evaluation map T b K (Λ h T i ) → K (Λ) , λ ( T ) λ (1) . We also obtain an evaluation map b K (Λ h T i ) → K (Λ[[ Γ kp ∞ ]]) , λ ( T ) λ ( γ − ) , as composition of T b K (Λ[[ Γ kp ∞ ]] h T i ) induced by T γ − T and the injectionΨ Λ[[ Γ kp ∞ ]] h T i : b K (Λ h T i ) → b K (Λ[[ Γ kp ∞ ]] h T i ) . Definition 5.1.
For any admissible covering ( f : Y → X, G ) and any A ∈ K ( X, Λ)we set e L G ( X/ F , A ) = L G ( X/ F , A ) Q ( f ! f ∗ A, . Since Q ( f ! f ∗ A, ∈ K (Λ[[ G ]]), we still have Theorem 5.2. d e L G ( X/ F , A ) = R Γ c ( X, f ! f ∗ A ) − in K loc0 ( H, Λ[[ G ]]) . We will now investigate the transformation properties of e L G ( X/ F , A ). Theorem 5.3.
Consider a separated scheme X of finite type over a finite field F .Let Λ be any adic Z p algebra and let A be in K ( X, Λ) . (1) Let Λ ′ be another adic Z p -algebra. For any complex M • in Λ[[ G ]] op - SP (Λ ′ ) ,we have Ψ M [[ G ]] δ • ( e L G ( X/ F , A )) = e L G ( X/ F , Ψ f M • ( A )) in K loc1 ( H, Λ ′ [[ G ]]) . (2) Let H ′ be a closed virtual pro- p -subgroup of H which is normal in G . Then Ψ Λ[[
G/H ′ ]] ( e L G ( X/ F , A )) = e L G/H ′ ( X/ F , A ) in K loc1 ( H ′ , Λ[[
G/H ′ ]]) . (3) Let U be an open subgroup of G and let F ′ be the finite extension corre-sponding to the image of U in Γ p ∞ . Then Ψ Λ[[ G ]] (cid:0) e L G ( X/ F , A ) (cid:1) = e L U ( Y U / F ′ , f ∗ U A ) in K ( H ∩ U, (Λ[[ U ]])) .Proof. In [Wit10, Thm. 8.4] we have already proved that L G ( X/ F , A ) satisfies thegiven transformation properties. To prove the same properties for Q ( f ∗ f ∗ A, Q ( A, T ) and Ψ and the equalitiesΨ M [[ G ]] δ • f ! f ∗ A = f ! f ∗ Ψ f M • ( A )Ψ Λ[[
G/H ′ ]] f ! f ∗ A = f H ′ ! f ∗ H ′ A Ψ Λ[[ G ]] f ! f ∗ A = R f U ! f U ! f U ∗ f ∗ U A with ( f U : Y → Y U , U ) the restriction of the covering to U [Wit10, Prop. 6.5, 6.7].For (3) it remains to notice that the evaluation Q ( A,
1) does not depend on thebase field F by Prop. 4.2. (cid:3) Proposition 5.4.
Consider the admissible covering ( f : X kp ∞ → X, Γ kp ∞ ) . Forany adic ring Λ and any A ∈ K ( X, Λ) , Q ( f ! f ∗ A, T ) = Q (Ψ Λ[[ Γ kp ∞ ]] ( A ) , γ − T ) in b K (Λ[[ Γ kp ∞ ]] h T i ) .Proof. We may assume that Λ is finite. Let s : X → F denote the structure map.From [Wit10, Prop. 7.2] it follows that L (R s ! f ! f ∗ A, T ) = L (R s ! Ψ Λ[[ Γ kp ∞ ]] ( A ) , γ − T ) . By applying this to x ∗ A for each closed point x : Spec k ( x ) → X of X and using f ! f ∗ x ∗ A = x ∗ f ! f ∗ A [Wit10, Prop. 6.4.(1)] we see that L ( f ! f ∗ A, T ) = L (Ψ Λ[[ Γ kp ∞ ]] ( A ) , γ − T ) . Hence, the images of Q ( f ! f ∗ A, T ) and Q (Ψ Λ[[ Γ kp ∞ ]] ( A ) , γ − T ) agree in the groupK (Λ[[ Γ kp ∞ ]][[ T ]]).If a : X ′ → X is a morphism in Sch sep F and ( f : X ′ → X, Γ kp ∞ ) is the cyclotomic Γ kp ∞ -covering of X ′ , then R a ! f ! f ∗ A = f ! f ∗ R a ! A by [Wit10, Prop. 6.4.(2)]. In particular, this applies to open and closed immersions.By induction on the dimension of X we can thus reduce to the case X integral and A = Ψ P • ( g ! g ∗ Z p ) for some finite connected Galois covering ( g : Y → X, G ) andsome complex P • in Z p [ G ] op - SP (Λ). With the complex P [[ Γ kp ∞ ]] • = Λ[[ Γ kp ∞ ]] ⊗ Λ P • in Z p [[ G × Γ kp ∞ ]]- SP (Λ[[ Γ kp ∞ ]]) we haveΨ P [[ Γ kp ∞ ]] • ( f ! f ∗ g ! g ∗ Z p ) = f ! f ∗ A, Ψ P [[ Γ kp ∞ ]] • (Ψ Λ[[ Γ kp ∞ ]] ( g ! g ∗ Z p )) = Ψ Λ[[ Γ kp ∞ ]] ( A ) . Applying Prop. 2.6 to R = Z p [[ Γ kp ∞ ]] h T i ∼ = Z p [ Z /k Z ][[ T ′ ]] h T i NIT L -FUNCTIONS 17 we may assume that Ψ P [[ Γ kp ∞ ]] h T i • factors through Det( Z p [[ G × Γ kp ∞ ]] h T i ). Weconclude Q ( f ! f ∗ A, T ) = Ψ P [[ Γ kp ∞ ]] h T i • ( Q ( f ! f ∗ g ! g ∗ Z p , T ))= Ψ P [[ Γ kp ∞ ]] h T i • ( Q ( g ! g ∗ Z p , γ − T ))= Q ( A, γ − T ) . (cid:3) The following theorem shows that e L G ( X/ F , A ) satisfies the right interpolationproperty. Theorem 5.5.
Let X be a scheme in Sch sep F and let ( f : Y → X, G ) be an admis-sible principal covering containing the cyclotomic Γ kp ∞ -covering. Furthermore, let Λ and Ω be adic Z p -algebras with Ω commutative. For every A ∈ K ( X, Λ) andevery M • in Λ[[ G ]] op - SP (Ω) , we have L (Ψ f M • ( A ) , T ) ∈ K (Ω h T i e S ) and Ψ Ω[[ Γ kp ∞ ]] Ψ M [[ G ]] δ • (cid:0) e L G ( X/ F , A ) (cid:1) = L (Ψ f M • ( A ) , γ − ) in K (Ω[[ Γ kp ∞ ]] S ) .Proof. As remarked above, the corresponding statement for L (R s ! Ψ f M • ( A ) , T ) and L G ( X/ F , A ) follows from [Wit10, Thm. 8.6]. Since L (Ψ f M • ( A ) , T ) = Q (Ψ f M • ( A ) , T ) L (R s ! Ψ f M • ( A ) , T ) , an application of Prop. 5.4 concludes the proof of the theorem. (cid:3) As in the case where p is not equal to the characteristic of F , the element L ( A, γ − ) interpolates the values L ( A ( ǫ r ) , A ( ǫ r ) denotes the twist of A by the r -th power of the cyclotomic character ǫ . However, different from thissituation, we obtain no information on the values L ( A, q n ). In particular, we stilllack a noncommutative analogue of the description of these values as obtained byMilne [Mil86] in the commutative case. References [Bur11] D. Burns,
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Malte WitteRuprecht-Karls-Universit¨at HeidelbergMathematisches InstitutIm Neuenheimer Feld 288D-69120 Heidelberg
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