On a Localisation Sequence for the K-Theory of Skew Power Series Rings
aa r X i v : . [ m a t h . K T ] J u l ON A LOCALISATION SEQUENCE FOR THE K -THEORY OFSKEW POWER SERIES RINGS MALTE WITTE
Abstract.
Let B = A [[ t ; σ, δ ]] be a skew power series ring such that σ isgiven by an inner automorphism of B . We show that a certain Waldhausenlocalisation sequence involving the K-theory of B splits into short split exactsequences. In the case that A is noetherian we show that this sequence is givenby the localisation sequence for a left denominator set S in B . If B = Z p [[ G ]]happens to be the Iwasawa algebra of a p -adic Lie group G ∼ = H ⋊ Z p , this set S is Venjakob’s canonical Ore set. In particular, our result implies that0 → K n +1 ( Z p [[ G ]]) → K n +1 ( Z p [[ G ]] S ) → K n ( Z p [[ G ]] , Z p [[ G ]] S ) → n ≥
0. We also prove the corresponding result for thelocalisation of Z p [[ G ]][ p ] with respect to the Ore set S ∗ . Both sequences playa major role in non-commutative Iwasawa theory. Introduction
Let B = A [[ t ; σ, δ ]] be a skew power series rings and assume that σ extends toan inner automorphism of B . In this article, we exploit an observation first madeby Schneider and Venjakob [SV06], namely the existence of a short exact sequence( ∗ ) 0 → B ˆ ⊗ A M → B ˆ ⊗ A M → M → B -module M . (In loc. cit. , M was also assumed to be finitelygenerated over A , but we will show below that this is not necessary.) This was used— first in the context of non-commutative Iwasawa theory by Burns [Bur09], thenin general by Schneider and Venjakob [SV10] — to construct a natural splitting ofthe boundary map K ( B S ) ∂ −→ K ( B, B S )in the K-theory localisation sequence of a regular noetherian skew power series ring B and its localisation B S with respect to a certain left denominator set S .Generalising this result, we show that by an application of Waldhausen’s ad-ditivity theorem [Wal85] to the above exact sequence and by a well-known resultin homotopy theory one obtains a natural splitting of the boundary map ∂ n forevery n . Using Waldhausen’s construction of K-theory for arbitrary Waldhausencategories also helps us to extend this result to non-regular skew power series andeven to situations where no suitable left denominator set S is known to exist.The main application of our result lies in the area of non-commutative Iwasawatheory. Let H be a p -adic Lie group and G a semi-direct product of H and Z p . Fixa topological generator γ of Z p . Then B = Z p [[ G ]] is a skew power series ring over A = Z p [[ H ]] with σ given by conjugation with γ , t = γ − δ = σ − id. The set S is Venjakob’s canonical Ore set and the sequenceK ( Z p [[ G ]]) → K ( Z p [[ G ]] S ) ∂ −→ K ( Z p [[ G ]] , Z p [[ G ]] S ) → Date : November 18, 2018.1991
Mathematics Subject Classification. is used to formulate the main conjecture of non-commutative Iwasawa theory (see[CFK + ∂ does exist even if G has p -torsion elements. It was part of the proof of the non-commutative mainconjecture for totally real fields [Kak11] to verify this fact for Galois groups ofadmissible p -adic Lie extensions.The authors of [CFK +
05] also use the localisation sequence for the left denomi-nator set S ∗ = [ n ≥ p n S. Using the same tools as above, we can show that the boundary morphismK n +1 ( Z p [[ G ]] S ∗ ) → K n ( Z p [[ G ]][ 1 p ] , Z p [[ G ]] S ∗ )splits as well for every n ≥ A of the skew power series ring B is always assumedto be noetherian. This restriction excludes important examples such as the com-pleted group ring of the absolute Galois group of a global field. Therefore, we willwork in a much larger generality by only assuming that A is pseudocompact as A - A -bimodule. For the construction of the left denominator set S we cannot disposeof the noetherian condition, but we can considerably weaken the extra conditionsrequired in [SV10].In those cases where a suitable S is not known to exist, we obtain a split exactsequence 0 → K n +1 ( B ) → K n +1 ( TSP A ( B )) → K n ( TSP A ( B )) → TSP A ( B ) and TSP A ( B ). For applications inIwasawa theory, this is still a perfectly sensible construction (see e. g. [Wit10]).The article is structured as follows. In Section 1 we recall properties of pseu-docompact modules and of the completed tensor product. For the latter, someextra care is needed because we do not assume that our rings are pseudocompactas modules over a commutative subring as it is done in [Bru66]. In Section 2 weintroduce skew power series ring, prove the existence of the short exact sequence( ∗ ) and then the splitting of the localisation sequence in terms of the Waldhausencategories TSP A ( B ) and TSP A ( B ). We then introduce the set S and prove theOre condition in the noetherian case. The K-groups of the Waldhausen categoriesmay then be replaced by K n ( B S ) and K n ( B, B S ). In Section 3 we consider thespecial case of completed group rings. In this section, we also prove our result forthe localisation with respect to S ∗ .All rings will be associative with identity. The opposite ring of a ring R will bedenoted by R op . An R -module is always understood to be a left unitary R -module.The author would like to thank A. Schmidt and O. Venjakob for their adviceand support. 1. Preliminaries on Pseudocompact Modules
Pseudocompact modules have been considered in [Gab62] and [Bru66]. At somepoints, it is also convenient to work in the larger category of pro-discrete modules.1.1.
First Properties.Definition 1.1.
Let A be a topological ring and let M be a topological A -module.(1) The module M is pro-discrete if M = lim ←− i ∈ I M i OCALISATION SEQUENCE 3 is the limit of a filtered system of discrete A -modules M i .(2) The module M is pseudocompact if in addition the A -modules M i are offinite length.For any topological A -module M , we will denote by U M the directed set of opensubmodules of M . Proposition 1.2.
Let M be a topological A -module. (1) M is pro-discrete if and only if for each U ∈ U M , M/U is a discrete A -module and M → lim ←− U ∈U M M/U is a topological isomorphism. (2) M is pseudocompact if and only if it is pro-discrete and for each U ∈ U M , M/U has finite length.Proof.
Clearly the given conditions are sufficient. To prove necessity, consider M = lim ←− i ∈ I M i with M i discrete and let φ i : M → M i denote the structure morphisms. The ker φ i form a cofinal system in U M and for each i , the induced homomorphism M/ ker φ i → M i is injective and M/ ker φ i is a discrete A -module. It has finite length if M i does.This is then also true for all U ∈ U M containing ker φ i . Since the composition M α −→ lim ←− U ∈U M M/U = lim ←− i ∈ I M/ ker φ i β −→ lim ←− i ∈ I M i is a topological isomorphism and β is injective, α is a topological isomorphism aswell. (cid:3) Remark . The proposition shows that a topological A -module M is pro-discreteprecisely if it is linearly topologised, separated, and complete.We will denote the category of pro-discrete A -modules (with continuous homo-morphisms) by PD ( A ), the category of pseudocompact A -modules by PC ( A ). Thecategory PD ( A ) is additive and has arbitrary limits, but is not abelian. The fulladditive subcategory PC ( A ) is much better behaved. Proposition 1.4.
The category PC ( A ) is abelian and has arbitrary exact productsas well as exact filtered limits.Proof. See [Gab62, §
5, Thm. 3] and note that the proof does not need A to bepseudocompact as left A -module. (cid:3) The Completed Tensor Product.Definition 1.5.
Let A be a topological ring. For M in PD ( A op ) and N in PD ( A ),we define the completed tensor product M ˆ ⊗ A N = lim ←− U ∈U M lim ←− V ∈U N M/U ⊗ A N/V where we give
M/U ⊗ A N/V the discrete topology.
Proposition 1.6.
The completed tensor product M ˆ ⊗ A N satisfies the followinguniversal property. (1) M ˆ ⊗ A N is pro-discrete as Z -module (with respect to the discrete topologyon Z ) and the canonical map M × N → M ˆ ⊗ A N, ( m, n ) m ˆ ⊗ n, is continuous and bilinear. MALTE WITTE (2)
For any pro-discrete Z -module T and any continuous and bilinear map f : M × N → T , there exists a unique continuous homomorphism ¯ f : M ˆ ⊗ A N → T with ¯ f ( m ˆ ⊗ n ) = f ( m, n ) .Proof. This follows easily from the universal property of the usual tensor product,see [Bru66, Sect. 2]. (cid:3)
Proposition 1.7.
Let M be a pro-discrete A op -module, let N be the limit of afiltered system ( N i ) i ∈ I of pro-discrete A -modules, and assume that all structuremaps φ i : N → N i are surjective. Then M ˆ ⊗ A lim ←− i ∈ I N i → lim ←− i ∈ I M ˆ ⊗ A N i is a topological isomorphism. In particular, the completed tensor product commuteswith arbitrary products of pro-discrete A -modules and with filtered limits of pseu-docompact A -modules.Proof. For each U ∈ U N i , the map φ i,U : N → N i /U is surjective and the set { ker φ i,U | i ∈ I , U ∈ U N i } is a cofinal subset of U N . Hence, M ˆ ⊗ A N = lim ←− i ∈ I lim ←− V ∈U Ni lim ←− U ∈U M M/U ⊗ A N i /V = lim ←− i ∈ I M ˆ ⊗ A N i as claimed.Finally, note that every direct product over a set I may be rewritten as thelimit of the products over finite subsets of I and that by Prop. 1.4 the maps φ i aresurjective if all N i are pseudocompact. (cid:3) Definition 1.8.
Let A and B be topological rings and M a topological A - B -bimodule, i. e. a topological abelian group with compatible topological A -moduleand B op -module structures. We call M (1) pro-discrete if M is pro-discrete as A -module and there exists a fundamentalsystem of neighbourhoods of 0 consisting of open B op -submodules,(2) pseudocompact if M is pseudocompact both as A -module and as B op -module.Note that a pro-discrete A - B -bimodule M is automatically pro-discrete as B op -module. Indeed, any open A -submodule U of M contains an open B op -submodule V . Hence, it also contains X a ∈ A aV , which is an open A - B -subbimodule of M . In particular, M ∼ = lim ←− U ∈U ′ M M/U where U ′ M denotes the filtered set of open subbimodules of M . The bimodule M ispseudocompact precisely if it is pro-discrete and for each U ∈ U ′ M , M/U is of finitelength as A -module and noetherian as B op -module [GW04, Thm. 8.12]. Proposition 1.9.
Let A and B be topological rings and M a pro-discrete A - B -bimodule. (1) If N is a pro-discrete B -module, then M ˆ ⊗ B N is a pro-discrete A -module. (2) If M is pseudocompact as A -module and N is a pseudocompact B -module,then M ˆ ⊗ B N is a pseudocompact A -module. OCALISATION SEQUENCE 5
Proof.
One reduces to the case that M and N carry the discrete topology. For thefirst assertion, one then needs to verify that M ⊗ B N with its discrete topology is atopological A -module. This is the case precisely if the annihilator of each elementis open in A . The relationsAnn A m ⊂ Ann A m ⊗ n and Ann A x ∩ Ann A y ⊂ Ann A x + y imply that this condition is satisfied.Further, if M is of finite length as A -module and N is a finitely generated B -module, then M ⊗ B N is again of finite length. Hence, the second assertionfollows. (cid:3) The completed tensor product is in general neither left exact nor right exact,but we obtain right exactness in the following situation.
Proposition 1.10.
Let A and B be topological rings and M be a pro-discrete A - B -bimodule which is pseudocompact as A -module. Then the functor PC ( B ) → PC ( A ) , N M ˆ ⊗ B N, is right exact.Proof. This follows from the exactness of filtered limits (Prop. 1.4), the right ex-actness of the usual tensor product, and Prop. 1.7. (cid:3)
Pseudocompact Rings.Definition 1.11.
The topological ring A is called(1) left pseudocompact if it is pseudocompact as a topological (left) A -module,(2) (left and right) pseudocompact if it is pseudocompact as a topological A - A -bimodule.Our terminology agrees with the notion of pseudocompact rings in [Bru66]. In[Gab62], the term “pseudocompact ring” refers to left pseudocompact rings in ourterminology. Nevertheless, the following results remain true. Proposition 1.12. If A is a left pseudocompact ring, then every pseudocompactmodule is quotient of a direct product of copies of A . Furthermore, the projectivesin PC ( A ) are the direct summands of arbitrary direct products of copies of A .Proof. See [Bru66, Cor. 1.3, Lemma 1.6]. (cid:3)
Proposition 1.13.
Let A be a left pseudocompact ring. The forgetful functorfrom the category of finitely presented, pseudocompact A -modules to the categoryof finitely presented, abstract A -modules is an equivalence of categories. Moreover,any abstract homomorphism from a finitely presented A -module to a pseudocompact A -module is automatically continuous.Proof. This follows as [NSW00, Prop. 5.2.22]. (cid:3)
For the following result, we must restrict to (left and right) pseudocompact rings.
Proposition 1.14.
Let A be a pseudocompact ring. (1) For any pseudocompact A -module N , the canonical homomorphism N → A ˆ ⊗ A N is an isomorphism. (2) If B is a topological ring, M a pseudocompact B - A -bimodule, and N afinitely presented A -module, then the canonical homomorphism M ⊗ A N → M ˆ ⊗ A N is an isomorphism. MALTE WITTE
Proof.
For the first assertion we choose a presentation Y i ∈ I A → Y j ∈ J A → N → N = A . In this case, the statement followsimmediately from the definition.The second assertion follows by choosing a finite presentation of N and applyingthe first assertion to the pseudocompact A op -module M . Since M is also pseu-docompact as B -module, the completed tensor product is again right exact suchthat M ⊗ A N → M ˆ ⊗ A N is indeed an isomorphism. (cid:3) Remark . Note that for the second assertion of the proposition, it does notsuffice to assume that N is pseudocompact and finitely generated. The proof of[Bru66, Lemma 2.1.(ii)] is erroneous since the kernel of M ⊗ A A n → M ⊗ A N is not necessarily closed.2. The Localisation Sequence
Skew Power Series Rings.
Let A be a pseudocompact ring with a contin-uous ring automorphism σ . Recall that a σ -derivation on A is a continuous map δ : A → A satisfying δ ( ab ) = σ ( a ) δ ( b ) + δ ( a ) b for all a, b ∈ A . Definition 2.1.
Let A be a (left and right) pseudocompact ring with a continuousautomorphism σ and a topologically nilpotent σ -derivation δ . Assume that B is aleft pseudocompact ring with an element t ∈ B such that(1) B contains A as a closed subring,(2) B is pseudocompact as A op -module,(3) as topological A -module, B is topologically freely generated by the powersof t : B = ∞ Y n =0 At n , (4) for any a ∈ A , ta = σ ( a ) t + δ ( a ) . Then B = A [[ t ; σ, δ ]] will be called a (left) skew power series ring over A . Remark . A skew power series ring does not exist for every choice of σ and δ . Ifit does exist, it is not clear that it is also a right skew power series ring, i. e. topo-logically free over the powers of t as a topological A op -module. A condition that isclearly necessary for the existence and pseudocompactness is that δ is topologicallynilpotent, i. e. that for each open ideal I of A , there exists a positive integer n suchthat δ n ( A ) ⊂ I . If A is noetherian, it is shown in [SV06] that a sufficient conditionfor the existence of a left and right skew power series ring is that δ is σ -nilpotent.Analysing the multiplication formula (4) in [SV06, Sect. 1] one sees that a sufficientcondition for the existence as left skew power series ring is that δ is topologicallynilpotent and that A admits a fundamental system of open neighbourhoods of 0 OCALISATION SEQUENCE 7 consisting of two-sided ideals stable under σ and δ . More general definitions of skewpower series rings are considered in [R¨us11, § Remark . If B = A [[ t ; σ, δ ]] is a left skew power series ring, a ∈ A × and b ∈ A ,then we may consider B also as skew power series ring B = A [[ t ′ ; σ ′ , δ ′ ]] in t ′ = at + b with the A -automorphism σ ′ : A → A, x aσ ( x ) a − , and the σ ′ -derivation δ ′ : A → A, x aδ ( x ) + bx − σ ′ ( x ) b. For this, we note that B → B, ∞ X n =0 f n t n ∞ X n =0 f n ( t ′ ) n , is an isomorphism of pseudocompact left A -modules and hence, B is also topologi-cally freely generated by the powers of t ′ .We fix a skew power series ring B = A [[ t ; σ, δ ]] and let B σ denote the pseudo-compact B - A -bimodule with the A op -structure given by a · b = bσ ( a )for a ∈ A op , b ∈ B .The following proposition extends [SV06, Prop. 2.2]. An important difference isthat it applies not only to B -modules finitely generated as A -modules, but to allpseudocompact B -modules. Proposition 2.4.
Let B be a skew power series ring over A . For any pseudocom-pact B -module M , there exists an exact sequence of pseudocompact B -modules → B σ ˆ ⊗ A M κ −→ B ˆ ⊗ A M µ −→ M → with κ ( b ˆ ⊗ m ) = bt ˆ ⊗ m − b ˆ ⊗ tm, µ ( b ˆ ⊗ m ) = bm. Proof.
We closely follow the argumentation of [SV06, proof of Prop. 2.2], replacinggenerators by topological generators. Only the proof of the injectivity of κ isdifferent.From Prop. 1.12 and the definition of skew power series rings one deduces that M is pseudocompact as A -module. Prop. 1.9 then implies that B ˆ ⊗ A M and B σ ˆ ⊗ A M are pseudocompact as B -modules. Using the universal property of the completedtensor product (Prop. 1.6) one shows that κ and µ are well defined and functorialcontinuous B -homomorphisms. Clearly, µ ◦ κ = 0. The map µ is always surjective,since M → B ˆ ⊗ A M, m ⊗ m, is a continuous A -linear splitting by Prop. 1.6. One then observes that ker µ istopologically generated as pseudocompact A -module by t i ˆ ⊗ m − ⊗ t i m = i − X k =0 κ ( t i − − k ˆ ⊗ t k m )where i is an positive integer and m runs through a topological generating systemof the pseudocompact A -module M . Hence, ker µ = im κ .We will now consider the special case M = B . The canonical isomorphism ofleft pseudocompact A -modules B ˆ ⊗ A B = B ˆ ⊗ A ∞ Y i =0 A = ∞ Y i =0 B = ∞ Y i =0 ∞ Y j =0 A MALTE WITTE shows that the system ( t i ˆ ⊗ t j ) i ≥ ,j ≥ is a topological basis of B ˆ ⊗ A B as pseudo-compact A -module. The same argument also works for B σ ˆ ⊗ A B . Let a i,j ∈ A with a i,j = 0 for i < j <
0. An easy inductive argument shows that κ ∞ X i =0 ∞ X j =0 a i,j t i ˆ ⊗ t j = ∞ X i =0 ∞ X j =0 ( a i − ,j − a i,j − ) t i ˆ ⊗ t j is zero if and only if a i,j = 0 for all i and j . Since the completed tensor productcommutes with arbitrary products, it follows that κ is injective for all topologicallyfree pseudocompact B -modules M .If M is an arbitrary pseudocompact B -module, Prop. 1.12 implies the existenceof an exact sequence 0 → N → F → M → B -modules with F topologically free. We obtain the followingcommutative diagram with exact rows and columns: 0 (cid:15) (cid:15) B σ ˆ ⊗ A N / / (cid:15) (cid:15) B ˆ ⊗ A N / / (cid:15) (cid:15) N / / (cid:15) (cid:15) / / B σ ˆ ⊗ A F / / (cid:15) (cid:15) B ˆ ⊗ A F / / (cid:15) (cid:15) F / / (cid:15) (cid:15) B σ ˆ ⊗ A M κ / / (cid:15) (cid:15) B ˆ ⊗ A M / / (cid:15) (cid:15) M / / (cid:15) (cid:15)
00 0 0By the Snake lemma, we see that κ is injective in general. (cid:3) K-theory.
We recall that the Quillen K-groups of a ring R may be calculatedvia Waldhausen’s S -construction from the Waldhausen categories of either perfector strictly perfect complexes [Wal85], [TT90]. Definition 2.5.
A complex of R -modules P • is strictly perfect if each P n is pro-jective and finitely generated and if P n = 0 for almost all n . We denote by SP ( R )the Waldhausen category of strictly perfect complexes with quasi-isomorphisms asweak equivalences and injections with strictly perfect cokernel as cofibrations. Definition 2.6.
A complex of R -modules P • is perfect if it is quasi-isomorphicto a strictly perfect complex. We denote by P ( R ) the Waldhausen category ofperfect complexes. Quasi-isomorphisms are weak equivalences and injections arecofibrations.Let A be a left pseudocompact ring and P • a strictly perfect complex. ByProp. 1.13, each P n carries a natural pseudocompact topology and thus becomesa projective object in PC ( A ). Moreover, if Q • is a complex of pseudocompact A -modules (with continuous differentials) and a : P • → Q • is a morphism of com-plexes, then a is automatically continuous. Definition 2.7.
Let A be a left pseudocompact ring. We call a complex P • inthe category of complexes over PC ( A ) topologically strictly perfect if P • is quasi-isomorphic to a strictly perfect complex, each P n is projective in PC ( A ), and OCALISATION SEQUENCE 9 P n = 0 for almost all n . We denote by TSP ( A ) the Waldhausen category of topo-logically strictly perfect complexes with quasi-isomorphisms as weak equivalencesand injections with topologically strictly perfect cokernel as cofibrations.By what was said above and the Waldhausen approximation theorem [TT90,Thm. 1.9.1] the natural functors SP ( A ) → TSP ( A ) , TSP ( A ) → P ( A )induce isomorphisms on the associated K-groups. The Waldhausen K-groups of SP ( A ) are known to agree with the Quillen K-groups of the ring A , see [TT90,Thm. 1.11.2, Thm. 1.11.7.]. Hence,K n ( A ) = K n ( SP ( A )) = K n ( TSP ( A )) = K n ( P ( A ))for all n . For us, it is most convenient to work with the category TSP ( A ). Lemma 2.8.
Let B be a left pseudocompact ring, A be a pseudocompact ring and M be a pseudocompact B - A -bimodule which is finitely generated projective as B -module. For any P • in TSP ( A ) , the complex M ˆ ⊗ A P • is in TSP ( B ) and thefunctor TSP ( A ) → TSP ( B ) , P • M ˆ ⊗ A P • is Waldhausen exact.Proof. Since the projectives in PC ( B ) are the direct summands of products ofcopies of B (Prop. 1.12), M ˆ ⊗ A A = M (Prop. 1.14.(1)), and the completed tensorproduct commutes with products (Prop. 1.7) we conclude that M ˆ ⊗ A P n is a projec-tive pseudocompact B -module for each n . Let a : Q • → P • be quasi-isomorphismwith Q • a strictly perfect complex of A -modules. Since we are in the situation ofProp. 1.10, we conclude that M ˆ ⊗ A Q • id ⊗ a −−−→ M ˆ ⊗ A P • is a quasi-isomorphism. On other hand, Prop. 1.14 implies that M ⊗ A Q • → M ˆ ⊗ A Q • is an isomorphism. Since M is finitely generated and projective as B -module, thecomplex M ⊗ A Q • is strictly perfect as complex of B -modules. This shows that M ˆ ⊗ A P • is an object of TSP ( B ).By Prop. 1.10, the completed tensor product with M is a right exact functor on PC ( A ). Since the objects in TSP ( A ) are bounded complexes of projective objectsin PC ( A ), it takes short exact sequences of complexes in TSP ( A ) to short exactsequences in TSP ( B ) and acyclic complexes to acyclic complexes. Therefore, itconstitutes a Waldhausen exact functor between the categories (see e. g. [Wit08,Lemma 3.1.8]). (cid:3) Definition 2.9.
Let A be a subring of B . We let SP A ( B ) denote the Waldhausensubcategory of SP ( B ) consisting of strictly perfect complexes of B -modules whichare perfect as complexes of A -modules. Furthermore, we let SP A ( B ) denote theWaldhausen category with the same objects and cofibrations as SP ( B ), but withnew weak equivalences: A morphism of complexes f : P • → Q • is a weak equiva-lence if the cone of f is in SP A ( B ). If additionally A and B are left pseudocompactrings such that B is pseudocompact as left A -module, we define likewise the Wald-hausen categories TSP A ( B ) and TSP A ( B ).We recall from [TT90, Thm. 1.8.2] that the natural Waldhausen functors SP A ( B ) → SP ( B ) → SP A ( B ) induce via the S -construction a homotopy fibre sequence of the associated topo-logical spaces and hence, a long exact sequence → K n ( SP A ( B )) → K n ( B ) → K n ( SP A ( B )) → . . . → K ( B ) → K ( SP A ( B )) → . The same is also true for the
TSP -version and by the approximation theorem[TT90, Thm. 1.9.1] the resulting long exact sequence is isomorphic to the oneabove.
Theorem 2.10.
Let B be a skew power series ring over a pseudocompact ring A and assume that there exists a unit γ ∈ B such that σ ( a ) = γaγ − for all a ∈ A . The long exact localisation sequence for TSP A ( B ) → TSP ( B ) → TSP A ( B ) splits into short split-exact sequences → K n +1 ( B ) → K n +1 ( TSP A ( B )) → K n ( TSP A ( B )) → for n ≥ .Proof. For any Waldhausen category W , let K ( W ) denote the topological spaceassociated to W via the S -construction [TT90, Def. 1.5.3]. Consider the homotopyfibre sequence K ( TSP A ( B )) → K ( TSP ( B )) → K ( TSP A ( B )) . from [TT90, Thm. 1.8.2].Note that right multiplication with γ − gives rise to an isomorphism of B - A -bimodules B → B σ . By Lemma 2.8 and since every projective pseudocompact B -module is also projective as pseudocompact A -module the completed tensor product TSP A ( B ) → TSP ( B ) , P • B ˆ ⊗ A P • is a Waldhausen exact functor. By Prop. 2.4(2.1) 0 → B ˆ ⊗ A P • κ ◦ γ − −−−−→ B ˆ ⊗ A P • → P • → TSP ( B ). The additivity theorem [TT90, Cor. 1.7.3] appliedto this sequence implies that the map K ( TSP A ( B )) → K ( TSP ( B )) induced by thecanonical inclusion is homotopic to the constant map defined by the base point. By[Bre93, Thm. 6.8], which applies equally well to homotopy fibre sequences insteadof fibre sequences, the assertion of the theorem follows. (cid:3) Remark . Let B = A [[ t ; σ, δ ]] be a skew power series ring with a unit γ ∈ B × as above. For a ∈ A × and b ∈ A , let B ′ = A [[ t ′ ; σ ′ , δ ′ ]] be the topological ring B with the modified skew power series ring structure as in Remark 2.3 and set γ ′ = aγ . Then the exact sequence (2 .
1) for B ′ is exactly the same as the one for B . In particular, one obtains the same splitting for the K-groups.Using the algebraic description of the first Postnikov section of the topologicalspace associated to a Waldhausen category [MT07, Def. 1.2] we can give the follow-ing explicit formula for a section s : K ( TSP A ( B )) → K ( TSP A ( B )): The groupK ( TSP A ( B )) is the abelian group generated by the objects of TSP A ( B ) modulothe relations generated by exact sequences and weak equivalences in TSP A ( B ).Moreover, there exists a canonical class [ w ] ∈ K ( TSP A ( B )) for each weak auto-equivalence w : P • → P • in TSP A ( B ). Definition 2.12.
In the situation of Theorem 2.10, set s B,γ ( P • ) = [ B ˆ ⊗ A P • κ ◦ γ − −−−−→ B ˆ ⊗ A P • ] − ∈ K ( TSP A ( B ))for each P • in TSP A ( B ). OCALISATION SEQUENCE 11
It is easy to check that s B,γ defines a group homomorphismK ( TSP A ( B )) → K ( TSP A ( B ))(see e. g. [Wit08, Thm 2.4.7]). Moreover, ∂ ( s B,γ ( P • )) is the cone of κ ◦ γ − by[Wit10, Thm. A.5] and hence, equal to P • in K ( TSP A ( B )). Remark . Note that the above description of the boundary homomorphism ∂ is compatible with the formula given in [WY92, p. 2]. Other authors prefer to use − ∂ instead.We record the following functorial behaviour. Proposition 2.14.
Let B i = A i [[ t i ; σ i , δ i ]] ( i = 1 , ) be two skew power series rings, M a pseudocompact B - B -bimodule which is finitely generated and projective as B -module, N ⊂ M a pseudocompact A - A -subbimodule which is finitely generatedand projective as A -module such that N ˆ ⊗ A B ∼ = M as A - B -bimodules. Thenthe completed tensor product with M induces Waldhausen exact functors T M : TSP A ( B ) → TSP A ( B ) , P • M ˆ ⊗ B P • ,T M : TSP A ( B ) → TSP A ( B ) , P • M ˆ ⊗ B P • . Assume further that there exists units γ i ∈ B i with σ i ( a ) = γ i aγ − i for all a ∈ A i and let one of the following two conditions be satisfied: (1) The inclusion N ⊂ M also induces an isomorphism of pseudocompact B - A -bimodules B ˆ ⊗ A N ∼ = M and for all n ∈ N , γ nγ − ∈ N, t n − γ nγ − t ∈ N. (2) A = A and B ⊂ B is a closed subring; t i = γ i − for i = 1 , ; thereexists a number k ∈ Z such that γ = γ k ; M = B and N = A .Then the following diagram is commutative: K ( TSP A ( B )) s B ,γ / / T M (cid:15) (cid:15) K ( TSP A ( B )) T M (cid:15) (cid:15) K ( TSP A ( B )) s B ,γ / / K ( TSP A ( B )) Proof.
Let P • be in TSP A ( B ). By definition, there exists a strictly perfectcomplex of A -modules Q • and a quasi-isomorphism Q • → P • in TSP ( A ). Hence, N ⊗ A Q • → N ˆ ⊗ A P • ∼ = M ˆ ⊗ B P • is a quasi-isomorphism of complexes of pseudocompact A -modules. This showsthat T M takes objects of TSP A ( B ) to objects of TSP A ( B ) and weak equiva-lences in TSP A ( B ) to weak equivalences in TSP A ( B ). It follows easily thatboth versions of T M are Waldhausen exact as claimed.Assume that condition (1) is satisfied. For n ∈ N set σ ( n ) = γ nγ − , δ ( n ) = t n − σ ( n ) t and consider the homomorphism of B - B -bimodules κ ′ : B ˆ ⊗ A N ˆ ⊗ A B → B ˆ ⊗ A N ˆ ⊗ A B ,b ˆ ⊗ n ˆ ⊗ b b γ − t ˆ ⊗ n ˆ ⊗ b − b γ − ˆ ⊗ σ ( n ) ˆ ⊗ t b − b γ − ˆ ⊗ δ ( n ) ˆ ⊗ b . Then the diagram M ˆ ⊗ B B ˆ ⊗ A P • id M ˆ ⊗ ( κ ◦ γ − ) / / M ˆ ⊗ B B ˆ ⊗ A P • M ˆ ⊗ A P • / / M ˆ ⊗ A P • B ˆ ⊗ A N ˆ ⊗ A B ˆ ⊗ B P •∼ = O O ∼ = (cid:15) (cid:15) κ ′ ˆ ⊗ id P • / / B ˆ ⊗ A N ˆ ⊗ A B ˆ ⊗ B P •∼ = O O ∼ = (cid:15) (cid:15) B ˆ ⊗ A M ˆ ⊗ B P • ( κ ◦ γ − ) ˆ ⊗ id P • / / B ˆ ⊗ A M ˆ ⊗ B P • commutes for every P • in TSP A ( B ). We conclude T M ( s B ,γ ( P • )) = [id M ˆ ⊗ ( κ ◦ γ − )] − = [ κ ′ ˆ ⊗ id P • ] − = [( κ ◦ γ − ) ˆ ⊗ id P • ] − = s B ,γ ( T M ( P • ))from the relations listed in [MT07, Def. 1.2].The proof in the case that condition (2) is satisfied follows essentially as in [BV11,Proof of Lemma 4.6]: Let A = A = A and set for P • in TSP A ( B )∆ i : B ˆ ⊗ A P • → B ˆ ⊗ A P • , x ˆ ⊗ y xγ − i ˆ ⊗ γ i y,κ : B ˆ ⊗ A P • → B ˆ ⊗ A P • , x ˆ ⊗ y xγ − t ˆ ⊗ y − xγ − ˆ ⊗ t = x ˆ ⊗ y − xγ − ˆ ⊗ γ y,κ : B ˆ ⊗ A P • → B ˆ ⊗ A P • , x ˆ ⊗ y xγ − t ˆ ⊗ y − xγ − ˆ ⊗ t = x ˆ ⊗ y − xγ − ˆ ⊗ γ y. Then ( B ˆ ⊗ A P • ) k → B ˆ ⊗ A P • , ( x i ) k − i =0 k − X i =0 ∆ i ( x i ) , is an isomorphism in TSP ( B ). Using this as an identification, κ is given by rightmultiplication with the k × k -matrix of B ˆ ⊗ A P • -endomorphisms id − id 0 . . .
00 . . . . . . . . . ...... . . . . . . . . . 00 . . . . . . . . . − id − ∆ k . . . = id 0 . . . . . . . . .
00 id 0 . . . . . . − ∆ k . . . . . . . − ∆ k id − ∆ k id − id 0 . . .
00 id − id . . . ...... 0 . . . . . . 0... ... . . . . . . − id0 0 . . . . Note that id − ∆ k = κ . The relations listed in [MT07, Def. 1.2] imply that theclass in K ( TSP A ( B )) of a triangular matrix as above is the product of the classesof the diagonal entries and the class of a composition of weak auto-equivalences isthe product of the classes of the weak auto-equivalences. Hence, T M ( s B ,γ ( P • )) = [ κ ] − = [id − ∆ k ] − = s B ,γ ( T M ( P • )) . (cid:3) OCALISATION SEQUENCE 13
Left Denominator Sets.
We recall that a subset S in a ring B is a leftdenominator set if(1) S is multiplicatively closed,(2) ( Ore condition ) for each s ∈ S , b ∈ B there exist s ′ ∈ S , b ′ ∈ B such that b ′ s = s ′ b , and(3) ( annihilator condition ) for each s ∈ S , b ∈ B with bs = 0 there exists s ′ ∈ S with s ′ b = 0.If S is a left denominator set, the (left) quotient ring B S of B exists and is flat asright B -module [GW04, Ch. 10].We would like to identify the K-groups of the Waldhausen category SP A ( B )with the K-groups of the quotient ring B S of B with respect to a left denominatorset S ⊂ B . Unfortunately, it is not clear to us that this is always possible. Below,we will show that there exists an essentially unique candidate for S . Under thecondition that A is noetherian plus a mild extra condition we show that this S doesindeed have the right properties. Definition 2.15.
Let A be a subring of B . We let S denote the set of elements b ∈ B such that B · b −→ B is perfect as complex of A -modules and hence, an object in SP A ( B ). Lemma 2.16.
The set S is multiplicatively closed and saturated, i. e. if any two ofthe elements a, b, ab ∈ B are in S , then so is the third.Proof. The commutative diagram B = / / · a (cid:15) (cid:15) B · a / / · ab (cid:15) (cid:15) B · b (cid:15) (cid:15) B · b / / B = / / B constitutes a distinguished triangle if one reads the columns as complexes. If ina distinguished triangle two of the three complexes are perfect, then so is thethird. (cid:3) Lemma 2.17.
Assume that there exists a left denominator set S ′ such that SP A ( B ) is precisely the subcategory of those complexes in SP ( B ) whose cohomology modulesare S ′ -torsion. Then the set S has the same property and is the saturation of S ′ ,i. e. it is the left denominator set consisting of those elements in B that become aunit in B S ′ .Proof. An element b ∈ B is a unit in B S ′ if and only if the complex B · b −→ B is S ′ -torsion and thus perfect. Hence, B S = B S ′ and a complex is S -torsion preciselyif it is S ′ -torsion. (cid:3) The following theorem extends [SV10, Thm. 2.25].
Theorem 2.18.
Assume that B is a skew power series ring over a noetherianpseudocompact coefficient ring A and I is an open two-sided ideal of A stable under σ and δ and contained in the Jacobson radical of A . Then S is a left denominatorset and that SP A ( B ) is precisely the subcategory of those complexes in SP ( B ) whosecohomology modules are S -torsion.Proof. Since A is noetherian, the ideal powers I k constitute a fundamental systemof open neighbourhoods of 0 [War93, Sect. VII]. Since I is stable under σ and δ ,we deduce that J k = ∞ Y n =0 I k t n is a closed two-sided ideal of B for each positive integer k . Consider the set S ′ = ∞ [ n =0 t n + J . Clearly, S ′ is multiplicatively closed. Furthermore, the annihilator Ann B t n + f iszero for f ∈ J . Indeed, if gt n + gf = 0 then reduction modulo J k shows that thecoefficients of g must lie in ∞ \ k =1 I k = 0 . For any f ∈ J we have( B/B ( t n + f )) /I ( B/B ( t n + f )) = B/Bt n + J which is clearly finitely generated as A/I -module. By the topological Nakayamalemma we conclude that
B/B ( t n + f ) is finitely generated as A -module.Assume that M is a B -module which is finitely generated over A and let m ∈ M . Since A is noetherian, the module Bm is also finitely generated over A . Inparticular, it is pseudocompact for the I -adic topology. Since t is topologicallynilpotent, there exists a positive integer n such that t n m ∈ J m , i. e. there exists a g ∈ J such that ( t n + g ) m = 0. Applying this to B/B ( t n + f ) we conclude that S ′ satisfies the Ore condition, whereas the annihilator condition is automaticallysatisfied. Hence, S ′ is a left denominator set.Let now P • be a strictly perfect complex of B -modules. Since B is a directproduct of free A -modules and A is noetherian, any finitely generated projective B -module is flat as A -module. Hence, P • has finite flat dimension over A and isperfect as complex of A -modules if and only if its cohomology groups are finitelygenerated as A -modules. It follows immediately that every complex in SP A ( B ) is S ′ -torsion.Conversely, assume that P • is S ′ -torsion. Without loss of generality we mayassume that P • is concentrated in the degrees 0 to n , that H n ( P ) = 0 and thatall P i for i > B -modules. We do induction on n . In the casethat the finitely generated projective B -module P happens to be S ′ -torsion wechoose a system of generators of P and an s ∈ S ′ that kills all the generators andobtain a surjection B k /B k s → P , which shows that P is also finitely generatedand projective as A -module. If n = 1 and H ( P ) = 0 we proceed similarly withH ( P ). If H ( P ) = 0 or if n > B k · s −→ B k to P • such that we get a surjection onto H n ( P ). The cone of thismorphism is strictly perfect as complex of B -modules, S ′ -torsion and concentratedin the degrees 0 to n − n >
1. If n = 1, it is concentrated in degrees − P • is in SP A ( B ). We now apply Lemma 2.17 to S ′ . (cid:3) Remark . If one drops the assumption that A is noetherian one can still showthat the annihilators of elements in the set S ′ are trivial. For this, one uses that I is transfinitely nilpotent [War93, Thm. 33.21]. Assuming that S ′ also satisfies theOre condition, the above proof shows that every S ′ -torsion complex is in SP A ( B ).However, we were not able to prove the converse nor verify the Ore condition inthis generality. Proposition 2.20.
Assume that S is a left denominator set such that SP A ( B ) isprecisely the subcategory of those complexes in SP ( B ) whose cohomology modulesare S -torsion. The Waldhausen exact functor SP A ( B ) → SP ( B S ) , P • B S ⊗ B P • , OCALISATION SEQUENCE 15 induces isomorphisms K n ( SP A ( B )) ∼ = K n ( B S ) for n > . The groups K n ( SP A ( B )) can be identified with the relative K -groups K n ( B, B S ) for all n ≥ .Proof. This is a direct consequence of the localisation theorem in [WY92]. (cid:3)
Corollary 2.21.
Let B be a skew power series ring over a noetherian pseudocom-pact ring A and let γ ∈ B be a unit such that σ ( a ) = γaγ − for every a ∈ A . Assume further that I is an open two-sided ideal of A whichis contained in the Jacobson radical and stable under σ and δ . Then S is a leftdenominator set in B and for every n ≥ n +1 ( B S ) ∼ = K n +1 ( B ) ⊕ K n ( B, B S ) Proof.
This follows from Theorem 2.10, Theorem 2.18 and Prop. 2.20. (cid:3)
Remark . If in the above corollary, B is itself a (left and right) pseudocompactring, then the existence of I is automatic. Indeed, the definitions of σ and δ can beextended to B and every two-sided ideal of B is then stable under σ and δ . Theintersections of the open two-sided ideals of B with A form a fundamental system ofneighbourhoods of 0 in A . Since A is noetherian, the Jacobson radical of A is open.Hence, there exists an open two-sided ideal J of B such that J ∩ A is contained inthe Jacobson radical.3. Applications to Completed Group Rings
Definition 3.1.
Let G be a pro-finite group and U G the directed set of open normalsubgroups. For any topological ring R we define the completed group ring of G over R to be the topological ring R [[ G ]] = lim ←− U ∈U G R [ G/U ] . We will now fix a pseudocompact coefficient ring R and a prime p which istopologically nilpotent in R . Furthermore, let G be a pro-finite group which is thesemi-direct product of a closed normal subgroup H and a closed subgroup Γ ∼ = Z p .We also fix a topological generator γ ∈ Γ. The following proposition is a harmlessgeneralisation of [Ven05, Ex. 5.1].
Proposition 3.2.
The completed group ring R [[ G ]] is a left and right pseudocom-pact skew power series ring over R [[ H ]] with t = γ − ,σ ( a ) = γaγ − ,δ ( a ) = σ ( a ) − a for a ∈ R [[ H ]] .Proof. It is obvious that the completed group rings are pseudocompact and that R [[ G ]] is pseudocompact as R [[ H ]]- R [[ H ]]-bimodule. Since H is normal, it is alsoeasy to verify that σ is a continuous ring automorphism of R [[ H ]] and that δ is acontinuous σ -derivation. It remains to show that R [[ G ]] = ∞ Y n =0 R [[ H ]] t n as R [[ H ]]-module. The topological nilpotence of p implies that R [[ H ]] is pro-discreteover Z p . One then verifies that R [[ H ]] ˆ ⊗ Z p Z p [[Γ]] → R [[ G ]] , a ˆ ⊗ x ax, is an isomorphism of pseudocompact R [[ H ]]-modules. By a classical result fromIwasawa theory [Was97, Thm. 7.1] Z p [[Γ]] → Z p [[ t ]] , γ t + 1 , defines a topological ring isomorphism. Hence, the system ( t n ) n ≥ is a topological R [[ H ]]-basis of R [[ G ]]. (cid:3) Corollary 3.3.
Let R , G , and H be as above. For any n ≥ , K n +1 ( TSP R [[ H ]] ( R [[ G ]])) ∼ = K n +1 ( R [[ G ]]) ⊕ K n ( TSP R [[ H ]] ( R [[ G ]])) . Proof.
See Theorem 2.10. (cid:3)
Corollary 3.4.
Suppose that R is a commutative noetherian pseudocompact ringwith p ∈ R topologically nilpotent and that G is a compact p -adic Lie group. Then R [[ H ]] and R [[ G ]] are noetherian, the set S of Definition 2.15 is a left denominatorset, and for any n ≥ , K n +1 ( R [[ G ]] S ) ∼ = K n +1 ( R [[ G ]]) ⊕ K n ( R [[ G ]] , R [[ G ]] S ) . Proof.
Every commutative noetherian pseudocompact ring is a finite direct productof commutative noetherian pseudocompact local rings with the topology given bythe powers of the maximal ideal [War93, Cor. 36.35]. Hence, we may suppose that R is local. As a corollary of Cohen’s structure theorem, we have R ∼ = O [[ X , . . . , X n ]] /I for a complete discrete valuation ring O with maximal ideal p O , some integer n ≥ I (which is finitely generated and therefore closed) [Bou89, Ch. IX, §
2, Prop. 1, Thm. 3]. Hence, R [[ G ]] is a factor ring of O [[ X , . . . , X n ]][[ G ]] ∼ = O [[ G × n Y k =1 Γ]] , Γ ∼ = Z p . So, we may assume that R = O . By [Sch11, Thm. 27.1] we can find an open p -valuable subgroup G ′ of G . By [Sch11, Thm. 33.4], O [[ G ′ ]] is noetherian. Since O [[ G ]] is finitely generated as left or right O [[ G ′ ]]-module, O [[ G ]] is also noetherian.Now apply Corollary 2.21 together with Remark 2.22. (cid:3) If R = Z p , it is not hard to verify that our set S coincides with Venjakob’scanonical Ore set S introduced in [CFK + R is unramified over Z p . In particular, one can replace in [Bur10,Thm. 2.2, Cor. 2.3] the base ring Z p by the valuation ring O of any finite extensionfield of Q p .Let G = H ⋊ Γ be a p -adic Lie group as above. Write SQ ab1 ( G ) for the set ofpairs τ = ( U τ , J τ ) with U τ ⊂ G an open subgroup, J τ ⊂ U τ ∩ H a subgroup whichis open in H and normal in U τ and such that G τ = U τ /J τ is an abelian p -adic Liegroup of rank 1. Furthermore, let d τ be the index of HU τ in G and H τ be the imageof H ∩ U τ in G τ . For each τ , we also fix once and for all a γ τ lying in a p -Sylowsubgroup of U τ such that γ − d τ γ τ ∈ H . We write Γ τ ∼ = Z p for the closed subgroupof U τ topologically generated by γ τ and identify it with its image in G τ . Note that H τ is a commutative finite group and that the choice of γ τ induces decompositions U τ = U τ ∩ H ⋊ Γ τ and G τ = H τ × Γ τ . OCALISATION SEQUENCE 17
From now on, we assume that R is commutative, noetherian, and pseudocompactwith p ∈ R topologically nilpotent. Since R [[ G τ ]] ˆ ⊗ R [[ U τ ]] R [[ G ]] ∼ = R [ H τ ] ˆ ⊗ R [[ J τ ]] R [[ G ]]as pseudocompact R [ H τ ]- R [[ G ]]-bimodules, the exact functor TSP ( R [[ G ]]) → TSP ( R [[ G τ ]]) , P • R [[ G τ ]] ˆ ⊗ R [[ U τ ]] P • induces exact functors TSP R [[ H ]] ( R [[ G ]]) → TSP R [ H τ ] ( R [[ G τ ]]) , TSP R [[ H ]] ( R [[ G ]]) → TSP R [ H τ ] ( R [[ G τ ]])by the first part of Prop. 2.14. We will denote by q τ the induced homomorphismsof K-groups.For each φ in the group of characters ˆ H τ of H τ , let R [ φ ] denote the finite exten-sion of R generated by the values of φ . Consider the ring homomorphism R [[ G τ ]] = R [[Γ τ ]][ H τ ] → R [ φ ][[Γ τ ]]which maps h ∈ H τ to φ ( h ). Thus, R [ φ ][[Γ τ ]] becomes a pseudocompact R [ φ ][[Γ τ ]]- R [[ G τ ]]-bimodule and we obtain an exact functor TSP ( R [[ G τ ]]) → TSP ( R [ φ ][[Γ τ ]]) , P • R [ φ ][[Γ τ ]] ˆ ⊗ R [[ G τ ]] P • . Since R [ φ ][[Γ τ ]] ∼ = R [ φ ] ˆ ⊗ R [ H τ ] R [[ G τ ]] as R [ φ ]- R [[ G τ ]]-bimodules we also get exactfunctors TSP R [ H τ ] ( R [[ G τ ]]) → TSP R [ φ ] ( R [ φ ][[Γ τ ]]) , TSP R [ H τ ] ( R [[ G τ ]]) → TSP R [ φ ] ( R [ φ ][[Γ τ ]]) . We will denote the induced homomorphisms of K-groups by q φ .Recall that R [ φ ][Γ τ and R [ φ ][[Γ τ ]] S are semilocal commutative rings. Hence, thedeterminant induces isomorphismsK ( TSP ( R [ φ ][[Γ τ ]])) = K ( R [ φ ][[Γ τ ]]) ∼ = R [ φ ][[Γ τ ]] × , K ( TSP R [ φ ] ( R [ φ ][[Γ τ ]])) = K ( R [ φ ][[Γ τ ]] S ) ∼ = R [ φ ][[Γ τ ]] × S . Consider the homomorphisms∆ : K ( R [[ G ]]) → Y τ ∈SQ ab1 ( G ) Y φ ∈ ˆ H τ R [ φ ][[Γ τ ]] × ∆ S : K ( R [[ G ]] S ) → Y τ ∈SQ ab1 ( G ) Y φ ∈ ˆ H τ R [ φ ][[Γ τ ]] × S whose ( τ, φ )-component is q φ ◦ q τ and note that they agree with the maps ∆ O ,G and ∆ O ,G,S from [Bur10, Thm. 6.1] in the case that R = O is the valuation ringof a finite extension of Q p . (We also note that they depend on the splittings G τ = H τ × Γ τ induced by the choices of the generators γ τ .) Corollary 3.5.
Let G = H ⋊ Γ be a p -adic Lie group and R be any commutative,noetherian, pseudocompact ring such that p ∈ R is topologically nilpotent. Then im ∆ S ∩ Y τ ∈SQ ab1 ( G ) Y φ ∈ ˆ H τ R [ φ ][[Γ τ ]] × = im ∆ . Proof.
We use the explicit description of the splitting s = s G,γ : K ( TSP R [[ H ]] ( R [[ G ]])) → K ( TSP R [[ H ]] ( R [[ G ]]))given in Definition 2.12. It suffices to prove that the diagramK ( TSP R [[ H ]] ( R [[ G ]])) s G,γ / / ( q φ ◦ q τ ) (cid:15) (cid:15) K ( TSP R [[ H ]] ( R [[ G ]])) ∆ S (cid:15) (cid:15) Q τ,φ K ( TSP R [ φ ] ( R [ φ ][[Γ τ ]])) ( s Γ τ ,γτ ) / / Q τ,φ K ( TSP R [ φ ] ( R [ φ ][[Γ τ ]]))commutes.For each integer d , we let V d denote the subgroup of G topologically generatedby H and γ d and let q d denote the homomorphisms of K-groups resulting fromthe application of Prop. 2.14 to the R [[ V d ]]- R [[ G ]]-bimodule M = R [[ G ]] and the R [[ H ]]- R [[ H ]]-subbimodule N = R [[ H ]]. Since M and N satisfy condition (2) ofProp. 2.14, we know that each q d commutes with s .Now U τ ⊂ V d τ for each τ ∈ SQ ab1 ( G ) and we may write q τ = q ′ τ ◦ q d τ with q ′ τ corresponding to the R [[ G τ ]]- R [[ V d ]]-bimodule M = R [[ G τ ]] ˆ ⊗ R [[ U τ ]] R [[ V d τ ]]and the R [[ H τ ]]- R [[ H ]]-subbimodule N = R [[ H τ ]] ˆ ⊗ R [[ U τ ∩ H ]] R [[ H ]] . Since M , N , γ = γ d τ , γ = γ τ , t i = γ i − i = 1 ,
2) satisfy condition (1) ofProp. 2.14 we conclude that q ′ τ commutes with s . Likewise, we also see that q φ commutes with s for each φ ∈ ˆ H τ . The commutativity of the above diagramfollows. (cid:3) Remark . The above corollary also holds for arbitrary pro-finite groups G = H ⋊ Γ and for non-noetherian R if one formulates it directly in terms of the K-groups K ( TSP R [[ H ]] ( R [[ G ]])) and K ( TSP R [[ H ]] ( R [[ G ]])).We will now assume that R = O is a complete discrete valuation ring of charac-teristic 0 and residue characteristic p . In [CFK + S ∗ = ∞ [ n =0 p n S. By the same method as in the proof of Theorem 2.10, we obtain the following result.
Theorem 3.7.
Assume that O is a complete discrete valuation ring of character-istic residue characteristic p and that G is a compact p -adic Lie group. Then K n +1 ( O [[ G ]] S ∗ ) ∼ = K n +1 ( O [[ G ]][ 1 p ]) ⊕ K n ( O [[ G ]][ 1 p ] , O [[ G ]] S ∗ ) for n ≥ and K ( O [[ G ]] S ∗ ) ∼ = K ( O [[ G ]][ 1 p ]) . Proof.
Since O [[ G ]] is noetherian, the rings A = O [[ G ]][ p ] and B = O [[ G ]] S ∗ arealso noetherian. By [Sch11, Thm. 33.4] O [[ G ′ ]] is Auslander regular for any open p -valuable subgroup G ′ of G . We may apply the argument of [FK06, Prop. 4.3.4]to deduce that A is also Auslander regular. Hence, the same is true for B . Let M ( R ) denote the abelian category of finitely generated modules over any noetherian OCALISATION SEQUENCE 19 ring R . Further, we let M ( A ) S ∗ denote the abelian category of finitely generated, S ∗ -torsion modules over A . We then have isomorphismsK n ( A ) ∼ = K n ( M ( A )) , K n ( B ) ∼ = K n ( M ( B )) , K n ( A, B ) ∼ = K n ( M ( A ) S ∗ )for all n ≥
0. The localisation sequence for Quillen G -theory shows that the mapK ( M ( A )) → K ( M ( B )) is a surjection.Let now M be a module in M ( A ) S ∗ and fix a system of generators m , . . . , m n .Define M ′ = n X i =1 O [[ G ]] m i . The O [[ G ]]-module M ′ is p -torsion-free and S ∗ -torsion, hence S -torsion and there-fore, finitely generated as O [[ H ]]-module. From Prop. 2.4 we obtain the exactsequence of finitely generated O [[ G ]]-modules0 → O [[ G ]] ⊗ O [[ H ]] M ′ κ −→ O [[ G ]] ⊗ O [[ H ]] M ′ µ −→ M ′ → . Inverting p we obtain a corresponding exact sequence for M which does not dependon the choice of the generators. By the Waldhausen additivity theorem and by[Bre93, Thm. 6.8] we conclude as in the proof of Theorem 2.10 that the localisationsequence splits into short split exact sequences. (cid:3) The following corollary extends [CFK +
05, Prop. 3.4].
Corollary 3.8.
Assume that G is a compact p -adic Lie group. Then K ( O [[ G ]]) → K ( O [[ G ]][ 1 p ]) = K ( O [[ G ]] S ∗ ) is a split injection. Hence, ∂ : K ( O [[ G ]] S ∗ ) → K ( O [[ G ]] , O [[ G ]] S ∗ ) is a surjection.Proof. Let P be an open normal pro- p -subgroup of G . The kernel of O [[ G ]] →O [ G/P ] is contained in the Jacobson radical of O [[ G ]] [Wit10, Prop. 3.2]. Hence,K ( O [[ G ]]) → K ( O [ G/P ]) is an isomorphism [Lam91, Thm. 25.3]. Let K = O [ p ]be the fraction field of O and consider the commutative diagramK ( O [[ G ]]) / / ∼ = (cid:15) (cid:15) K ( O [[ G ]][ p ]) (cid:15) (cid:15) K ( O [ G/P ]) e / / K ( K [ G/P ])By [Ser77, Ch. 16, Thm. 34], the homomorphism e is split injective. (cid:3) The localisation sequence for O [[ G ]] → O [[ G ]] S ∗ does not split in general. Forexample, let G = H × Z p with H a finite group. In this case, the kernel ofK ( Z p [[ G ]]) i −→ K ( Z p [[ G ]][ 1 p ]) ⊂ K ( Z p [[ G ]] S ∗ )contains the group SK ( Z p [ H ]), which can be nontrivial if p divides the order of H .(In fact, the other inclusion ker( i ) ⊂ SK ( Z p [ H ]) is also true by [Wit09, Prop. 5.3].)It is tempting to hope that it does split if G has no element of order p , but thisseems to be difficult to prove even for commutative groups G , in which case it isrelated to the Gersten conjecture. For the first K-group however, we can restatethe following result obtained by Burns and Venjakob: Corollary 3.9.
Let O be a discrete valuation ring of characteristic with a residuefield k of characteristic p . Assume that G is a compact p -adic Lie group without p -torsion. Then K ( O [[ G ]] S ∗ ) ∼ = K ( O [[ G ]]) ⊕ K ( O [[ G ]] , O [[ G ]] S ) ⊕ K ( k [[ G ]]) . Proof.
This is [BV11, Thm 2.1] extended to general complete discrete valuationrings O of characteristic 0 and residue characteristic p combined with our newknowledge that K ( O [[ G ]]) injects into K ( O [[ G ]] S ). (cid:3) References [Bou89] N. Bourbaki,
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