The homology of SL 2 of discrete valuation rings
aa r X i v : . [ m a t h . K T ] J u l THE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS
KEVIN HUTCHINSON, BEHROOZ MIRZAII, FATEMEH Y. MOKARI
Abstract.
Let A be a discrete valuation ring with field of fractions F and (sufficientlylarge) residue field k . We prove that there is a natural exact sequence H (SL ( A ) , Z (cid:2) (cid:3) ) → H (SL ( F ) , Z (cid:2) (cid:3) ) → RP ( k ) (cid:2) (cid:3) → , where RP ( k ) is the refined scissors congruence group of k . Let Γ ( m A ) denote thecongruence subgroup consisting of matrices in SL ( A ) whose lower off-diagonal entry liesin the maximal ideal m A . We also prove that there is an exact sequence0 → P ( k ) (cid:2) (cid:3) → H (Γ ( m A ) , Z (cid:2) (cid:3) ) → H (SL ( A ) , Z (cid:2) (cid:3) ) → I ( k ) (cid:2) (cid:3) → , where I ( k ) is the second power of the fundamental ideal of the Grothendieck-Witt ringGW( k ) and P ( k ) is a certain quotient of the scissors congruence group (in the sense ofDupont-Sah) P ( k ) of k . Introduction
The purpose of this article is to study the low-dimensional homology of certain lineargroups and to demonstrate what we hope is interesting behaviour.Let A be a discrete valuation ring with field of fractions F and residue field k . k maybe finite, but for the validity of our proofs should be sufficiently large : if | k | = p d , then d ( p − >
6. Our first main result (Theorem 2.1 below) is:
Theorem A . There is a natural exact sequence H (SL ( A ) , Z (cid:2) (cid:3) ) → H (SL ( F ) , Z (cid:2) (cid:3) ) → RP ( k ) (cid:2) (cid:3) → , where RP ( k ) is the refined scissors congruence group of the field k . Dupont and Sah defined the scissors congruence group P ( F ) of a field F (also calledthe pre-Bloch group of F ) in [3]. It is an abelian group given by generators and relations.They related P ( C ) and its subgroup B ( C ) to H (SL ( C ) , Z ). In fact, there is a naturalisomorphism H (SL ( C ) , Z ) ∼ = K ind3 ( C ), the indecomposable K of C . Some time later,Suslin showed in [21] how to generalize the result of Dupont and Sah to arbitrary (infinite)fields, identifying K ind3 ( F ) with B ( F ) modulo a certain well-understood torsion subgroup(for a precise statement, see Theorem 1.1 below).However, when the field F is not quadratically closed, the natural surjective homomor-phism H (SL ( F ) , Z ) → K ind3 ( F ) has a nontrivial, and often quite large, kernel (whichwe denote by H (SL ( F ) , Z ) ). To give an analogous description of H (SL ( F ) , Z ) onemust replace the scissors congruence group P ( F ) of Dupont-Sah with the refined scissorscongruence group RP ( F ) (and its subgroup RB ( F )), as shown by the first author in [4]. The refined scissors congruence group of a commutative ring A is defined by a presentationanalogous to P ( A ) but as a module over the group ring R A := Z [ A × / ( A × ) ] rather thanas an abelian group. The homology groups H • (SL ( A ) , Z ) are naturally R A -modules andthis module structure plays a central role in all of our calculations. Thus, for example, theexact sequence of Theorem A is a sequence of R A -modules. Theorem A generalizes the main result of [8], where a (somewhat more precise) result wasproved in the case of complete discrete valuations (with residue characteristic not equal2). We note furthermore that there is an analogous behaviour of second homology groups,replacing the functor RP with the first Milnor-Witt K -theory functor K MW1 : Supposethat 1 / ∈ A and that k is infinite. Then there is a natural short exact sequence0 → H (SL ( A ) , Z ) → H (SL ( F ) , Z ) → K MW1 ( k ) → . This follows from the facts that H (SL ( R ) , Z ) can be identified naturally with K MW2 ( R )for local domains R with infinite residue field (by Schlichting [16]) and that there is anexact localization sequence in Milnor-Witt K -theory0 → K MW2 ( A ) → K MW2 ( F ) → K MW1 ( k ) → A containing 1 / Theorem A to beinjective in general. It is certainly injective in the case of a complete discrete valuation (by[8]). The general question is related to the Gersten conjecture for K .Our second main result concerns the calculation of the second homology of the congru-ence group Γ ( m A ) := (cid:26)(cid:18) a bc d (cid:19) ∈ SL ( A ) | c ∈ m A (cid:27) where m A is the maximal ideal of the discrete valuation ring A . We show the following (fora more precise statement see Theorem 4.16 below): Theorem B.
The inclusion Γ ( m A ) → SL ( A ) induces an exact sequence (of R A -modules) → P ( k ) (cid:2) (cid:3) → H (Γ ( m A ) , Z (cid:2) (cid:3) ) → H (SL ( A ) , Z (cid:2) (cid:3) ) → I ( k ) (cid:2) (cid:3) → . Here I ( k ) denotes the second power of the fundamental ideal I ( k ) of the Grothendieck-Witt ring GW( k ) of the field k and P ( k ) is a certain quotient of the scissors congruencegroup P ( k ).The map H (SL ( A ) , Z ) ∼ = K MW2 ( A ) → I ( A ) → I ( k ) is well-known from Milnor-Witt K -theory. We give an explicit formula for the map P ( k ) (cid:2) (cid:3) → H (Γ ( m A ) , Z (cid:2) (cid:3) ): Let d : A × → Γ ( m A ) denote the inclusion a diag( a, a − ). The map sends a generator[¯ a ] of P ( k ), a ∈ A × \ { } , to 1 / d ( a ) ∧ d (1 − a ). This elementin turn is known to map to [ a ][1 − a ] ∈ K MW2 ( A ) (cid:2) (cid:3) ∼ = H (SL ( A ) , Z (cid:2) (cid:3) ) (see, for ex-ample, [7, Corollary 4.2]), which is of course 0 by the Steinberg relation in Milnor-Witt K -theory. However, the kernel of the quotient map P ( k ) → P ( k ) is in general small andoften trivial. In particular, we show how to calculate P ( k ) in the case where F is a globalfield (and hence k is finite). For example, when F = Q and A = Z ( p ) ( p ≥
11) we have
HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 3 P ( F p ) (cid:2) (cid:3) = P ( F p ) (cid:2) (cid:3) when p p + 1) ′ where m ′ is the odd part of the integer m . Remark.
We have had to state all of our main results over Z (cid:2) (cid:3) because of 2-torsionambiguities in several of the fundamental results that we rely on, and because our methodsof proof (eg. the character-theoretic local-global principle) require us to invert 2. We donot know what modifications we should expect to have to make to the results presented ifwe want them to be valid over Z instead. Overview of the article.
In Section 1 we review the relevant facts about scissors con-gruence groups and their relation to the third homology of SL , and to indecomposable K .In Section 2 we prove Theorem A (Theorem 2.1 below). We first must define the R A -homomorphism ∆ π : H (SL ( F ) , Z ) → RP ( k ) (depending on a choice of uniformizer π ). Up to some known results about K , Theorem A can be reduced to an exact se-quence involving scissors congruence groups (Theorem 2.3 ). This in turn is proved usinga character-theoretic local-global principle for modules over group rings Z [ G ] where G is a(multiplicative) elementary abelian 2-group (recalled from [9]).Our second main theorem, Theorem B (Theorem 4.16 below) is proved by a carefulcomparison of the Mayer-Vietoris homology exact sequence associated to the amalgamatedproduct decomposition SL ( F ) ∼ = SL ( A ) ∗ Γ ( m A ) SL ( A ) with a spectral sequence relatingthe homology of SL ( A ) to (refined) scissors congruence groups. Section 3 reviews thebasic facts about the amalgamated product decomposition and the associated long exactMayer-Vietoris sequence. In particular, we detail how this is a sequence of R F -modules,because this module structure plays an essential role in our subsequent calculations.In Section 4, we give the technical details of the proof of Theorem B. First, we use The-orem A to identify im( δ ) with P ( k ) (cid:2) (cid:3) (as an R F -module). Here δ : H (SL ( F ) , Z (cid:2) (cid:3) ) → H (Γ ( m A ) , Z (cid:2) (cid:3) ) is the connecting homomorphism of the Mayer-Vietoris sequence. Theremainder of the section is devoted to the explicit formula for δ and the calculation of kerneland cokernel of H (Γ ( m A ) , Z (cid:2) (cid:3) ) → H (SL ( A ) , Z (cid:2) (cid:3) ). Both of these require calculationswith spectral sequences associated to the action of GL ( A ) on complexes of vectors. Indeed,the connecting homomorphism δ is shown to be essentially identifiable with a d -differentialfrom such a spectral sequence.Finally, in Section 5 we calculate P ( k ) (cid:2) (cid:3) in the case that F is a global field. We usethis to obtain more precise calculations for the groups SL ( Z ( p ) ) ⊂ SL ( Q ). Terminology and Notation.
In this article all rings are commutative, except possiblygroup rings, and have an identity element. For a ring A , A × will denote its group of units.For an abelian group or module M , M (cid:2) (cid:3) denotes M ⊗ Z Z (cid:2) (cid:3) . Aknowledgements.
BM and FM would like to acknowledge the support of S ao Paulo Re-search Foundation FAPESP (Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo) fortheir visit to University College Dublin during the year 2019 (Grant Numbers 2017/26310-7
KEVIN HUTCHINSON, BEHROOZ MIRZAII, FATEMEH Y. MOKARI and 2018/03561-7 respectively), which part of this work is done. BM and FM also wouldlike to thank KH and University College Dublin for their hospitality and for very friendlyworking environment.1.
Review: Scissors congruence groups and the third homology of SL Classical scissors congruence groups and a Bloch-Wigner exact sequence.
For a ring A , let W A := { a ∈ A × : 1 − a ∈ A × } . The scissors congruence group P ( A ) of A is the quotient of the free abelian group generatedby symbols [ a ], a ∈ W A , by the subgroup generated by elements X a,b := [ a ] − [ b ] + h ba i − h − a − − b − i + h − a − b i , where a, b, a/b ∈ W A . Let S Z ( A × ) := ( A × ⊗ A × ) / h a ⊗ b + b ⊗ a : a, b ∈ A × i . We denote the elements of P ( A ) and S Z ( A × ) represented by [ a ] and a ⊗ b again by [ a ] and a ⊗ b , respectively. By direct computation one sees that λ : P ( A ) → S Z ( A × ) , [ a ] a ⊗ (1 − a ) , is a well-defined homomorphism. The kernel of λ is called the Bloch group of A and isdenoted by B ( A ).In fact we have the following: If A is a field or a local ring whose residue field has morethan five elements, then we have the exact sequence0 → B ( A ) → P ( A ) → S Z ( A × ) → K M ( A ) → , (see [15, Lemma 4.2]), where K M ( A ) is the second Milnor K -group of A .Recall that for a local ring A there is a natural homomorphism of graded commutativerings K M • ( A ) → K • ( A ), from Milnor K -theory to K -theory. The indecomposable K of A , K ind3 ( A ), is defined to be the cokernel of the map K M ( A ) → K ( A ). Over a local ring (ormore generally a ring with many units) the Bloch group and the indecomposable part ofthe third K -group are deeply connected. Theorem 1.1 (Bloch-Wigner exact sequence) . Let A be either a field or a local domainwhose residue field has at least elements. Then we have a natural exact sequence → Tor Z ( µ ( A ) , µ ( A )) ∼ → K ind3 ( A ) → B ( A ) → , where Tor Z ( µ ( A ) , µ ( A )) ∼ is the unique nontrivial extension of Tor Z ( µ ( A ) , µ ( A )) by Z / if char( A ) = 2 and is equal to Tor Z ( µ ( A ) , µ ( A )) if char( A ) = 2 .Proof. The case of infinite fields has been proved in [21] and the case of finite fields hasbeen settled in [4]. The case of local rings has been dealt in [15]. (cid:3)
A Bloch-Wigner exact sequence also exists over a ring with many units [13], [14].
HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 5
Refined scissors congruence groups.
Let A be a ring. Let G A := A × / ( A × ) andset R A := Z [ G A ]. The element of G A represented by a ∈ A × is denoted by h a i . We set hh a ii := h a i − ∈ R A .Let RP ( A ) be the quotient of the free R A -module generated by symbols [ a ], a ∈ W A ,by the R A -submodule generated by elements Y a,b := [ a ] − [ b ] + h a i h ba i − h a − − i (cid:20) − a − − b − (cid:21) − h − a i (cid:20) − a − b (cid:21) , where a, b, a/b ∈ W A . We have a natural surjective map RP ( A ) → P ( A ) and from thedefinition it follows immediately that P ( A ) ≃ RP ( A ) G A = H ( G A , RP ( A )) . Let I A be the augmentation ideal of the group ring R A . By direct, but tedious, compu-tation one can show that the map λ : RP ( A ) → I A , [ a ]
7→ hh a iihh − a ii is a well-defined R A -homomorphism. If we consider S Z ( A × ) as a trivial G A -module, thenthe map λ : RP ( A ) → S Z ( A × ) , [ a ] a ⊗ (1 − a ) , is a homomorphism of R A -modules. In fact λ is the composite RP ( A ) → P ( A ) λ → S Z ( A × ).The refined scissors congruence group of A is defined as the R A -module RP ( A ) := ker( λ : RP ( A ) → I A ) . The refined Bloch group of A is defined as the R A -module RB ( A ) := ker( λ | RP ( A ) ) (see[5], [4]). Proposition 1.2. ([8, Proposition 2.9])
Let A be a ring. Then (i) RP ( A ) → P ( A ) induces the isomorphism RP ( A ) h i G A ≃ P ( A ) h i , (ii) RB ( A ) → B ( A ) induces the isomorphism RB ( A ) h i G A ≃ B ( A ) h i , (iii) RB ( A ) → RP ( A ) induces the isomorphism I A RB ( A ) h i ≃ I A RP ( A ) h i . Over finite fields the Bloch group and the refined Bloch group are the same.
Proposition 1.3. ([4, Section 7]) If k is a finite field (with at least elements), then G k acts trivially on RB ( k ) . In particular RB ( k ) ≃ B ( k ) . Moreover RB ( k ) (cid:2) (cid:3) = RP ( k ) (cid:2) (cid:3) ≃ P ( k ) (cid:2) (cid:3) = B ( k ) (cid:2) (cid:3) . For more results on finite fields we refer the reader to [4, Section 7].
KEVIN HUTCHINSON, BEHROOZ MIRZAII, FATEMEH Y. MOKARI
A refined Bloch-Wigner exact sequence.
For any ring A , H • (SL ( A ) , Z ) is nat-urally a module over the ring R A as follows: Given a ∈ A × , choose M ∈ GL ( A ) withdeterminant a . Then h a i · z := ( C M ) ∗ ( z ) where C M denotes conjugation by M . As we willsee, this module structure plays a central role in all of our calculations below.For any local ring, whose residue field has at least three elements, there is a naturalhomomorphism of R A -modules H (SL ( A ) , Z ) → K ind3 ( A ) , (in which K ind3 ( A ) has the trivial R A -module structure). This is surjective if A is an infinitefield [11, Section 5].Throughout this article, we will say that a field F is sufficiently large if either F isinfinite or | F | = p d with ( p − d >
6. Thus F is sufficiently large if and only if | F | / ∈{ , , , , , , , , , , } . Theorem 1.4. ([8, Theorem 3.22])
Let A be a local domain with sufficiently large residuefield. Then there is a commutative diagram of R A -modules with exact rows (where all termsin the lower row are trivial R A -modules): Z ( µ ( A ) , µ ( A )) h i H (SL ( A ) , Z h i ) RB ( A ) h i
00 Tor Z ( µ ( A ) , µ ( A )) h i K ind3 ( A ) h i B ( A ) h i . = Using this and taking G A -coinvariants on the top row, one deduces: Proposition 1.5. ([8, Corollary 3.23])
Let A be a local domain with sufficiently largeresidue field. Then H (SL ( A ) , Z (cid:2) (cid:3) ) G A ≃ K ind3 ( A ) (cid:2) (cid:3) . In particular we have the exact sequence → I A H (SL ( A ) , Z (cid:2) (cid:3) ) → H (SL ( A ) , Z (cid:2) (cid:3) ) → K ind3 ( A ) (cid:2) (cid:3) → . Note that for any R A -module M , we have I A M = ker ( M → M G A ).Let H (SL ( A ) , Z ) be the kernel of the map H (SL ( A ) , Z ) → K ind3 ( A ). Theorem 1.4and Propositions 1.2 and 1.5 imply that: Corollary 1.6.
Let A be a local domain with sufficiently large residue field. Then H (SL ( A ) , Z (cid:2) (cid:3) ) = I A H (SL ( A ) , Z (cid:2) (cid:3) ) ≃ I A RP ( A ) (cid:2) (cid:3) = I A RB ( A ) (cid:2) (cid:3) . HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 7
Special elements in P ( A ) and RP ( A ) . Let A be a local ring. For an element a ∈ W A , let { a } := [ a ] + [ a − ] ∈ P ( A ). If U ,A := 1 + m A , then A × = W A ∪ U ,A . For u ∈ U ,A , we define { u } := { ua } − { a } , for any a ∈ W A . It is easy to see that this definitionis well-defined. By a direct calculation one can show that the map ψ : A × → P ( A ) , a
7→ { a } , is a homomorphism and ( A × ) is in its kernel, i.e. { ab } = { a } + { b } and { c } = 0 for all a, b, c ∈ A × . Let e P ( A ) := P ( A ) / K A , where K A is the subgroup of P ( A ) generated by the elements ψ ( a ) = { a } , a ∈ A × . Since λ ( { a } ) = ( − a ) ⊗ a , we have the natural homomorphism˜ λ : e P ( A ) → e S Z ( A × ) , where e S Z ( A × ) := S Z ( A × ) / h ( − a ) ⊗ a | a ∈ A × i . We set e B ( A ) := ker(˜ λ ).We now consider two different liftings to RP ( A ) of the family of elements { a } in P ( A ).For a ∈ W A , let ψ ( a ) := [ a ] + h− i [ a − ] and ψ ( a ) := h − a i (cid:0) h a i [ a ] + [ a − ] (cid:1) . For u ∈ U ,A = A × \W A , we define ψ i ( u ) := ψ i ( ua ) − h u i ψ i ( a ) , where a ∈ W a . By [8, Lemma 4.2] this is independent of the choice of a ∈ W A , so thedefinition is well defined. Now one can show [8, Proposition 4.3] that the maps ψ i : A × → RP ( A ) , a ψ i ( a ) , i = 1 , ψ i ( ab ) = h a i ψ i ( b ) + ψ i ( a ), for all a, b ∈ A × . For basicproperties of ψ i ( a ) see [8, Section 4] and [5, Section 3].Let K ( i ) A , i = 1 ,
2, denote the R A -submodule of RP ( A ) generated by the set { ψ i ( a ) | a ∈ A × } . One can show that λ ( ψ i ( a )) = p + − hh a ii = hh− a iihh a ii , where p + − := h− i + 1 ∈ R A . Thus λ ( K ( i ) A ) = p + − I A ⊆ I A . Moreover ker( λ | K ( i ) A ) isannihilated by 4 [8, Lemma 4.6]. Let g RP ( A ) := RP ( A ) / K (1) A . Then ˜ λ : g RP ( A ) → I A /p + − I A , and ˜ λ : g RP ( A ) → e S Z ( A × )induced by λ and λ respectively, are well-defined R A -homomorphism. We set g RP ( A ) := ker(˜ λ ) , g RB ( A ) := ker (˜ λ ) . It is easy to see that RP ( A ) → P ( A ) induces the natural maps g RP ( A ) → e P ( A ) , g RP ( A ) → e P ( A ) , g RB ( A ) → e B ( A ) . KEVIN HUTCHINSON, BEHROOZ MIRZAII, FATEMEH Y. MOKARI
Proposition 1.7. ([8, Corollary 4.7])
Let A be a local ring such that its residue field hasat least elements. Then the natural maps RP ( A ) → g RP ( A ) and RB ( A ) → g RB ( A ) aresurjective with kernel annihilated by . In particular RP ( A ) (cid:2) (cid:3) ≃ g RP ( A ) (cid:2) (cid:3) , RB ( A ) (cid:2) (cid:3) ≃ g RB ( A ) (cid:2) (cid:3) . Let A be a local ring. As in [21, Lemma 1.3] one can show that the element [ a ] + [1 − a ] ∈B ( A ) is independent of the choice of a ∈ W A . We denote this constant by c A := [ a ] + [1 − a ] , a ∈ W A . This constant has order dividing 6.In [8, Lemma 4.9], it has been shown that [ a ] + h− i [1 − a ] + hh − a ii ψ ( a ) is in RB ( A )and is independent of the choice of a ∈ W A . We denote this constant by C A := [ a ] + h− i [1 − a ] + hh − a ii ψ ( a ) , a ∈ W A . Under the homomorphism RP ( A ) → P ( A ), C A maps to c A . One can show [8, Lemma 4.9]that 3 C A = ψ ( −
1) and 6 C A = 0 . Refined scissors congruence group with generators and relations.
The fol-lowing theorem gives a description of the structure of RP ( A ). Theorem 1.8. ([9, Corollary 4.4] , [8, Proposition 5.4]) Let A be either a field with at leastfour elements or a local ring whose residue field has more than elements. Then RP ( A ) (cid:2) (cid:3) = e + − RP ( A ) (cid:2) (cid:3) , where e + − := p + − / h− i + 1) / . In particular h− i ∈ R A acts trivially on RP ( A ) (cid:2) (cid:3) (and hence also on H (SL ( A ) , Z (cid:2) (cid:3) ) ). It follows from this theorem that RP ( A ) (cid:2) (cid:3) is a quotient of RP ( A ) (cid:2) (cid:3) and henceadmits a simple explicit presentation as a R A -module. Proposition 1.9.
For a ring A let RP ′ ( A ) be the R A -module with generators denoted bysymbols [ a ] ′ , a ∈ W A , subject to the following relations: (i) [ a ] ′ − [ b ] ′ + h a i h ba i ′ − h a − − i (cid:20) − a − − b − (cid:21) ′ − h − a i (cid:20) − a − b (cid:21) ′ = 0 for all a, b, a/b ∈ W A , (ii) h− i [ a ] ′ = [ a ] ′ for all a ∈ W A , (iii) [ a ] ′ + [ a − ] ′ = 0 for all a ∈ W A .If A is either a field with at least four elements or a local ring where its residue field hasmore that elements, then the R A -module homomorphism RP ′ ( A ) (cid:2) (cid:3) → RP ( A ) (cid:2) (cid:3) , [ a ] ′ g ( a ) is an isomorphism, where g ( a ) := p + − [ a ] + hh − a ii ψ ( a ) .Proof. See [9, Corollary 4.5] and [8, Remark 5.6]. (cid:3)
HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 9
The key identity.
We recall the following key identity : Lemma 1.10. ([5, Theorem 3.12] , [8, Theorem 4.14]) Let B be a field or a local ring whoseresidue field has at least elements. Then for any a ∈ B × we have hh a ii C B = ψ ( a ) − ψ ( a ) in RP ( B ) . Corollary 1.11. If B is a local ring whose residue field has at least elements, then forall a ∈ W B , we have hh a ii C B = h a − ihh− a ii [ a ] in g RP ( B ) .Proof. Since 3 C B + ψ ( −
1) = 0, we have 2 C B = − C B in g RP ( B ). Thus hh a ii C B = ψ ( a ) in g RP ( B ). However, 0 = ψ ( a ) = [ a ] + h− i [ a − ] in g RP ( B ). Hence [ a − ] = −h− i [ a ]. Thusin g RP ( B ) we have ψ ( a ) = h − a i (cid:0) h a i [ a ] + [ a − ] (cid:1) = h − a i ( h a i [ a ] − h− i [ a ])= h a − i ( h− a i − a ] = h a − ihh− a ii [ a ] . (cid:3) We will need the following refinement of [8, Lemma 6.1].
Lemma 1.12.
Let B be a local ring with maximal ideal m B and residue field k . Let L B denote the R B -submodule of RP ( B ) generated by the elements [ au ] − [ a ] and hh u ii C B , a ∈ W B , u ∈ U ,B = 1 + m B . Then there is a short exact sequence of R B -modules → L B → g RP ( B ) → g RP ( k ) → . Proof.
Clearly the functorial map, p say, g RP ( B ) → g RP ( k ) is surjective and L B ⊆ ker( p ).Let Q ( B ) := g RP ( B ) / L B .We claim that U ,B ⊂ B × acts trivially on Q ( B ): Let a ∈ W B . By Corollary 1.11[ a ] = h− a i [ a ] − h a − ihh a ii C B . Since for any u ∈ U ,B , au ∈ W B , again by Corollary 1.11[ au ] = h− au i [ au ] − h au − ihh au ii C B . However in Q ( B ) we have h au − ihh au ii C B = h a − ih u ′ ihh au ii C B = h a − ihh a ii C B , where u ′ = (1 − au ) / (1 − a ) ∈ U ,B . (Note that in above we use the formula hh au ii = hh a ii + h a ihh u ii and the fact that for all w ∈ U ,B , h w i C B = C B in Q ( B ).) Thus in Q ( B )0 = [ a ] − [ au ] = h− a i [ a ] − h− au i [ au ] for all a ∈ W B , u ∈ U ,B . It follows that h− a i [ a ] = h− au i [ au ] = h− au i [ a ] in Q ( B ) for all a ∈ W B , u ∈ U ,B . Multiplying both sides by h− a i , we deduce that[ a ] = h u i [ a ]in Q ( B ) for all u ∈ U ,B , a ∈ W B , proving the claim.It follows that the R B -module structure on Q ( B ) induces a R k -module structure, since k × ≃ B × /U ,B .Thus there is a well-defined R k -module homomorphism g RP ( k ) → Q ( B ), [¯ a ] [ a ] + L B ,giving an inverse to the map ¯ p : Q ( B ) → g RP ( k ). (cid:3) The third homology of SL of a discrete valuation ring Throughout this section A will be a discrete valuation ring with maximal ideal m A andresidue field k . Let F be the field of fractions of A and let v = v A : F × → Z be theassociated discrete valuation. We fix a uniformizer π of the valuation, i.e. a generator of m A . Moreover let U n,A := 1 + m nA .In this section we will prove the following main theorem: Theorem 2.1.
Let A be a discete valuation ring with field of fractions F and sufficientlylarge residue field k . There is an R A -map ∆ π : H (SL ( A ) , Z ) → g RP ( k ) such that thesequence H (SL ( A ) , Z (cid:2) (cid:3) ) → H (SL ( F ) , Z (cid:2) (cid:3) ) ∆ π −→ RP ( k ) (cid:2) (cid:3) → is an exact sequence of R A -modules. Induction of modules.
From the natural maps
A ֒ → F and A ։ k we obtain thehomomorphisms of groups G A ֒ → G F , G A ։ G k . Thus any R F -module or any R k -module has a natural R A -module structure. For any R k -module M we define the induced R F -moduleInd Fk M := R F ⊗ R A M. Note that, since M is an R A - R k -bimodule, it follows that Ind Fk M is an R k -module in anatural way: h ¯ a i . ( x ⊗ m ) = x ⊗ h ¯ a i m = h a i x ⊗ m . Lemma 2.2. ([9, Lemma 5.4])
For any R k -module M , the R k -homomorphisms ρ : Ind Fk M → M, h uπ r i ⊗ m
7→ h ¯ u i m, and ρ π : Ind Fk M → M, h uπ r i ⊗ m ( h ¯ u i m if r is odd if r is even,for all r ∈ Z , u ∈ A × , induce the isomorphism of R k -modules (and so R A -modules) Ind Fk M ( ρ ,ρ π ) ≃ −→ M ⊕ M, I F Ind Fk M ( ρ ,ρ π ) ≃ −→ I k M ⊕ M. HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 11
The specialization homomorphism.
There is a natural surjective specializationintroduced and developed in [5, Section 4.3] and [9, Section 5] defined as follows: S v : RP ( F ) → Ind Fk g RP ( k ) , [ a ] ⊗ [¯ a ] if v ( a ) = 01 ⊗ C k if v ( a ) > − (1 ⊗ C k ) if v ( a ) < R F -modules. It induces a well-defined map S v : g RP ( F ) → Ind Fk g RP ( k ) . Furthermore this induces a well-defined specialization S v : RP ( F ) → Ind Fk g RP ( k ) . The composite RP ( F ) S v −→ Ind Fk g RP ( k ) ( ρ ,ρ π ) ≃ −→ g RP ( k ) ⊕ g RP ( k )induces two natural homomorphisms of R A -modules δ π := ρ π ◦ S π : RP ( F ) → g RP ( k ) , δ := ρ ◦ S π : RP ( F ) → g RP ( k ) . Furthermore the restriction of these maps to I F RP ( F ) induce homomorphisms of R A -modules δ π : I F RP ( F ) → g RP ( k ) , δ : I F RP ( F ) → I k g RP ( k ) . By direct computation one sees that the composite I A RP ( A ) → I F RP ( F ) δ π → g RP ( k )is trivial.The main theorem (Theorem 2.1) will follow from the following result (whose proof willoccupy Subsections 2.3 to 2.6): Theorem 2.3.
Let A be a discrete valuation ring with field of fractions F and sufficientlylarge residue field k . Then I A RP ( A ) (cid:2) (cid:3) → I F RP ( F ) (cid:2) (cid:3) δ π → RP ( k ) (cid:2) (cid:3) → is an exact sequence of R A -modules. Characters and a local global principal.
Here we review a character-theoreticlocal-global principle for modules over elementary 2-torsion abelian groups, developed in[8, Section 3]. This theory will be used extensively in the proof of Theorem 2.3.Let G be a multiplicative abelian group in which g = 1 for all g ∈ G . Set R := Z [ G ].The group b G := Hom( G, {± } ) is called the group of characters of G .For a character χ ∈ b G , let R χ be the ideal of R generated by the set { g − χ ( g ) | g ∈ G } .In fact R χ is the kernel of the ring homomorphism ρ χ : R → Z sending g ∈ G to χ ( g ) forany g ∈ G . Let R χ := R / R χ , which is isomorphic to Z . If M is an R -module, we let M χ := M/ R χ M . Furthermore, given m ∈ M we will denoteits image in M χ by m χ .In particular, if χ is the trivial character, then R χ is the augmentation ideal I G and M χ = M G . Now we list some basic, but useful, properties of the ideals R χ which we willuse:(i) For any χ ∈ b G , R χ (cid:2) (cid:3) = ( R χ ) (cid:2) (cid:3) .(ii) If χ , χ ∈ b G are distinct, then R χ (cid:2) (cid:3) + R χ (cid:2) (cid:3) = R (cid:2) (cid:3) .(iii) If χ , χ ∈ b G are distinct and if M is a R -module, then R χ M (cid:2) (cid:3) ∩ R χ M (cid:2) (cid:3) = R χ R χ M (cid:2) (cid:3) . Lemma 2.4. ([8, Corollaries 3.7, 3.8])
The functors M
7→ R χ M and M M χ are exacton the category of R (cid:2) (cid:3) -modules. In particular the functors M M G and M
7→ I G M areexact on the category of R (cid:2) (cid:3) -modules. Here is the character-theoretic local-global principle:
Theorem 2.5. ([8, Proposition 3.10])
Let f : M → N be a homomorphism of R (cid:2) (cid:3) -modules. For any χ ∈ b G , let f χ be the induced homomorphism M χ → N χ . (i) f is injective if and only if f χ is injective for all χ ∈ b G . (ii) f is surjective if and only if f χ is surjective for all χ ∈ b G . (iii) f is an isomorphism if and only if f χ is an isomorphism for all χ ∈ b G . Here we fix some notations for later use. For g ∈ G , consider the orthogonal idempotents e g + := g + 12 , e g − := g − , in R (cid:2) (cid:3) . If M is a R (cid:2) (cid:3) -module, then we have the decomposition M = e g + M ⊕ e g − M. For a ring A , G A = A × / ( A × ) is a 2-torsion group. If M is a R A (cid:2) (cid:3) -module we set M + := e − M = M/e − − M, M − := e − − M = M/e − M. The reduction and the specialization maps.
Let L v denote the R F -submoduleof g RP ( F ) generated by the elements [ u ], u ∈ U ,A . HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 13
Lemma 2.6.
There is a commutative diagram of R A -modules with exact rows and columns L A g RP ( A ) g RP ( k ) 00 L v g RP ( F ) Ind Fk g RP ( k ) 0 g RP ( k )0 S v δ π Furthermore, the lower row in this diagram is a sequence of R F -modules.Proof. The top row is Lemma 1.12. The lower row is [10, Proposition 3.2]. The columnexact sequence is Lemma 2.2. The commutativity of the diagram and the exactness of thecolumn are immediate from the definitions. (cid:3)
If we tensor this diagram with Z (cid:2) (cid:3) and then multiply the rows by e − ∈ R A (cid:2) (cid:3) , thenby Theorem 1.8 and Proposition 1.7 we obtain the commutative diagram with exact rowsand column 00 L + A (cid:2) (cid:3) RP ( A ) (cid:2) (cid:3) RP ( k ) (cid:2) (cid:3) L + v (cid:2) (cid:3) RP ( F ) (cid:2) (cid:3) Ind Fk RP ( k ) (cid:2) (cid:3) RP ( k ) (cid:2) (cid:3) S v δ π Now if we apply the exact functor I A · − to the top row and the exact functor I F · − tothe lower row (see Lemma 2.4) we arrive at the following result. Corollary 2.7.
There is a commutative diagram of R A -modules with exact rows and col-umn I A L + A (cid:2) (cid:3) I A RP ( A ) (cid:2) (cid:3) I k RP ( k ) (cid:2) (cid:3) I F L + v (cid:2) (cid:3) I F RP ( F ) (cid:2) (cid:3) I F (cid:0) Ind Fk RP ( k ) (cid:2) (cid:3)(cid:1) RP ( k ) (cid:2) (cid:3) S v δ π Furthermore, the lower row is an exact sequence of R F -modules. Given Corollary 2.7, Theorem 2.3 is equivalent to the following: • The natural R A -homomorphism α : I A L + A (cid:2) (cid:3) → I F L + v (cid:2) (cid:3) is surjective.By the character-theoretic local-global principle (Theorem 2.5), this in turn is equivalentto the statement: • The homomorphisms α χ : (cid:0) I A L + A (cid:2) (cid:3)(cid:1) χ → (cid:0) I F L + v (cid:2) (cid:3)(cid:1) χ are surjective for all χ ∈ b G A .We need the following lemma. Lemma 2.8.
Let M be a R A (cid:2) (cid:3) -module and N a R F (cid:2) (cid:3) -module. Then (i) ( I A M ) χ = 0 , where χ is the trivial character on G A . (ii) ( M + ) χ = 0 = ( N + ) χ if χ ∈ b G A and χ ( −
1) = − . (iii) ( I A M + ) χ ≃ M χ and ( I F N + ) χ ≃ N χ if χ ∈ b G A , χ = χ and χ ( −
1) = 1 .Proof. (i) The triviality of ( I A M ) χ follows from the fact that I A (cid:2) (cid:3) = I A (cid:2) (cid:3) .(ii) First note that ( M + ) χ = ( e − M ) χ = e − M/e − R χA M . If χ ( −
1) = −
1, then e − ∈R χA (cid:2) (cid:3) . Since e − is an idempotent, we have e − M = e − e − M ⊆ e − R χA M . Thus( M + ) χ = 0. Similarly ( N + ) χ = 0.(iii) We have the decomposition of R A -modules I A M = e − I A M ⊕ e − − I A M . Let χ = χ and χ ( −
1) = 1. Then clearly e − − ∈ R χA (cid:2) (cid:3) . Thus with a similar argument as in the proofof (ii), we have ( e − − I A M ) χ = 0. Therefore ( I A M + ) χ = ( I A M ) χ . Now it is easy to showthat the natural map β : ( I A M ) χ → M χ , induced by inclusion, is an isomorphism. In factsince χ = χ , there is a ∈ A × such that χ ( a ) = −
1. Thus hh a ii m = ( h a i − χ ( a )) m − m HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 15 and hence β ( hh a ii m ) = − m . This shows that β is surjective. The injectivity of β followsfrom the equality I A R χA M = I A M ∩ R χA M .The proof of the isomorphism ( I F N + ) χ ≃ N χ is similar. Just note that here we shoulduse the fact R χA I F M = R χA M ∩ I F M which follows from the fact that R χA (cid:2) (cid:3) + I F (cid:2) (cid:3) = R F (cid:2) (cid:3) . (cid:3) A decomposition lemma.
For any χ ∈ b G A , we define two characters χ + , χ − ∈ b G F (depending on our chosen uniformizer π ) as follows: For u ∈ A × , r ∈ Z , χ + ( π r u ) := χ ( u ) χ − ( π r u ) := ( − r χ ( u ) . For example if χ ∈ b G A is the trivial character, then χ is the trivial character on G F and χ − = χ v , where χ v ∈ b G F is the character associated to the valuation v on F given by χ v ( a ) = ( − v ( a ) . Lemma 2.9.
Let M be a R F (cid:2) (cid:3) -module. Let χ ∈ b G A . Then the homomorphism M χ → M χ + ⊕ M χ − , m χ ( m χ + , m χ − ) is an isomorphism of R A -modules.Proof. As a R F -module we have the decomposition M = e π + M ⊕ e π − M . Thus M χ = ( e π + M ) χ ⊕ ( e π − M ) χ . We show that ( e π + M ) χ ≃ M χ + . The proof of ( e π − M ) χ ≃ M χ − is similar. We have( e π + M ) χ = ( M/e π − M ) χ = ( M/e π − M ) / R χA ( M/e π − M )= ( M/e π − M ) / (( R χA M + e π − M ) /e π − M ) ≃ M/ ( R χA M + e π − M ) . Since M χ + = M/ R χ + F M , we must show that R χ + F M = R χA M + e π − M . The ideal R χ + F isgenerated by h uπ i − χ ( u ) and h u i − χ ( u ), u ∈ A × . Clearly R χA M + e π − M ⊆ R χ + F M . Since h uπ i − χ ( u ) = 2 h u i e π − + ( h u i − χ ( u )), we see that R χ + F M ⊆ R χA M + e π − M . (cid:3) Let N := L v (cid:2) (cid:3) . By Lemma 2.9, ( I F N + ) χ ≃ ( I F N + ) χ ⊕ ( I F N + ) χ − , where χ ∈ b G A is the trivial character. Since χ is the trivial character on G F , ( I F N + ) χ = 0 (Lemma2.8 (i)). Moreover χ − = χ v and thus ( I F N + ) χ − = ( I F N + ) χ v = I F ( N + ) χ v . By the proofof [10, Theorem 3.7], ( L + v (cid:2) (cid:3) ) χ v = 0. Therefore( I F L + v (cid:2) (cid:3) ) χ = 0 . On the other hand ( I A L + A (cid:2) (cid:3) ) χ = 0. Thus by applying Lemma 2.8 and Lemma 2.9,Theorem 2.3 is equivalent to the following statement: • The homomorphisms α χ : ( L A (cid:2) (cid:3) ) χ ( L v (cid:2) (cid:3) ) χ = ( L v (cid:2) (cid:3) ) χ + ⊕ ( L v (cid:2) (cid:3) ) χ − aresurjective for all χ ∈ b G A satisfying χ = χ and χ ( −
1) = 1.
Since the R F -module L v is generated by the elements [ u ], u ∈ U ,A , it follows that forany ψ ∈ b G F , ( L v (cid:2) (cid:3) ) ψ is generated as a Z -module by the elements [ u ] ψ for u ∈ U ,A . ThusTheorem 2.3 is entirely equivalent to the following: Proposition 2.10.
For all u ∈ U ,A and for all χ ∈ b G A satisfying χ = χ and χ ( −
1) = 1 ,the elements ([ u ] χ + , and (0 , [ u ] χ − ) lie in im( α χ ) . Proposition 2.10 (and hence Theorem 2.3) follows from Lemmas 2.11 and 2.19 andCorollaries 2.16 and 2.18 below.
Lemma 2.11.
Let χ ∈ b G A . Suppose that χ | U n,A = 1 for some n ≥ , where U n,A = 1 + m nA .Then ( L v (cid:2) (cid:3) ) χ + = ( L v (cid:2) (cid:3) ) χ − = 0 .Proof. For δ ∈ { + , −} the characters χ δ ∈ b G F have the property that χ δ | U n,A = 1. Thelemma now follows from (the proof of) [10, Theorem 3.11]. (The reader may readily verifythat the hypothesis U n,A ⊂ U ,A in this theorem is only used in the proof to deduce that χ | U n,A = 1 for any given χ .) (cid:3) Obtainable elements.
For the remainder of this section we fix χ ∈ b G A satisfying(1) χ = χ ,(2) χ ( −
1) = 1,(3) For all n ≥ u ∈ U n,A with χ ( u ) = − u ∈ U ,A is obtainable if ([ u ] χ + ,
0) and (0 , [ u ] χ − ) both lie in im( α χ ). Wemust show that all u ∈ U ,A are obtainable. We begin with some elementary observations:For u ∈ U ,A we set ℓ ( u ) := v (1 − u ) ∈ N . Thus ℓ ( u ) = n if and only if u ∈ U n,A \ U n +1 ,A . Lemma 2.12. (i)
For any u ∈ U ,A with ℓ ( u ) = n there exists w ∈ U ,A satisfying w ≡ u mod U n +1 ,A and χ ( w ) = − χ ( u ) . (ii) For any u ∈ U ,A such that χ ( u ) = − we have (for δ ∈ { + , −} ) [ u ] χ δ = χ δ (1 − u )( C F ) χ δ in ( L v (cid:2) (cid:3) ) χ δ . (iii) If u ∈ U ,A and if χ ( u ) = 1 and χ δ (1 − u ) = − for some δ ∈ { + , −} , then [ u ] χ δ = 0 . (iv) If u ∈ U ,A , then χ − (1 − u ) = (cid:26) χ + (1 − u ) , ℓ ( u ) even − χ + (1 − u ) , ℓ ( u ) odd.Proof. (i) Choose any u ∈ U n,A \ U n +1 ,A . Choose z ∈ U n +1 ,A with χ ( z ) = − w = uz . Then w ≡ u mod U n +1 ,A and χ ( w ) = − χ ( u ).(ii) By Corollary 1.11 we have hh u ii C F = h u − ihh− u ii [ u ] in g RP ( F ) . Thus for any ψ ∈ b G F we have( ψ ( u ) − C F ) ψ = ψ ( u − ψ ( − u ) − u ] ψ in g RP ( F ) ψ . HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 17 If ψ ( u ) = − ψ ( −
1) = 1, this gives − C F ) ψ = − ψ (1 − u )[ u ] ψ . Now apply to ψ = χ δ and multiply both sides by − / − u , we have [1 − u ] χ δ = ( C F ) χ δ . However, ( C F ) χ δ = [ u ] χ δ +[1 − u ] χ δ .(iv) We have 1 − u = wπ ℓ ( u ) for some w ∈ A × . Therefore χ − (1 − u ) = χ ( w )( − ℓ ( u ) = χ + (1 − u )( − ℓ ( u ) . (cid:3) We immediately deduce:
Corollary 2.13.
Let u ∈ U ,A . (i) If χ ( u ) = − and ℓ ( u ) is even, then [ u ] χ = ± ( C F ) χ in (cid:0) L v (cid:2) (cid:3)(cid:1) χ . (ii) If χ ( u ) = − and ℓ ( u ) is odd, then [ u ] χ = ± (cid:16) ( C F ) χ + , − ( C F ) χ − (cid:17) in (cid:0) L v (cid:2) (cid:3)(cid:1) χ = (cid:0) L v (cid:2) (cid:3)(cid:1) χ + ⊕ (cid:0) L v (cid:2) (cid:3)(cid:1) χ − . (iii) If χ ( u ) = 1 and ℓ ( u ) is even and χ + (1 − u ) = − , then [ u ] χ = 0 . (iv) If χ ( u ) = 1 and ℓ ( u ) is odd, then [ u ] χ = (cid:0) [ u ] χ + , [ u ] χ − (cid:1) = (cid:26) (cid:0) [ u ] χ + , (cid:1) , χ + (1 − u ) = 1 (cid:0) , [ u ] χ − (cid:1) , χ + (1 − u ) = − . Corollary 2.14.
Let u ∈ U ,A . If ℓ ( u ) is even and χ ( u ) = − , then [ u ] χ ∈ im( α χ ) .Proof. By Corollary 2.13 (i), we only need to show that ( C F ) χ ∈ im( α χ ). Now if x ∈ U ,A with χ ( x ) = −
1, then hh x ii C A ∈ L A and hence α χ ( hh x ii C A ) = − C F ) χ = ( C F ) χ = (cid:16) ( C F ) χ + , ( C F ) χ − (cid:17) ∈ im( α χ ) . (cid:3) Lemma 2.15. If u ∈ U ,A and if χ ( u ) = 1 and ℓ ( u ) is odd, then [ u ] χ ∈ im( α χ ) .Proof. Let n = ℓ ( u ). Since U n,A /U n +1 ,A ≃ k and since | k | >
2, there exists w with ℓ ( w ) = ℓ ( uw ) = n . Furthermore, by Lemma 2.12 (i), we can choose w such that χ ( w ) = − a := 1 − w − uw , so that w = 1 − a − au . We claim that a ∈ W A : First note that v (1 − w ) = ℓ ( w ) = ℓ ( uw ) = v (1 − uw ) and hence a ∈ A × . Now 1 − a = w − uw − uw = w · − u − uw which lies in A × since v (1 − u ) = ℓ ( u ) = ℓ ( uw ) = v (1 − uw ). This proves the claim. For δ ∈ { + , −} we have0 = [ a ] χ δ − [ au ] χ δ + χ ( a )[ u ] χ δ − χ ( a ) χ (1 − a )[ uw ] χ δ + χ (1 − a )[ w ] χ δ in RP ( F ) (cid:2) (cid:3) χ δ .Since χ ( uw ) = χ ( w ) = − − w = a (1 − uw ), we have (by Lemma 2.12) that χ ( a )[ uw ] χ δ + [ w ] χ δ = ( − χ ( a ) χ δ (1 − uw ) + χ δ (1 − w )) ( C F ) χ δ = 0 . Thus ([ a ] − [ au ]) χ δ = χ ( a )[ u ] χ δ for each δ ∈ { + , −} and hence α χ ([ a ] − [ au ]) = χ ( a )[ u ] χ . (cid:3) Combining this with Corollary 2.13 (iv), we immediately deduce:
Corollary 2.16. If u ∈ U ,A satisfies ℓ ( u ) is odd and χ ( u ) = 1 , then u is obtainable. Lemma 2.17.
The elements (cid:16) ( C F ) χ + , (cid:17) and (cid:16) , ( C F ) χ − (cid:17) lie in im( α χ ) .Proof. Let u, w ∈ U ,A satisfy: ℓ ( u ) is odd, ℓ ( w ) > ℓ ( u ) is even and χ ( u ) = χ ( w ) = − ℓ ( uw ) = ℓ ( u ) is odd and χ ( uw ) = 1. Let z := (1 − u ) / (1 − uw ).We claim that z ∈ U ,A and ℓ ( z ) = ℓ ( w ) − ℓ ( u ) (and hence is odd): On the one hand, v (1 − u ) = ℓ ( u ) = ℓ ( uw ) = v (1 − uw ) implies that z ∈ A × . On the other hand,1 − z = u − uw − uw = u · − w − uw and v (1 − w ) = ℓ ( w ) > ℓ ( uw ) = ℓ ( u ) = v (1 − uw ). So v (1 − z ) = ℓ ( w ) − ℓ ( u ) > δ ∈ { + , −} . In L v (cid:2) (cid:3) χ δ we have0 = [ u ] χ δ − [ uw ] χ δ + χ ( u )[ w ] χ δ + χ δ (1 − u )[ wz ] χ δ + χ δ (1 − u )[ z ] χ δ (using χ δ (1 − u − ) = χ δ ( u ) χ δ (1 − u ) = − χ δ (1 − u ) since χ ( u ) = − χ ( z ) = − χ ( z ) = 1: Case 1: χ ( z ) = −
1. Then we write[ uw ] χ δ − χ ( u )[ w ] χ δ − χ δ (1 − u )[ wz ] χ δ = [ u ] χ δ + χ δ (1 − u )[ z ] χ δ . Let X := ([ uw ] − χ ( u )[ w ] − χ (1 − u )[ wz ]) χ and observe that X ∈ im( α χ ) by Corollaries 2.14 and 2.16, since ℓ ( w ) is even and χ ( w ) = − ℓ ( uw ) , ℓ ( wz ) are odd and χ ( uw ) = 1 = χ ( zw ). Thus X = (cid:16) [ u ] χ + + χ + (1 − u )[ z ] χ + , [ u ] χ − + χ − (1 − u )[ z ] χ − (cid:17) ∈ im( α χ ) . HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 19
However since ℓ ( u ), ℓ ( z ) are odd and χ ( u ) = χ ( z ) = −
1, by Lemma 2.12 (ii) we have[ u ] χ δ + χ δ (1 − u )[ z ] χ δ = ( χ δ (1 − u ) + χ δ (1 − u ) χ δ (1 − z )) ( C F ) χ δ = χ δ (1 − u ) (1 + χ δ (1 − z )) ( C F ) χ δ . Again, since ℓ ( z ) is odd, χ − (1 − z ) = − χ + (1 − z ) and hence 1 + χ δ (1 − z ) takes the value0 for one value of δ and 2 for the other. It follows that X is either ± (cid:16) ( C F ) χ + , (cid:17) or ± (cid:16) , ( C F ) χ − (cid:17) and the Lemma follows. Case 2: χ ( z ) = 1. In this case we have χ ( wz ) = − ℓ ( wz ) is odd. We write[ uw ] χ δ − χ ( u )[ w ] χ δ − χ δ (1 − u )[ z ] χ δ = [ u ] χ δ + χ δ (1 − u )[ wz ] χ δ and the argument of Case 1 applies by interchanging z and wz . (cid:3) Combining Lemma 2.17 with Corollary 2.13 (i) and (ii) we immediately deduce:
Corollary 2.18. If u ∈ U ,A and χ ( u ) = − , then u is obtainable. We deal with the last remaining case:
Lemma 2.19. If u ∈ U ,A satisfies χ ( u ) = 1 and ℓ ( u ) is even, then u is obtainable.Proof. Choose any w with ℓ ( w ) = 1. Since ℓ ( u ) ≥ ℓ ( uw ) = ℓ ( w ) = 1. Consider z := (1 − w ) / (1 − uw ). Then z ∈ U ,A and ℓ ( z ) = ℓ ( u ) − uz ∈ U ,A and ℓ ( uz ) = ℓ ( u ) − δ ∈ { + , −} , in L v (cid:2) (cid:3) χ δ we have relations[ u ] χ δ = ± [ w ] χ δ ± [ uw ] χ δ ± [ uz ] χ δ ± [ z ] χ δ and since all the terms on the right are obtainable (since they satisfy ℓ ( x ) is odd), it followsthat u is obtainable also. (cid:3) Thus we have completed the proof of Theorem 2.3.2.7.
Proof of Theorem 2.1.
We will need the following result from K -theory: Lemma 2.20.
Let A be a discrete valuation ring with field of fractions F and residue field k . Then the homomorphism K ind3 ( A ) → K ind3 ( F ) is surjective. Proof.
Consider the following commutative diagram with exact columns: K M ( A ) K M ( F ) K M ( k ) 0 K ( A ) K ( F ) K ( k ) 0 K ind3 ( A ) K ind3 ( F )0 0 ∼ = The rows are exact by the localization exact sequences for K -theory and Milnor K -theory,together with the fact that the homomorphism K M ( F ) → K M ( k ) is surjective. (For aproof, see [22, V.6.6.2].) The indicated right vertical arrow is an isomorphism by Mat-sumoto’s theorem. The lemma follows immediately. (cid:3) Now let ∆ π : H (SL ( F ) , Z ) → g RP ( k )be the composite H (SL ( F ) , Z ) → RP ( F ) δ π −→ g RP ( k ). By [8, Lemma 7.1], the compos-ite H (SL ( A ) , Z (cid:2) (cid:3) ) → H (SL ( F ) , Z (cid:2) (cid:3) ) ∆ π −→ g RP ( k ) (cid:2) (cid:3) is the zero map. Thus we have the commutative diagram I A RP ( A ) h i I F RP ( F ) h i RP ( k ) h i H (SL ( A ) , Z h i ) H (SL ( F ) , Z h i ) RP ( k ) h i K ind3 ( A ) K ind3 ( F )0 0 δ π =∆ π in which the columns are exact (by Proposition 1.5). The top row is exact (by Theorem2.3) and the middle row is a complex. Since the bottom horizontal arrow is surjective byLemma 2.20, a straightforward diagram chase establishes the exactness of the middle row.Thus Theorem 2.1 is proved. HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 21 The Mayer-Vietoris exact homology sequence of SL ( A )As in Section 2, let A be a discrete valuation ring with maximal ideal m A . Let k , F , v = v A be the residue field, field of fractions and discrete valuation of A . Let π be a fixedchoice of uniformizer.In this section we review the relevant facts about the tree associated to the discretevaluation v , the resulting decomposition of SL ( F ) as an amalgamated product and theMayer-Vietoris exact sequence in homology. We show that Mayer-Vietoris is naturally asequence of R F -modules and explicitly describe this action on the relevant terms of thesequence.3.1. Rank two lattices and the associated tree.
Let F be the usual F -vector spaceof dimension 2. Any rank two free A -submodule of F is called a lattice . If x ∈ F × and L is a lattice of F , Lx is also a lattice of F . Thus F × acts on the set of lattices of F .Two lattices L and L ′ are said to be equivalent if there is an element x ∈ F × such that L = L ′ x . In other words, two lattices are equivalent if their orbits under the action of F × are the same. The set of equivalence classes of lattices of F will be denoted by V . Thefollowing review follows the standard account in [19].Let L and L ′ be two lattices of F . By the Invariant Factor Theorem, there is an A -basis { e , f } of L and integers a, b such that { e π a , f π b } is an A -basis for L ′ . The integers a, b doesnot depend on the choice of bases for L, L ′ and | a − b | depends only on the equivalenceclasses Λ , Λ ′ of L and L ′ . Furthermore L ′ ⊆ L if and only if a, b ∈ Z ≥ , in which case L/L ′ ≃ A/π a A ⊕ A/π b A. The distance between two classes Λ , Λ ′ ∈ V is defined as d (Λ , Λ ′ ) := | a − b | . If L is a lattice, each class Λ ′ ∈ V has exactly one representative L ′ , satisfying L ′ ⊆ L and L ′ * Lπ (or equivalently L ′ ⊆ L and L ′ is maximal in Λ ′ with this property). In this casewe have L/L ′ ≃ A/π n A , where n = d (Λ , Λ ′ ). In particular(i) d (Λ , Λ ′ ) = 0 if and only if Λ = Λ ′ ,(ii) d (Λ , Λ ′ ) = 1 if and only if there are representatives L ′ ⊆ L of Λ ′ and Λ such that L/L ′ ≃ k .Two elements Λ , Λ ′ of V are siad to be adjacent if d (Λ , Λ ′ ) = 1. In this way we candefine a combinatorial graph structure on V . We denote this graph, whose vertices are theelements of V , by T V . The structure of this graph is known: Theorem 3.1. ([19, pp. 70-72])
The graph T V is a tree. Moreover the edges with origin Λ correspond bijectively to the points of P ( k ) . Amalgamated product decomposition.
The action of G := SL ( F ) on V has twoorbits say with representatives Λ and Λ which are the equivalence classes of L := A ⊕ A and L := A ⊕ πA , respectively. The stabilizer of the vertices Λ , Λ and the edge (Λ , Λ ) are G Λ = G L = { g ∈ G : g Λ = Λ } = SL ( A ) ,G Λ = G L = { g ∈ G : g Λ = Λ } = SL ( A ) g π =: g π SL ( A ) g − π ,G (Λ , Λ ) = G ( L ,L ) = { g ∈ G : g (Λ , Λ ) = (Λ , Λ ) } = G Λ ∩ G Λ =: Γ ( m A ) , where g π = (cid:18) − π (cid:19) [19, Section 1.3, Chap. II]. Observe that g π SL ( A ) g − π = n(cid:18) a bππ − c d (cid:19) | (cid:18) a bc d (cid:19) ∈ SL ( A ) o , Γ ( m A ) = n(cid:18) a bc d (cid:19) ∈ SL ( A ) | c ∈ m A o . Note that for h π := (cid:18) π
00 1 (cid:19) , we have G Λ = h − π SL ( A ) h π = SL ( A ) h − π . Serre’s theory ofTrees [19, Chap. II] allows us to deduce: Theorem 3.2. (Ihara) The group SL ( F ) is the sum of the subgroups SL ( A ) and SL ( A ) g π amalgamated along their common intersection Γ ( m A ) : SL ( F ) = SL ( A ) ∗ Γ ( m A ) SL ( A ) g π . The action of GL ( F ) on the singular complex of the tree. Let G denoteSL ( F ), as above, and let e G = GL ( F ). The group e G acts transitively on the set ofvertices, V , of the tree T V . The matrix g π = (cid:18) − π (cid:19) ∈ e G transforms L into L and L into L π . Therefore it transforms Λ to Λ , Λ to Λ and the edge (Λ , Λ ) into itsopposite (Λ , Λ ). The collection E + := { ( g Λ , g Λ ) | g ∈ G } gives a set of oriented edges of T V . Since T V is contractible, its singular complex0 → Z [ E + ] → Z [ V ] → Z → Z [ G ]-modules. We let e G act on Z [ E + ] as follows: g. (Λ , Λ ′ ) := ( +( g Λ , g Λ ′ ) if ( g Λ , g Λ ′ ) ∈ E + − ( g Λ , g Λ ′ ) if ( g Λ ′ , g Λ) ∈ E + . For g ∈ e G , let ǫ ( g ) ∈ { , } ⊂ Z be defined by v ◦ det( g ) ≡ ǫ ( g ) (mod 2) . In fact we have
Lemma 3.3. If Λ ∈ V and g ∈ e G = GL ( F ) , then v ◦ det( g ) ≡ d (Λ , g Λ) (mod 2) .Proof.
See [19, Corollary, p. 75]. (cid:3)
HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 23
Thus if e ∈ E + , g ∈ e G we have g.e = ( + ge if ǫ ( g ) = 0 − ge if ǫ ( g ) = 1 . With this action, the abovesequence is a sequence of Z [ e G ]-modules. Thus the corresponding long exact homologysequence · · · → H i ( G, Z [ E + ]) → H i ( G, Z [ V ]) → H i ( G, Z ) δ −→ H i − ( G, Z [ E + ]) → · · · is a sequence of e G -modules, letting e G act on G by conjugation. Since the restriction of thisaction to G is trivial, it follows that the homology sequence is a sequence of F × -modules,because e G/G det ≃ F × . Since furthermore diagonal matrices act trivially, ( F × ) acts triviallyon the sequence. Therefore the above long exact sequence is a sequence of R F -modules.3.4. The Mayer-Vietoris exact sequence.
Let G = SL ( A ), G := SL ( A ) g π andΓ = Γ ( m A ). The group G i is the stabilizer in G of the Λ i , for i = 0 , G/G ⊔ G/G ←→ V, gG i ↔ g Λ i , i = 0 , G/ Γ ←→ E + , g Γ ↔ ( g Λ , g Λ ) . With these identifications, the singular complex of T V has the form0 → Z [ G/ Γ] → Z [ G/G ] ⊕ Z [ G/G ] → Z → . The resulting long exact sequence has the form · · · → H i ( G, Z [ G/ Γ]) → H i ( G, Z [ G/G ]) ⊕ H i ( G, Z [ G/G ]) → H i ( G, Z ) δ −→ H i − ( G, Z [ G/ Γ]) → · · · . Hence by the Shapiro Lemma, we have the Mayer-Vietoris homology exact sequence · · · −→ H i (Γ , Z ) ( i ∗ , − i ′∗ ) −−−−−→ H i ( G , Z ) ⊕ H i ( G , Z ) j ∗ + j ′∗ −−−−→ H i ( G, Z ) δ −→ H i − (Γ , Z ) −→ · · · which is an exact sequence of R F -modules. Here i : Γ → G , i ′ : Γ → G , j : G → G and j ′ : G → G are the usual inclusion maps. Thus we have Theorem 3.4.
For any discrete valuation ring A we have the Mayer-Vietoris exact se-quence of R F -modules · · · → H i (SL ( A ) , Z ) ⊕ H i (SL ( A ) , Z ) β −→ H i (SL ( F ) , Z ) δ −→ H i (Γ ( m A ) , Z ) α −→ H i − (SL ( A ) , Z ) ⊕ H i − (SL ( A ) , Z ) β −→ H i − (SL ( F ) , Z ) −→ · · · , where β ( z, z ′ ) = j ∗ ( z ) + h π i j ∗ ( z ′ ) and α ( x ) = ( i ∗ ( x ) , − i ∗ ( h π i • x )) . Here i : Γ ( m A ) → SL ( A ) and j : SL ( A ) → SL ( F ) are the usual inclusions and h π i • x is the action of h π i on H n (Γ , Z ) induced by conjugation. Proof.
Based on what we have discussed in the above, to finish the proof we only need todescribe the maps α and β . If C π : G → G , is the conjugation map C π ( g ) = g π gg − π , thenfrom the commutative diagram H i (Γ , Z ) H i ( G , Z ) H i ( G, Z ) H i (Γ , Z ) H i ( G , Z ) H i ( G, Z ) i ∗ ( C π | Γ ) ∗ j ∗ ( C π | G ) ∗ C π ∗ i ′∗ j ∗ ′ we see that h π i j ∗ ( z ) = j ′∗ ( C π ∗ ( z )) and i ′ ( h π i • x ) = C π ∗ ◦ i ∗ ( x ). If we replace x with h π i • x ,then we have i ′∗ ( x ) = C π ∗ ◦ i ∗ ( h π i • x ). This completes the proof. (cid:3) Explicit action of GL ( F ) on the singular complex of the tree. Now we makeexplicit the action of e G on (basis elements of) Z [ G/G ] ⊕ Z [ G/G ] and Z [ G/ Γ]. For any g ∈ e G , let det( g ) = π s ( g )+ ǫ ( g ) u g , where s ( g ) ∈ Z , ǫ ( g ) ∈ { , } and u g ∈ A × . Then we can write g = (cid:18) π π (cid:19) s ( g ) R ( g ) (cid:18) u g
00 1 (cid:19) g ǫ ( g ) π , where R ( g ) ∈ G . As we have seen before g π . Λ = Λ and g π . Λ = Λ . So using theidentification of the sets G/G ⊔ G/G ←→ V , we have g π G = G and g π G = G . So gG i = R ( g ) G i + ǫ ( g ) . More generally g. ( xG i ) = R ( gx ) .G i + ǫ ( gx ) . In particular for ˜ u := (cid:18) u
00 1 (cid:19) , where u ∈ A × , we have ˜ uG i = G i , i = 0 ,
1, since R (˜ u ) = I and ǫ (˜ u ) = 0. Moreover, g π (Λ , Λ ) = (Λ , Λ ) and thus g π . Γ = − Γ. Using this, we have g. Γ = ( − ǫ ( g ) R ( g ) . Γ and so, more generally, g. ( x Γ) = ( − ǫ ( gx ) R ( gx ) . Γ . (3.1)So with these explicitly defined actions, the exact sequence0 → Z [ G/ Γ] → Z [ G/G ] ⊕ Z [ G/G ] → Z → , with the maps x Γ ( xG , − xG ) and ( nxG , myG ) n + m , is a sequence of e G -modules. Hence the resulting Mayer-Vietoris exact sequence, obtained by applying thefunctor H • ( G, − ) is a sequence of R F -modules.3.6. The action of R F on H • ( G , Z ) ⊕ H • ( G , Z ) . Now we study the action of G F onthe terms of the Mayer-Vietoris exact sequence. Proposition 3.5.
Let M = Z [ G/G ] ⊕ Z [ G/G ] and let C ′ π : G → G be given by g g − π gg π . Then with the isomorphism C ′ π ∗ : H • ( G , Z ) → H • ( G , Z ) , the action of h π i ∈ R F on H • ( G, M ) ≃ H • ( G , Z ) ⊕ H • ( G , Z ) is to interchange the two factors. HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 25
Proof.
First let study the action of G F on H • ( G, M ) on the level of chain complex. Let P • → Z be a right resolution of Z over e G . Let a ∈ F × and choose g ∈ e G with det( g ) = a .Then the multiplication by h a i is given at the level of chains by the morphism P • ⊗ G M → P • ⊗ G M, x ⊗ m xg − ⊗ gm. It follows that u ∈ A × acts in the standard way, i.e. via conjugation by ˜ u , on each of thefactors H • ( G , Z ) and H • ( G , Z ). Furthermore the isomorphism H • ( G , Z ) ≃ −→ H • ( G , Z )induced by C ′ π • : P • ⊗ G Z → P • ⊗ G Z , x ⊗ xg π ⊗ A × -module isomorphism,since det( g π ˜ ug − π ) = det(˜ u ) = u .Now since det( g π ) = π , we look at the action of g π on P • ⊗ G M . Let i and i be theinclusions Z [ G/G ] → M and Z [ G/G ] → M . The diagrams P • ⊗ G M P • ⊗ G MP • ⊗ G Z [ G/G ] P • ⊗ G Z [ G/G ] x ⊗ G xg − π ⊗ G P • ⊗ G Z xg − π ⊗ P • ⊗ G Z P • ⊗ G Z x ⊗ x ⊗ C ′ π • i • i • ≃ ≃ = and P • ⊗ G M P • ⊗ G MP • ⊗ G Z [ G/G ] P • ⊗ G Z [ G/G ] xg − π ⊗ G x ⊗ G P • ⊗ G Z xg − π ⊗ P • ⊗ G Z P • ⊗ G Z x ⊗ x ⊗ C ′ π • i • i • ≃ ≃ = commute. This implies the claim. (cid:3) Corollary 3.6.
For any n , there is an isomorphism of R F -modules H n ( G, Z [ G/G ] ⊕ Z [ G/G ]) ≃ Ind FA H n ( G , Z ) = R F ⊗ R A H n ( G , Z ) . In particular the Mayer-Vietoris exact sequence has the following form as an sequence of R F -modules · · · → Ind FA H i (SL ( A ) , Z ) β → H i (SL ( F ) , Z ) δ → H i (Γ ( m A ) , Z ) α → Ind FA H i − (SL ( A ) , Z ) β → · · · The two actions of R F on H n (Γ , Z ) . We begin by noting that there is a naturalaction of R F on H • (Γ , Z ) compatible with the inclusion Γ ֒ → G : Let e Γ := F × Γ e U h g π i , where e U := n(cid:18) u
00 1 (cid:19) | u ∈ A × o . Then e Γ is a subgroup of e G (observe that g π = − πI ). Notethat if g ∈ e Γ, then g = (cid:18) π π (cid:19) s ( g ) R ( g ) (cid:18) u − g
00 1 (cid:19) g ǫ ( g ) π , where now R ( g ) ∈ Γ. There is a map of group extensions1 Γ e Γ F × G e G F × . det =det Thus Z [ F × ] acts on H • (Γ , Z ) in such a way that the map H • (Γ , Z ) → H • ( G, Z ) is a mapof Z [ F × ]-modules. However, clearly ( F × ) acts trivially also on the first term, so that thisis a map of R F -modules.We denote this action of R F on H n (Γ , Z ) by h a i • x (where h a i ∈ G F and x ∈ H n (Γ , Z )).We will refer to it as the natural action of R F on H • (Γ , Z ).However, this action of R F on H n (Γ , Z ) is not the same as the R F -module action associ-ated to the Mayer-Vietoris sequence (as in Theorem 3.4). We now describe the relationshipbetween these two actions, the natural action and the Mayer-Vietoris action:Given an R F -module M and a character χ ∈ b G F , we can define the χ -twisted R F -module M ( χ ) by h a i ∗ χ m := χ ( a ) h a i m. Recall that the discrete valuation v induces a character χ v ∈ b G F defined by χ v ( a ) :=( − v ( a ) . Theorem 3.7.
The action of R F on H • (Γ , Z ) which is induced from the Mayer-Vietorisexact sequence is the χ v -twist of the natural action, i.e. h a i x = ( − v ( a ) h a i • x for any h a i ∈ G F and x ∈ H n (Γ , Z ) .Proof. We begin by recalling the R F -action associated to the Mayer-Vietoris sequence on H n (Γ , Z ): Let C • → Z be a right projective resolution of Z over Z [ ˜ G ]. We can use the HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 27 complex C • ⊗ G Z [ G/ Γ] to calculate H • ( G, Z [ G/ Γ]) ∼ = H • (Γ , Z ). Let g ∈ ˜ G . Then the actionof the class of det( g ) on H • (Γ , Z ) is induced from the map C • ⊗ G Z [ G/ Γ] → C • ⊗ G Z [ G/ Γ] ,z ⊗ x Γ zg − ⊗ g · ( x Γ) = ( − ǫ ( gx ) zg − ⊗ R ( gx )Γusing (3.1) above.Now let h a i ∈ G F be represented by uπ ǫ ∈ F × , where u ∈ A × and ǫ ∈ { , } . Let g = g ( a ) := ˜ ug ǫπ ∈ e Γ. Thus R ( g ) = 1 and g Γ = ( − ǫ Γ and hence the Mayer-Vietorisaction of h a i on H • (Γ , Z ) is induced by the map z ⊗ Γ ( − ǫ zg − ⊗ Γ.The map C • ⊗ Γ Z → C • ⊗ G Z [ G/ Γ] , z ⊗ z ⊗ Γinduces an isomorphism on homology by Shapiro’s lemma. We conclude the proof of thetheorem by noting that for any g ∈ ˜Γ, the map induced on H • (Γ , Z ) by conjugation by g is described at the level of the complex C • ⊗ Γ Z by z ⊗ zg − ⊗ (cid:3) Example 3.8.
Here we show that the action of R F on H (Γ , Z ) is trivial, provided that k has at least 4 elements.The natural map θ : Γ → T k given by (cid:18) a bc d (cid:19) (cid:18) ¯ a
00 ¯ a − (cid:19) induces the isomorphism H (Γ , Z ) ≃ H ( T k , Z ) ≃ k × , (see the proof Theorem 4.11 below). Since H (Γ , Z ) = Γ / Γ ′ , it follows that if the diagonalof two matrices in Γ are the same, then they represent the same element of H (Γ , Z ).To study the action of R F , it is enough to study the action of the elements h u i , u ∈ A × and h π i . If x ∈ H (Γ , Z ) = Γ / Γ ′ is represented by (cid:18) a bc d (cid:19) , then by Theorem 3.7 we have h u i .x = (cid:16)(cid:18) u
00 1 (cid:19)(cid:18) a bc d (cid:19)(cid:18) u −
00 1 (cid:19)(cid:17) ( − v ( u ) Γ ′ = (cid:18) a ubu − c d (cid:19) Γ ′ = (cid:18) a bc d (cid:19) Γ ′ = x, and h π i .x = (cid:16) g π (cid:18) a bc d (cid:19) g − π (cid:17) ( − v ( π ) Γ ′ = (cid:18) d − cπ − − πb a (cid:19) − Γ ′ = (cid:18) d − b − c a (cid:19) − Γ ′ = x. Thus the action of R F on H (Γ , Z ) is trivial.4. The connecting homomorphism and the structure of H (Γ ( m A ) , Z (cid:2) (cid:3) )In this section we will give an explicit formula for the connecting homomorphism δ : H (SL ( F ) , Z (cid:2) (cid:3) ) = H ( G, Z ) → H (Γ , Z ) = H (Γ ( m A ) , Z (cid:2) (cid:3) )and a description of its image. Moreover we describe the kernel and the cokernel of thenatural map H (Γ ( m A ) , Z (cid:2) (cid:3) ) → H (SL ( A ) , Z (cid:2) (cid:3) ) . The image of δ . We will begin by using Theorem 2.1 to identify coker( β ) ∼ = im( δ ) asan R F -module. In order to do this, we will need to compare the homomorphism ∆ π , whichis a R A -module map, with an R F -homomorphism denoted ∆ ′ π which we now describe:Let M be a R k -module. We equip M with two R F -module structure h a i m := h u a i m, h a i m := ( − v ( a ) h u a i m, where u a ∈ A × is the unique element satisfying a = u a π v ( a ) . The R k -module M equippedwith these R F -structures are denoted by M ( v ) and M { v } , respectively. Example 4.1.
Consider P ( k ) with the trivial G k -action. Then P ( k ) { v } is a (nontrivial) R F -module with the definition given by h a i [ x ] := ( − v ( a ) [ x ] . But P ( k )( v ) becomes trivial as R F -module.It is easy to see that the natural maps ρ ′ : Ind Fk M → M ( v ) , h a i ⊗ m
7→ h u a i m and ρ ′ π : Ind Fk M → M { v } h a i ⊗ m ( − v ( a ) h u a i m are R F -homomorphisms (both depending on the choice of the uniformizer π ). Lemma 4.2. ([9, Lemma 6.5])
Let M be an R k (cid:2) (cid:3) -module. Then there are natural de-compositions of R F (cid:2) (cid:3) -modules Ind Fk M ≃ M ( v ) ⊕ M { v } and I F Ind Fk M ≃ ( I k M )( v ) ⊕ M { v } . In other words, I F Ind Fk M ≃ I k M ⊕ M made into an R F -module by letting h π i acts as theidentity on the first factor and as multiplication by − on the second factor. Now let δ ′ : RP ( F ) → g RP ( k )( v ) , and δ ′ π : RP ( F ) → g RP ( k ) { v } be the composite maps RP ( F ) S v −→ Ind Fk g RP ( k ) ρ ′ −→ g RP ( k )( v ) , and RP ( F ) S v −→ Ind Fk g RP ( k ) ρ ′ π −→ g RP ( k ) { v } , respectively. These maps define well-defined R F -homomorphisms δ ′ : RP ( F ) → g RP ( k )( v ) , and δ ′ π : RP ( F ) → g RP ( k ) { v } . The restriction of these maps to I F RP ( F ) induce homomorphisms of R F -modules δ ′ : I F RP ( F ) → I k g RP ( k )( v ) , δ ′ π : I F RP ( F ) → g RP ( k ) { v } . HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 29
Proposition 4.3.
Let η π and η ′ π be the composite maps I F RP ( F ) δ π −→ g RP ( k ) → e P ( k ) , I F RP ( F ) δ ′ π −→ g RP ( k ) { v } → e P ( k ) { v } , respectively. Then η ′ π = − η π .Proof. Let ˜ θ be the natural map g RP ( k ) → e P ( k ). The maps η π and η ′ π are the compositemaps I F RP ( F ) S v −→ I F Ind Fk g RP ( k ) ρ π −→ g RP ( k ) ˜ θ −→ e P ( k ) , and I F RP ( F ) S v −→ I F Ind Fk g RP ( k ) ρ ′ π −→ g RP ( k ) { v } ˜ θ −→ e P ( k ) { v } , respectively. Let hh a ii [ x ] ∈ I F RP ( F ) and let a = u a π v ( a ) , u a ∈ A × . Then S v ( hh a ii [ x ]) = hh a ii ⊗ [ x ]. If v ( a ) is even, then ρ π ( hh a ii ⊗ [ x ]) = 0and ρ ′ π ( hh a ii ⊗ [ x ]) = ρ ′ π ( h a i ⊗ [ x ]) − ρ ′ π (1 ⊗ [ x ]) = ( − v ( a ) h u a i [ x ] − [ x ] = h u a i [ x ] − [ x ] . Since the action of G k on P ( k ) is trivial, we have ˜ θ ( h u a i [ x ] − [ x ]) = 0. If v ( a ) is odd, then ρ π ( hh a ii ⊗ [ x ]) = ρ π ( h a i ⊗ [ x ]) − ρ π (1 ⊗ [ x ]) = h u a i [ x ] − h u a i [ x ] , and ρ ′ π ( hh a ii ⊗ [ x ]) = ρ ′ π ( h a i ⊗ [ x ]) − ρ ′ π (1 ⊗ [ x ]) = ( − v ( a ) h u a i [ x ] − [ x ] = −h u a i [ x ] − [ x ] . Clearly ˜ θ ( h u a i [ x ]) = [ x ] and ˜ θ ( −h u a i [ x ] − [ x ]) = − x ]. This proves the claim. (cid:3) Corollary 4.4.
The composite I A RP ( A ) → I F RP ( F ) δ ′ π −→ g RP ( k ) { v } → e P ( k ) { v } is the zero map.Proof. This follows immediately from Proposition 4.3 and the fact that the composite I A RP ( A ) → I F RP ( F ) δ π −→ g RP ( k )is the zero map. (cid:3) Corollary 4.5.
Let k be a sufficiently large finite field. Then the sequence I A RP ( A ) (cid:2) (cid:3) → I F RP ( F ) (cid:2) (cid:3) δ ′ π −→ P ( k ) { v } (cid:2) (cid:3) → is exact.Proof. This follows from Proposition 4.3, Theorem 2.3 and Proposition 1.3. (cid:3)
We can now define ∆ ′ π : The R F -homomorphism∆ ′ π : H (SL ( F ) , Z ) → g RP ( k ) { v } is the composite H (SL ( F ) , Z ) → RP ( F ) δ ′ π −→ g RP ( k ) { v } . Proposition 4.6. If k is sufficiently large, then the diagram of R A -modules H (SL ( A ) , Z (cid:2) (cid:3) ) H (SL ( F ) , Z (cid:2) (cid:3) ) j ∗ ( H (SL ( A ) , Z (cid:2) (cid:3) )) H (SL ( F ) , Z (cid:2) (cid:3) ) RP ( k ) (cid:2) (cid:3) h π i j ∗ j ∗ p ∗ ∆ π ∆ ′ π commutes, where j : SL ( A ) → SL ( F ) is the natural inclusion and p : SL ( A ) → SL ( k ) is induced by the quotient map A → k .Proof. For any local domain R with sufficiently large residue field, we have the exactsequence H ( T R , Z (cid:2) (cid:3) ) → H (SL ( R ) , Z (cid:2) (cid:3) ) → RB ( R ) (cid:2) (cid:3) → , where the left side map is induced by the inclusion T R := n(cid:18) a a − (cid:19) | a ∈ R × o ֒ → SL ( R )(see the proof of the refined Bloch-Wigner exact sequence in [8, Theorem 3.22]). Thus itis enough to prove the commutativity of the diagram RB ( A ) (cid:2) (cid:3) RB ( F ) (cid:2) (cid:3) j ∗ ( RB ( A ) (cid:2) (cid:3) ) RB ( F ) (cid:2) (cid:3) RP ( k ) (cid:2) (cid:3) , h π i j ∗ j ∗ p ∗ δ π ∆ ′ π where j ∗ : RB ( A ) → RB ( F ) is induced by the natural inclusion map A ֒ → F and p ∗ : RB ( A ) → RP ( k ) is induced by the quotient map A → k .If x = P h u i [ a ] ∈ RB ( A ) (cid:2) (cid:3) , then ∆ ′ π ◦ j ∗ ( x ) = P ( − v ( u ) h ¯ u i [¯ a ] = P h ¯ u i [¯ a ] = x = p ∗ ( x ).Thus lower triangle of the diagram commutes. If ˜ β := h π i j ∗ : RB ( A ) → RB ( F ), then δ π ◦ ˜ β ( x ) = ρ π ◦ S v ( h π i j ∗ ( x )) = ρ π ◦ S v ( P h πu i [ a ]) = ρ π ( P h πu i ⊗ [ a ]) = P h u i [ a ] = p ∗ ( x ).This proves the commutativity of the upper triangle of the diagram. (cid:3) We are now in a position to describe the R F -module coker( β ) ∼ = im( δ ):Let P ( k ) denote the cokernel of the composite homomorphism K ind3 ( A ) → K ind3 ( k ) →P ( k ). We make P ( k ) into an R F -module by taking the χ v -twist of the trivial modulestructure. Theorem 4.7.
There is a natural isomorphism of R F -modules im( δ ) ∼ = P ( k ) . Proof.
Since ∆ π is an isomorphism by Theorem 2.1, it follows from the definition of β andProposition 4.6 thatim( δ ) ∼ = coker( β ) ∼ = coker (cid:0) p ∗ : H (SL ( A ) , Z (cid:2) (cid:3) ) → RP ( k ) (cid:2) (cid:3)(cid:1) HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 31 as an R F -module. Now note that the homomorphism I A H (SL ( A ) , Z (cid:2) (cid:3) ) ∼ = I A RP ( A ) (cid:2) (cid:3) → I k RP ( k ) (cid:2) (cid:3) is surjective, since the generators hh ¯ u ii g ( x ) of I k RP ( k ) (cid:2) (cid:3) can be lifted by Proposition1.9. The statement of the theorem thus follows from the commutative diagram with exactrows:0 I A H (SL ( A ) , Z (cid:2) (cid:3) ) H (SL ( A ) , Z (cid:2) (cid:3) ) K ind3 ( A ) (cid:2) (cid:3) I k RP ( k ) (cid:2) (cid:3) RP ( k ) (cid:2) (cid:3) P ( k ) (cid:2) (cid:3) . p ∗ (cid:3) We will require the following corollary in our calculation of δ below: Corollary 4.8.
If we endow H (Γ ( m A ) , Z ) with the natural action of G F then im( δ ) ⊂ H (Γ ( m A ) , Z (cid:2) (cid:3) ) G F . In particular, the map C π induced from conjugating by g π ∈ ˜Γ is the identity map on im( δ ) .Proof. On the one hand, by Theorem 4.7, im( δ ) is isomorphic as an R F -module to P ( k )with the χ v -twist of the trivial R F -structure. On the other hand, by Theorem 3.7, im( δ ) isa submodule of H (Γ ( m A ) , Z (cid:2) (cid:3) ) with the Mayer-Vietoris R F -structure, which is the χ v -twist of the natural structure. It follows that im( δ ) is trivial in the natural R F -structure. (cid:3) The spectral sequence E • , • ( A, R ) . Next we turn to the explicit calculation of δ ,which by the results above, is essentially a homomorphism P ( k ) → H (Γ ( m A ) , Z ). In factwe will show that it can be identified with a d -differential in a certain spectral sequence.We begin by describing this spectral sequence.First, let us describe the general context. Let G be a group and let L • be a complex ofleft G -modules: L • : · · · −→ L ∂ −→ L ∂ −→ L −→ . The n -th hyperhomology group of G with coefficients in L • , denoted by H n ( G, L • ), isdefined as the n -th homology of the total complex of the double complex F • ⊗ G L • , where F • → Z is a right projective resolution of Z over the group ring Z [ G ]. This double complexinduces two spectral sequences both converging to the hyperhomology groups H • ( G, L • ),as follow: E p,q = H p ( G, H q ( L • )) ⇒ H p + q ( G, L • )and E p,q = H q ( G, L p ) ⇒ H p + q ( G, L • )(see [2, §
5, Chap. VII]). By easy analysis of the spectral sequence E p,q ( G ) we get: Lemma 4.9.
Let L • be exact for ≤ i ≤ n . If M := H ( L • ) , then H i ( G, L • ) ≃ H i ( G, M ) for ≤ i ≤ n . Let R be a ring. A (column) vector u = (cid:18) u u (cid:19) ∈ R is said to be unimodular if u R + u R = R . Equivalently, u = (cid:18) u u (cid:19) is said to be unimodular if there exists a vector v = (cid:18) v v (cid:19) such that the matrix ( u , v ) := (cid:18) u v u v (cid:19) is an invertible matrix. Note that thematrix ( u , v ) is invertible if and only if u , v are a basis of R .For any non-negative integer n , let L n ( R ) be the free abelian group generated by theset of all ( n + 1)-tuples ( R v , . . . , R v n ), where every v i ∈ R is unimodular and any twodistinct vectors v i , v j are a basis of R .We consider L n ( R ) as a left GL ( R )-module (and so SL ( R )-module) in a natural way.If necessary, we convert these actions to a right actions by the definition m.g := g − m .Let us define the n -th differential operator ∂ n : L n ( R ) → L n − ( R ), n ≥
1, as analternating sum of face operators which throws away the i -th component of generators.Hence we have the complex L • ( R ) : · · · −→ L ( R ) ∂ −→ L ( R ) ∂ −→ L ( R ) −→ . Let ∂ = ǫ : L ( R ) → Z be defined by P i n i ( R v i ) P i n i . Lemma 4.10. ([8, Lemma 3.21]) If R is a local ring with residue field k , then the complex L • ( R ) is exact for ≤ i < | k | and H ( L • ( R )) ≃ Z . In particular for ≤ i < | k | , H i (SL ( R ) , L • ( R )) ≃ H i (SL ( R ) , Z ) . Let A → R be a homomorphism of rings. Then L • ( R ) → Z is a complex of (left)GL ( A )-modules (and so SL ( A )-modules) in a natural way, where Z is considered astrivial module over GL ( A ). Thus we have the spectral sequences E p,q ( A, R ) = H q (SL ( A ) , L p ( R )) ⇒ H p + q (SL ( A ) , L • ( R )) , E p,q ( A, R ) = H q (GL ( A ) , L p ( R )) ⇒ H p + q (GL ( A ) , L • ( R )) . When A = R and A → R is the identity map, these spectral sequences have been studiedextensively in [4], [8] and [21], [13], [15].We suppose now that the maps A × → R × and W A → W R are surjective. We set I := ker( A → R ) . (Of course, the case of interest in this article is is the quotient map A → A/ m A , where A is a discrete valuation ring with maximal ideal m A .)To study the spectral sequence E • , • ( A, R ), we must study the action of SL ( A ) on thesets of basis of L i ( R ) for 0 ≤ i ≤
4. Let e := (cid:18) (cid:19) , e := (cid:18) (cid:19) ∈ R . HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 33
It is easy to see that SL ( A ) acts transitively on the sets of generators of L i ( R ) for i = 0 , A × → R × . We choose ( e R ) and ( e R, e R ) asrepresentatives of the orbit of the generators of L ( R ) and L ( R ), respectively. Theorbits of the action of SL ( A ) on L ( R ), L ( R ) and L ( R ) are represented by h a i [ ] := ( e R, e R, ( e + a e ) R ) , h a i ∈ G R , h a i [ x ] := ( e R, e R, ( e + a e ) R, ( e + ax e ) R, h a i ∈ G R , x ∈ W R , and h a i [ x, y ] := ( e R, e R, ( e + a e ) R, ( e + ax e ) R, ( e + ay e ) R ) , h a i ∈ G R , x, y, x/y ∈ W R , respectively. Therefore L ( R ) ≃ Ind SL ( A )Γ ( I ) Z , L ( R ) ≃ Ind SL ( A )Γ ( I ) Z , L ( R ) ≃ M h a i∈G R Ind SL ( A )Γ ( I ) Z h a i [ ] ,L ( R ) ≃ M h a i∈G R M x ∈W R Ind SL ( A )Γ ( I ) Z h a i [ x ] , L ( R ) ≃ M h a i∈G R M x,y,x/y ∈W R Ind SL ( A )Γ ( I ) Z h a i [ x, y ] , where Γ ( I ) := n(cid:18) a bc d (cid:19) ∈ SL ( A ) : c ∈ I o , Γ ( I ) := n(cid:18) a bc d (cid:19) ∈ SL ( A ) : b, c ∈ I o , Γ ( I ) := n(cid:18) a bc d (cid:19) ∈ SL ( A ) : b, c, a − d ∈ I o . Thus by Shapiro’s lemma we have E ,q ( A, R ) ≃ H q (Γ ( I ) , Z ) , E ,q ( A, R ) ≃ H q (Γ ( I ) , Z ) , E ,q ( A, R ) ≃ M h a i∈G R H q (Γ ( I ) , Z ) ,E ,q ( A, R ) ≃ M h a i∈G R M x ∈W R H q (Γ ( I ) , Z ) , E ,q ( A, R ) ≃ M h a i∈G R M x,y,x/y ∈W R H q (Γ ( I ) , Z ) . In particular E , ( A, R ) ≃ Z , E , ( A, R ) ≃ Z , E , ( A, R ) ≃ Z [ G R ].If Z is the free abelian group generated by the symbols [ x ], x ∈ W R , and Z is the freeabelian group generated by the symbols [ x, y ], x, y, x/y ∈ W R , then in case of q = 0, wehave E , ( A, R ) ≃ Z [ G R ] Z , E , ( A, R ) ≃ Z [ G R ] Z . Now we study the differentials of the spectral sequence:It is not difficult to see that d ,q = H q ( σ ) − H q (inc) , where σ : Γ ( I ) → Γ ( I ) is conjugation by w = (cid:18) −
11 0 (cid:19) , i.e. σ ( X ) = wXw − . Furthermore d ,q | h a i⊗ H q (Γ ( I ) , Z ) = H q ( η a ) − H q ( η ′ a ) + H q (inc) , where η a , η ′ a : Γ ( I ) → Γ ( I ) are conjugation by (cid:18) − a − (cid:19) and (cid:18) − a − (cid:19) , respectively. Inparticular d , : Z [ G R ] → Z is the usual augmentation map P n i h a i i 7→ P n i .The action of GL ( A ) on L • ( R ) and the extension1 → SL ( A ) → GL ( A ) det → A × → , induces an action of A × on E • , ( A, R ) = L • ( R ) SL ( A ) . Since A × I = Z (GL ( A )) actstrivially on E p, ( A, R ), ( A × ) = { a | a ∈ A × } also will act trivially on E • , ( A, R ). Thus E p, ( A, R ), p ≥
0, has a natural G A -module structure. Moreover the differentials d p, : E p, ( A, R ) → E p − , ( A, R ) are G A -homomorphisms.By a direct calculation one sees that the G A -homomorphism d , : Z [ G R ] Z → Z [ G R ] isgiven by [ x ]
7→ hh x iihh − x ii , while the G A -homomorphism d , : Z [ G R ] Z → Z [ G R ] Z is given by[ x, y ] Y x,y = [ x ] − [ y ] + h x i h yx i − h x − − i (cid:20) − x − − y − (cid:21) − h − x i (cid:20) − x − y (cid:21) . Finally from the above calculations we have E , ( A, R ) = 0 , E , ( A, R ) = I ( R ) , E , ( A, R ) ≃ RP ( R ) , where I ( R ) := I R / h hh a iihh − a ii : a ∈ W R i is called the fundamental ideal of R . (In fact, when R is a local ring, it is the fundamentalideal in the Grothendieck-Witt ringGW( R ) ∼ = Z [ G R ] / hhh u iihh − u ii | u ∈ W R i . )The commutative diagram of ring homomorphism A id A −−−−→ A id A y y A −−−−→ R y y id R R id R −−−−→ R induces morphisms of spectral sequences E • , • ( A, A ) → E • , • ( A, R ) → E • , • ( R, R ) HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 35 which give us the commutative diagram I ( A ) d , ( A,A ) −−−−−−→ E , ( A, A ) y y I ( R ) d , ( A,R ) −−−−−−→ E , ( A, R ) id I ( R ) y y I ( R ) d , ( R,R ) −−−−−−→ E , ( R, R ) . Note that E ,q ( A, A ) = H q ( B A , Z ) , E ,q ( A, A ) = H q ( T A , Z ) , E ,q ( A, A ) = M h a i∈G A H q ( µ ( A ) I , Z ) , where T A = n(cid:18) a a − (cid:19) | a ∈ A × o , B A := n(cid:18) a b a − (cid:19) | a ∈ A × , b ∈ A o . It is easy to see that E , ( A, A ) ≃ G A ⊕ H ( A × , A ) and E , ( R, R ) ≃ G R ⊕ H ( R × , R ),where the action of A × on A (reps. R × on R ) is given by ( a, b ) a b .By direct calculation one can show that both of the maps d , : I ( A ) → G A ⊕ H ( A × , A )and d , : I ( R ) → G R ⊕ H ( R × , R ) are given by hh a ii 7→ h a i (see [12, Lemma 5]). Thus E , ( A, A ) ≃ I ( A ) and E , ( R, R ) ≃ I ( R ). Now from the surjectivity of I ( A ) → I ( R )and the above diagram we obtain the isomorphism E , ( A, R ) ≃ I ( R ) . The second homology of SL of a local ring.Proposition 4.11. Let A be a local ring whose residue field has at least four elements.Let I be a proper ideal of A and set R := A/I . Then we have a natural exact sequence H (Γ ( I ) , Z ) → H (SL ( A ) , Z ) → I ( R ) → . Proof.
By Lemma 4.10, H i (SL ( A ) , L • ) ≃ H i (SL ( A ) , Z ) for 0 ≤ i ≤
3. Moreover thenatural map A × → R × is surjective.Thus to prove the claim it is enough to prove that E , ( A, R ) = 0. This is the homologyof the complex L h a i∈G R H (Γ ( I ) , Z ) d , ( A,R ) −−−−−−→ H (Γ ( I ) , Z ) d , ( A,R ) −−−−−−→ H (Γ ( I ) , Z ) . Let Γ( I ) := n(cid:18) a bc d (cid:19) ∈ SL ( A ) : a − , d − , b, c ∈ I o . From the commutative diagram ofextensions 1 −→ Γ( I ) −→ Γ ( I ) −→ µ ( R ) I −→ y y y −→ Γ( I ) −→ Γ ( I ) −→ T R −→ y y y −→ Γ( I ) −→ Γ ( I ) −→ B R −→ E • , • ( A, R ) → E • , • ( R, R ) we obtain the commuta-tive diagram L h a i∈G R H (Γ( I ) , Z ) H (Γ( I ) , Z ) T R H (Γ( I ) , Z ) B R L h a i∈G R H (Γ ( I ) , Z ) H (Γ ( I ) , Z ) H (Γ ( I ) , Z ) L h a i∈G R H ( µ ( R ) , Z ) H ( T R , Z ) H ( B R , Z ) , d , ( A,R ) d , ( A,R ) d , ( R,R ) d , ( R,R ) where the maps on top row is induced by the maps on the middle row. Now the trivialityof E , ( A, R ) follows from the following facts:(i) The natural map Γ ( I ) → T R induces the isomorphism H (Γ ( I ) , Z ) ≃ H ( T R , Z ) ≃ R × . (ii) The elements of H (Γ( I ) , Z ) T R = Γ( I ) / [Γ( I ) , Γ ( I )] are represented by matrices (cid:18) a a − (cid:19) , a − ∈ I .In fact (i) implies that H (Γ ( I ) , Z ) = Γ ( I ) / [Γ ( I ) , Γ ( I )] injects into H ( B R , Z ). Nowby applying the Snake lemma to the diagram H (Γ( I ) , Z ) T R H (Γ ( I ) , Z ) H ( T R , Z ) 00 0 H (Γ ( I ) , Z ) H ( B R , Z ) d , ( A,R ) d , ( R,R ) we obtain the exact sequence H (Γ , Z ) T R → ker( d , ( A, R )) → H ( µ ( R ) , Z ) → . The map d , ( A, R ) | H (Γ( I ) , Z ) H (Γ( I ) , Z ) → H (Γ( I ) , Z ) T R takes the element representedby (cid:18) a a − (cid:19) to the element represented by the same matrix, thus by (ii) this map is HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 37 surjective. Therefore the claim follows from the commutativity of the above diagram. Nowwe prove (i) and (ii).
Proof of (i) : Let Λ ( I ) be the kernel of the natural mapΓ ( I ) → T R , (cid:18) a bc d (cid:19) (cid:18) ¯ a
00 ¯ a − (cid:19) . Since T R is abelian, [Γ ( I ) , Γ ( I )] ⊆ Λ ( I ). Let (cid:18) a bc d (cid:19) ∈ Λ ( I ). Since A is local, there is z ∈ ( a − A ⊂ I such that x := a + bz ∈ A × . Now if y := c + dz ∈ I , then (cid:18) a bc d (cid:19) = (cid:18) x x − (cid:19)(cid:18) xy (cid:19)(cid:18) x − b (cid:19)(cid:18) − z (cid:19) . Since x + I = a + I = 1 + I , x − ∈ I . Let x = 1 + t for some t ∈ I . Then (cid:18) x x − (cid:19) = (cid:18) − x − t (cid:19)(cid:18) (cid:19)(cid:18) t (cid:19)(cid:18) − x − (cid:19) . Let λ ∈ A × such that λ − ∈ A × . This is possible since the residue field of A has at leastfour elements. Now the above formulas together with the commutator formulas (cid:18) s (cid:19) = h(cid:18) λ λ − (cid:19) , (cid:18) s/ ( λ − (cid:19)i , (cid:18) s (cid:19) = h(cid:18) λ λ − (cid:19) , (cid:18) s/ ( λ −
1) 1 (cid:19)i ,s ∈ A , imply that Λ ( I ) ⊆ [Γ ( I ) , Γ ( I )]. Thus H (Γ ( I ) , Z ) ≃ T R ≃ H ( T R , Z ) . Proof of (ii) : First note that H (Γ( I ) , Z ) T R = H (Γ( I ) , Z ) Γ ( I ) ≃ Γ( I ) / [Γ( I ) , Γ ( I )] . If (cid:18) a bc d (cid:19) ∈ Γ( I ), then as in above we find z ∈ I , such that (cid:18) a bc d (cid:19) = (cid:18) x x − (cid:19)(cid:18) xy (cid:19)(cid:18) x − b (cid:19)(cid:18) − z (cid:19) , where x := a + bz ∈ A × and y := c + dz ∈ I . Now the claim (ii) follows from the abovecommutator formulas. (cid:3) The group E , ( A, R ) and the differential d , ( A, R ) . The morphism of spectralsequences E • , • ( A, A ) → E • , • ( A, R ) induces the commutative diagram RP ( A ) d , ( A,A ) −−−−−−→ E , ( A, A ) y y RP ( R ) d , ( A,R ) −−−−−−→ E , ( A, R ) . Since W A → W R is surjective, RP ( A ) → RP ( R ) is surjective. It is not difficult to showthat E , ( A, A ) = 0. Thus d , ( A, A ) is trivial. This implies that d , ( A, R ) : RP ( R ) → E , ( A, R )is trivial too. Therefore E , ( A, R ) ≃ RP ( R ) . Now we would like to calculate the differential d , ( A, R ) : RP ( R ) → E , ( A, R ). (Notethat E , ( A, R ) is a quotient of H (Γ ( I ) , Z ).) For this we need to put an extra conditionon A . Proposition 4.12. ([8, Proposition 3.19])
Let A be a local domain with residue field k . If k is finite we assume that it has p d elements such that ( p − d > . Then the natural map H (inc) : H ( T A , Z ) → H ( B A , Z ) is an isomorphism. So let the natural map H (inc) : H ( T A , Z ) → H ( B A , Z ) be an isomorphism. Byanalyzing the morphism of spectral sequences E • , • ( A, A ) → E • , • ( A, A ) , one can show (see the proof of [10, Proposition 6.1]) that the diagram RP ( A ) A × ∧ A × A × ∧ µ ( A ) P ( A ) S Z ( A ) d , ( A,A ) λ is commutative, where the vertical map on the right side is injective and is given by a ∧ b a ⊗ b ) . Under the map RP ( A ) → P ( A ), g ( a ) = p + − [ a ] + hh − x ii ψ ( a ) maps to 2[ a ]. This showsthat in the above diagram d , ( A, A ) maps g ( a ) to (1 − a ) ∧ a . (Note that when A is alocal ring where its residue field has more than 10 elements, then the set { g ( a ) : a ∈ W A } generates RP ( A )[ ].) Now by considering the commutative diagram RP ( A ) A × ∧ A × A × ∧ µ ( A ) RP ( R ) H (Γ ( I ) , Z ) /K d , ( A,A ) d , ( A,R ) we see that under the map d , ( A, R ), g ( a ) c (cid:16)(cid:18) (1 − a ) 00 (1 − a ) − (cid:19) , (cid:18) a a − (cid:19)(cid:17) (mod K ) . HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 39
For a, b ∈ A × , c (cid:16)(cid:18) a a − (cid:19) , (cid:18) b b − (cid:19)(cid:17) is the image of a ∧ b under the composite A × ∧ A × ≃ −→ T A ∧ T A ≃ −→ H ( T A , Z ) H (inc) −→ H (Γ ( I ) , Z ) . We deduce:
Proposition 4.13.
Let A be a local domain with residue field k . If k is finite we assumethat it has p d elements such that ( p − d > . Let I be a proper ideal of A and set R := A/I .Then there is a natural exact sequence H (SL ( A ) , Z ) −→ RP ( R ) α −→ H (Γ ( I ) , Z ) K −→ H (SL ( A ) , Z ) −→ I ( R ) → , where K is a certain subgroup of H (Γ ( I ) , Z ) and α ( g ( a )) = c (cid:16)(cid:18) (1 − a ) 00 (1 − a ) − (cid:19) , (cid:18) a a − (cid:19)(cid:17) (mod K ) . Proof.
By Lemma 4.10, H i (SL ( A ) , L • ) ≃ H i (SL ( A ) , Z ), for 0 ≤ i ≤
3. Moreover byProposition 4.12, H ( T A , Z ) ≃ H ( B A , Z ). Since the maps A × → R × and W A → W R aresurjective, the claim follows of an easy analysis of the spectral sequence E • , • ( A, R ). (cid:3) Comparing two exact sequences.
Again, let A be a discrete valuation ring withmaximal ideal m A . Let k , F , v = v A and π be as in Section 2.By Theorem 3.4 we have the Mayer-Vietoris exact sequence of G F -modules H (SL ( A ) , Z ) ⊕ H (SL ( A ) , Z ) j ∗ + h π i j ∗ −−−−−→ H (SL ( F ) , Z ) δ −→ H (Γ ( m A ) , Z ) ( i ∗ , − i ∗ ( h π i• )) −−−−−−−−−→ H (SL ( A ) , Z ) ⊕ H (SL ( A ) , Z ) j ∗ + h π i j ∗ −−−−−→ H (SL ( F ) , Z )From this and Proposition 4.11 we obtain the exact sequence H (SL ( A ) , Z ) h π i j ∗ −→ H (SL ( F ) , Z ) j ∗ H (SL ( A ) , Z ) δ −→ H (Γ ( m A ) , Z ) i ∗ −→ H (SL ( A ) , Z ) → I ( k ) → . On the other hand, by Proposition 4.13, we have the exact sequence H (SL ( A ) , Z ) → RP ( k ) d , ( A,k ) −−−−−−→ E , ( A, k ) → H (SL ( A ) , Z ) → I ( k ) → . The following proposition compares these two exact sequences.
Proposition 4.14. If k is sufficiently large then the diagram with exact rows H (SL ( A ) , Z (cid:2) (cid:3) ) H (SL ( F ) , Z (cid:2) (cid:3) ) j ∗ H (SL ( A ) , Z (cid:2) (cid:3) ) H (Γ ( m A ) , Z (cid:2) (cid:3) ) H (SL ( A ) , Z (cid:2) (cid:3) ) RP ( k ) (cid:2) (cid:3) E , ( A, k ) (cid:2) (cid:3) h π i j ∗ = δ ∆ π d , ( A,k ) commutes. Proof.
The commutativity of the left square of the diagram is proved in Proposition 4.6.To prove the commutativity of other square we fix some notations.Let R be a local ring with sufficiently large residue field and let G be a group that actson the complex L • ( R ). Let P • → Z be a free resolution of Z over G . This is also a freeresolution of Z over any subgroup H of G .For any Z (cid:2) (cid:3) [ G ]-module M and any subgroup H of G , Let T R • ( H, M ) be the totalcomplex of the double complex D R,H • , • := P • ⊗ H (cid:0) L • ( R ) ⊗ Z M (cid:1) . On the one hand the hyperhomology of H with coefficients in the complex L • ( R ) ⊗ Z M is the homology of the total complex T R • ( H, M ). Thus by Lemma 4.9, H i ( T R • ( H, M )) ≃ H i ( H, L • ( R ) ⊗ Z M ) ≃ H i ( H, M ) , for 0 ≤ i ≤ . On the other hand from the double complex D R,H • , • we obtain the spectral sequence E p,q = H q ( H, L p ( R ) ⊗ Z M ) ⇒ H p + q ( H, L • ( R ) ⊗ Z M ) . For some of the basic properties of the total complex T R • ( H, M ) which we will use, see [6,Section 8.2]Let G := SL ( F ), G = SL ( A ), G = SL ( A ) g π and Γ = Γ ( m A ). Consider the exactsequence of R F -modules0 → Z (cid:2) (cid:3) [ G/ Γ] → Z (cid:2) (cid:3) [ G/G ] ⊕ Z [ G/G ] → Z (cid:2) (cid:3) → → T F • ( G, Z (cid:2) (cid:3) [ G/ Γ]) → T F • ( G, Z (cid:2) (cid:3) [ G/G ]) ⊕ T F • ( G, Z (cid:2) (cid:3) [ G/G ]) → T F • ( G, Z (cid:2) (cid:3) → . The long exact sequence associated to this exact sequences of complexes gives us theMayer-Vietoris exact sequence of R F -modules H ( G , Z (cid:2) (cid:3) ) ⊕ H ( G , Z (cid:2) (cid:3) ) β → H ( G, Z (cid:2) (cid:3) ) δ → H (Γ , Z (cid:2) (cid:3) ) α → H ( G , Z (cid:2) (cid:3) ) ⊕ H ( G , Z (cid:2) (cid:3) )(see Theorem 3.4) which from it we obtain the upper exact sequence in our diagram: H (SL ( A ) , Z (cid:2) (cid:3) ) ˜ β −→ H (SL ( F ) , Z (cid:2) (cid:3) ) H (SL ( A ) , Z (cid:2) (cid:3) ) δ −→ H (Γ , Z (cid:2) (cid:3) ) i ∗ −→ H (SL ( A ) , Z (cid:2) (cid:3) ) . HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 41
Consider the commutative diagram of complexes0 → T F • ( G, Z (cid:2) (cid:3) [ G/ Γ]) T F • ( G, Z (cid:2) (cid:3) [ G/G ]) ⊕ T F • ( G, Z (cid:2) (cid:3) [ G/G ]) → T F • ( G, Z (cid:2) (cid:3) ) → T F • (Γ , Z (cid:2) (cid:3) ) T F • ( G , Z (cid:2) (cid:3) ) T A • (Γ , Z (cid:2) (cid:3) ) T A • ( G , Z (cid:2) (cid:3) ) T k • (Γ , Z (cid:2) (cid:3) ) T k • ( G , Z (cid:2) (cid:3) ) . ( i • , The double complex which is used to construct T k • ( G , Z (cid:2) (cid:3) ) is used to construct thespectral sequence E • , • ( A, k ) (cid:2) (cid:3) . This spectral sequence is used to prove the exactness ofthe lower sequence in our diagram.Let ω ∈ im( δ ) = ker( i ∗ ) and let ω be represented by z ⊗ ∈ P ⊗ Γ Z (cid:2) (cid:3) . We know that ω ∈ H (Γ , Z (cid:2) (cid:3) ) = H ( P • ⊗ Γ Z (cid:2) (cid:3) ) ≃ H ( T A • (Γ , Z (cid:2) (cid:3) )) . Let ˜ z = (0 , , z ⊗ e A ) be its corresponding element in T A (Γ , Z (cid:2) (cid:3) ) = L i =0 P i ⊗ Γ L − i ( A ) (cid:2) (cid:3) . Since H (Γ , Z (cid:2) (cid:3) ) ≃ H ( T A • (Γ , Z (cid:2) (cid:3) )) ≃ H ( T F • (Γ , Z (cid:2) (cid:3) )) , we may also assume that˜ z = (0 , , z ⊗ e F ) ∈ T F (Γ , Z (cid:2) (cid:3) ) = L i =0 P i ⊗ Γ L − i ( F ) Z (cid:2) (cid:3) . In the above diagram ˜ z ∈ T F (Γ , Z (cid:2) (cid:3) ) maps to ˜ z ⊗ Γ ∈ T F ( G, Z (cid:2) (cid:3) [ G/ Γ]) which representthe element ω ∈ H (Γ , Z (cid:2) (cid:3) ) = H ( T F • ( G, Z (cid:2) (cid:3) [ G/ Γ]).Since under the map H ( T A • (Γ , Z (cid:2) (cid:3) )) = H (Γ , Z (cid:2) (cid:3) ) i ∗ −→ H ( G , Z (cid:2) (cid:3) ) = H ( T A • ( G , Z (cid:2) (cid:3) )) ,ω maps to zero, it follows that the image of ˜ z in T A ( G , Z (cid:2) (cid:3) ) is a boundary. Thus thereexists x = ( x , x , x , x ) ∈ T A ( G , Z (cid:2) (cid:3) ) = L i =0 P i ⊗ G L − i ( A ) (cid:2) (cid:3) such that d ( x ) = ˜ z ,where d = ( d h + ( − p d v ). More precisely,(0 , , z ⊗ e A ) = d ( x ) = ( d h ( x ) + d v ( x ) , d h ( x ) − d v ( x ) , d h ( x ) + d v ( x )) . From d h ( x ) + d v ( x ) = 0 it follows that d , ( x ) = 0 and thus x ∈ E , ( A, k ) (cid:2) (cid:3) . From d h ( x ) − d v ( x ) = 0, it follows that d , ( x ) = 0 and thus x represent an element α of E , ( A, k ) (cid:2) (cid:3) = RP ( k ) (cid:2) (cid:3) . Through the natural morphism D A,G • , • → D k,G • , • , x represent an element α of E , ( A, k ) (cid:2) (cid:3) = RP ( k ) (cid:2) (cid:3) , such that d , ( α ) is the homology class in E , ( A, k ) (cid:2) (cid:3) = H (Γ , Z (cid:2) (cid:3) ) /K represented by z ⊗ e A . Thus d , ( α ) = ω + K ∈ E , ( A, k ) (cid:2) (cid:3) . The conjugation isomorphism C π : G → G induces the isomorphism of complexes C π • : T F • ( G , Z (cid:2) (cid:3) ) → T F • ( G , Z (cid:2) (cid:3) ) . Under this the element ˜ z = (0 , , z ⊗ e F ) ∈ T F ( G , Z (cid:2) (cid:3) ) = L i =0 P i ⊗ G L − i ( F ) (cid:2) (cid:3) maps to C π (˜ z ) = (0 , , C π ( z ) ⊗ e F ) ∈ T F ( G , Z (cid:2) (cid:3) ) = L i =0 P i ⊗ G L − i ( F ) (cid:2) (cid:3) . Thisimage is a boundary map and therefore d ( C π ( x )) = C π (˜ z ), where C π ( x ) = ( C π ( x ) , C π ( x ) , C π ( x ) , C π ( x )) ∈ T F ( G , Z (cid:2) (cid:3) ) . By Corollary 4.8, C π acts trivially on im( δ ). Note that the following diagram is com-mutative H (Γ , Z (cid:2) (cid:3) ) G F H ( G , Z (cid:2) (cid:3) ) H ( G , Z (cid:2) (cid:3) ) . i ∗ i ′∗ C π ∗ Hence we may replace the element C π (˜ z ) = (0 , , C π ( z ) ⊗ e F ) ∈ T F ( G , Z (cid:2) (cid:3) ) with˜ z = (0 , , z ⊗ e F ) ∈ T F ( G , Z (cid:2) (cid:3) ).Now under the map T F ( G, Z (cid:2) (cid:3) [ G/ Γ]) → T F ( G, Z (cid:2) (cid:3) [ G/G ]) ⊕ T F ( G, Z (cid:2) (cid:3) [ G/G ]) , we have ˜ z ⊗ Γ (˜ z ⊗ G , ˜ z ⊗ G ) . By the calculation above, this is the boundary of( x ⊗ G , C π ( x ) ⊗ G ) ∈ T F ( G, Z (cid:2) (cid:3) [ G/G ]) ⊕ T F ( G, Z (cid:2) (cid:3) [ G/G ]) . Under the map T F ( G, Z (cid:2) (cid:3) [ G/G ]) ⊕ T F ( G, Z (cid:2) (cid:3) [ G/G ]) → T F ( G, Z (cid:2) (cid:3) ) , this elements maps to Ω := C π ( x ) − x . By construction the cycle Ω represent an elementof H ( G, Z (cid:2) (cid:3) ), which maps to ω ∈ H (Γ , Z (cid:2) (cid:3) ) under the connecting map δ . Now underthe map H (SL ( F ) , Z (cid:2) (cid:3) ) → RP ( F ) (cid:2) (cid:3) , the homology class of Ω maps to h π i α − α ∈ RB ( F ) (cid:2) (cid:3) . By definition δ π ( h π i α − α ) = α .Therefore d , ◦ ∆ π (Ω) = d , ( α ) = ω + K = p ◦ δ (Ω) . HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 43
Now if Ω ′ is another element of H (SL ( F ) , Z (cid:2) (cid:3) ) thats map to ω by δ , then Ω ′ = Ω + h π i j ∗ ( y ) for some y ∈ H (SL ( A ) , Z (cid:2) (cid:3) ). Now using the commutativity of the left sidesquare in our diagram, it is easy to see that d , ◦ ∆ π (Ω ′ ) = d , ( α ) = ω + K = p ◦ δ (Ω ′ ) . This completes the proof of the proposition. (cid:3)
Corollary 4.15. If k is sufficiently large, then we have the exact sequence H (SL ( A ) , Z (cid:2) (cid:3) ) →RP ( k ) (cid:2) (cid:3) d → H (Γ ( m A ) , Z (cid:2) (cid:3) ) i ∗ → H (SL ( A ) , Z (cid:2) (cid:3) ) → I ( k ) (cid:2) (cid:3) → where d is given by the formula d ( g ( a )) = c (cid:16)(cid:18) − a
00 (1 − a ) − (cid:19) , (cid:18) a a − (cid:19)(cid:17) . Proof.
Set Γ = Γ ( m A ). By Proposition 4.14 we have the commutative diagram H (SL ( A ) , Z (cid:2) (cid:3) ) H (SL ( F ) , Z (cid:2) (cid:3) ) j ∗ H (SL ( A ) , Z (cid:2) (cid:3) ) H (Γ , Z (cid:2) (cid:3) ) H i (SL ( A ) , Z (cid:2) (cid:3) ) H (SL ( A ) , Z (cid:2) (cid:3) ) RP ( k ) (cid:2) (cid:3) E , ( A, k ) (cid:2) (cid:3) H i (SL ( A ) , Z (cid:2) (cid:3) ) , h π i j ∗ = δ ∆ π i ∗ = d , ( A,k ) i ∗ which implies that E , ( A, k ) (cid:2) (cid:3) ≃ H (Γ , Z (cid:2) (cid:3) ). Now the claim follows from Proposi-tion 4.13. (cid:3) The following is Theorem B in the introduction.
Theorem 4.16.
Let A be a discrete valuation ring with sufficiently large residue field k .Then the inclusion Γ ( m A ) → SL ( A ) induces the exact sequence od R A -modules → P ( k ) (cid:2) (cid:3) → H (Γ ( m A ) , Z (cid:2) (cid:3) ) → H (SL ( A ) , Z (cid:2) (cid:3) ) → I ( k ) (cid:2) (cid:3) → , where the left homomorphism is given by [ a ] c (cid:16)(cid:18) − a
00 (1 − a ) − (cid:19) , (cid:18) a a − (cid:19)(cid:17) .Proof. By the results above, the diagram RP ( k ) (cid:2) (cid:3) P ( k ) (cid:2) (cid:3) H (Γ ( m A ) , Z (cid:2) (cid:3) ) d commutes, where the vertical arrow sends g ( a ) to 2[ a ]. (cid:3) If k is finite, then I ( k ) = 0 and we deduce: Corollary 4.17. If k is a sufficiently large finite field, then we have the exact sequence → P ( k ) (cid:2) (cid:3) → H (Γ ( m A ) , Z (cid:2) (cid:3) ) → H (SL ( A ) , Z (cid:2) (cid:3) ) → . The case of global fields
In this final section we will study the group P ( k ) when F is a global field.5.1. The e -invariant of a field. Let E be a separably closed field. The group Aut( E )acts naturally on µ E , the group of roots of unity in E , where σ ∈ Aut( E ) sends ζ to σ ( ζ ). This action gives a surjective map Aut( E ) → Aut( µ E ). Note that µ E ≃ Q / Z ifchar( E ) = 0 and µ E ≃ ( Q / Z )[ p ] if char( E ) = p > i ∈ Z , we define µ E ( i ) as µ E turned into Aut( E )-module by letting σ ∈ Aut( E ) actsas ζ σ i ( ζ ) . On the other hand, Aut( E ) acts on K n ( F ) for any n ≥ σ : E → E induces theisomorphism σ ∗ : K n ( E ) → K n ( E ) and for any x ∈ K n ( E ) the action is given by σ.x = σ ∗ ( x ).If n is odd, then K n ( E ) tors ≃ µ E (if n is even, K n ( E ) is uniquely divisible). Clearly theaction of Aut( E ) on K n ( E ) induces an action of Aut( E ) on K n ( E ) tors . It is known thatfor i > K i − ( E ) tors is isomorphic to µ E ( i ) as Aut( E )-module [22, Proposition 1.7.1,Chap. VI].Let F be a field and F sep be its separable closure. Let G F := Gal( F sep /F ). Sincethe natural map K n ( F ) → K n ( F sep ) is a homomorphism of G F -modules with G F actingtrivially on K n ( F ) it follows that there is a natural map K n ( F ) tors → ( K n ( F sep ) tors ) G F .The e -invariant of ( K i − -group of ) F is the composition K i − ( F ) tors −→ ( K i − ( F sep ) tors ) G F ≃ −→ µ F sep ( i ) G F . If µ F sep ( i ) G F is finite, it is cyclic and we write w i ( F ) for its order. Thus µ F sep ( i ) G F ≃ Z /w i ( F ) . We will need the e -invariant of fields for the third K -group. Thus from now on we willdiscuss only this special case. Example 5.1. (Finite Fields) For a finite field F q , w ( F q ) = q −
1. One the other hand K ( F q ) ≃ Z / ( q − e -invariant e : K ( F q ) tors −→ µ F q (2) G F q is an isomorphism [22, Example 2.1.1, Chap. VI].From now on we assume that A is a discrete valuation ring, with field of fractions F andresidue field k . Let v = v A : F × → Z be the valuation associated to A . HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 45
The e -invariant of local and global fields and the indecomposable K . Let p be a prime number. For any torsion abelian group G , we let G [ p ′ ] denote the subgroup ofelements of order prime to p . In fact G [ p ′ ] ≃ G ⊗ Z [ p ] ≃ G [ p ]. Thus we have a canonicalsurjective map G → G [ p ′ ] . A (non-Archimedean ) local field E is a field which is complete in respect with a discretevaluation v : E × → Z and has finite residue field. In this case the discrete valuation ringis a complete ring. It is a classical result that a local field is either a finite extension of therational p-adic field Q p or is isomorphic to F q (( x )) for some finite field F q (see [18]). Lemma 5.2 (Local Fields) . If E is a local field, then we have the isomorphism e : K ind3 ( E )[ p ′ ] tors ≃ −→ µ E sep (2)[ p ′ ] G E , where p is the characteristic of the residue field.Proof. If E is finite over Q p with residue field F q , then w ( E ) = p n w ( F q ) = p n ( q − n ≥ e : K ( E ) tors −→ µ E sep (2) G E ≃ Z /w ( E )is surjective and induces an isomorphism on ℓ -primary torsion subgroups K ( E )( l ) ≃ Z /w ( ℓ )2 ( E ) for any prime ℓ = p [22, Example 2.3.2, Chap. VI].If E = F q (( t )), then K ( E ) tors ≃ K ( F q ) [22, Theorem 7.2, Chap. VI]. This shows that w ( E ) = w ( F q (( t ))) = q − e -invariant e : K ( E ) tors −→ µ E sep (2) G E ≃ Z /w ( E )is an isomorphism.Since for any local filed E , K M ( E ) is uniquely divisible [22, Proposition 7.1, Chap. VI],the e -invariant factors through K ind3 ( E ) tors . Therefore we get the desired isomorphism. (cid:3) Corollary 5.3.
Let F be local. If char( k ) = p , then we have the commutative diagram K ind3 ( A )[ p ′ ] tors µ F sep (2)[ p ′ ] G F K ind3 ( k ) µ ¯ k (2) G k , ≃ ≃≃ where the vertical maps are induced by the quotient map A → k . Moreover all the mapsinvolved are isomorphism. Lemma 5.4 (Global Fields) . For any global field E , the e -invariant factors through thegroup K ind3 ( E ) and gives the isomorphism e : K ind3 ( E ) tors ≃ −→ µ E sep (2) G E ≃ Z /w ( E ) . Proof.
If char( E ) = 0, then E is an algebraic number field. For this the claim follows from[22, Corollary 5.3, Chap. VI].If char( E ) = p >
0, then E is finite over a field of the form F q ( t ), where q is p -power.In this case K M ( E ) = 0 [1] and the natural map K ( F q ) → K ( E ) is an isomorphism [22,Theorem 6.8, Chap. VI]. Therefore K ( E ) = K ind3 ( E ) and clearly the e-invariant is anisomorphism. (cid:3) The image of the map K ind3 ( F ) → P ( k ) .Theorem 5.5. Let F be a global field such that char( k ) ∤ w ( F ) . Then (i) There is a natural splitting of the inclusions K ind3 ( F ) tors → K ind3 ( F ) , call it p F : K ind3 ( F ) → K ind3 ( F ) tors . (ii) The map K ind3 ( F ) ≃ K ind3 ( A ) → K ind3 ( k ) factors through p F . (iii) The image of K ind3 ( F ) ≃ K ind3 ( A ) → K ind3 ( k ) is cyclic of order w ( F ) .Proof. Consider the commutative diagram K ind3 ( F ) tors µ F sep (2) G F K ind3 ( F ) K ind3 ( k ) K ind3 ( F v ) K ind3 ( F v ) tors [ p ′ ] µ F sep v (2)[ p ′ ] G Fv e F iπ απ v e ′ Fv where F v is the completion of F with respect to the valuation and p = char( k ). Note that α is coming from Corollary 5.3 and π is the composite K ind3 ( F ) ≃ K ind3 ( A ) → K ind3 ( k ).Moreover the maps e F , e F v and α are isomorphism by Lemmas 5.4, 5.2 and Corollary 5.3,respectively.(i) From the commutativity of the above diagram one can take p F = e − F ◦ i − ◦ α ◦ π .(ii) By condtruction π = α − ◦ i ◦ e F ◦ p F . Hence it factors through p F .(iii) Since π factors throught p F , we haveim( K ind3 ( A ) → K ind3 ( k )) = im( K ind3 ( A ) tors → K ind3 ( k )) ≃ im( µ F sep (2) G F i ֒ → µ F sep v (2)[ p ′ ] G Fv ) . This completes the proof. (cid:3)
HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 47
Proposition 5.6.
Let F be a global field such that char( k ) ∤ w ( F ) . Then the image ofthe natural map K ind3 ( A ) → P ( k ) is cyclic of order gcd (cid:0) w ( F ) , ( | k | + 1) / (cid:1) if char( k ) = 2 and is gcd (cid:0) w ( F ) , | k | + 1 (cid:1) if char( k ) = 2 .Proof. The map K ind3 ( A ) → P ( k ) factors through K ind3 ( k ). Let η be the map K ind3 ( A ) → K ind3 ( k ). By Theorem 5.5, the image of η is cyclic of order w ( F ). By Theorem 1.1, wehave the Bloch-Wigner exact sequence0 → Tor Z ( k × , k × ) ∼ β → K ind3 ( k ) → P ( k ) → S Z ( k × ) → K ( k ) → , where Tor Z ( k × , k × ) ∼ is the extension of Tor Z ( k × , k × ) by Z / k ) = 2 and is isomor-phic to Tor Z ( k × , k × ) if char( k ) = 2.Let k = F q . If char( k ) = 2, then Tor Z ( k × , k × ) ∼ is a cyclic group of order 2( q − K ind3 ( k ) = K ( k ) is cyclic of order q −
1. Thus K ind3 ( k ) / im( β ) is cyclic oforder ( q + 1) / η ) → K ind3 ( k ) / im( β ) ֒ → P ( k ) . Since the image of im( η ) → K ind3 ( k ) / im( β ) is cyclic of order gcd (cid:0) w ( F ) , ( | k | + 1) / (cid:1) weare done. If char( k ) = 2, then Tor Z ( k × , k × ) is a cyclic group of order q −
1. The rest ofproof is similar to the above argument. (cid:3)
The second homology of Γ ( m A ) of a DVR in a global field. For a naturalnumber n , let n ′ be the odd part of n , i.e. n ′ = n/ r , where 2 r is highest power of 2thats divide n . By combining Lemma 4.16, Theorem 2.1 and Proposition 5.6 we obtainthe following result. Theorem 5.7.
Let F be a global field such that char( k ) ∤ w ( F ) . If k is sufficiently largethen we have the exact sequence → P ( k ) (cid:2) (cid:3) → H (Γ ( m A ) , Z (cid:2) (cid:3) ) → H (SL ( A ) , Z (cid:2) (cid:3) ) → , where P ( k ) (cid:2) (cid:3) is cyclic of order ( | k | + 1) ′ and P ( k ) (cid:2) (cid:3) is the quotient of P ( k ) (cid:2) (cid:3) by theunique cyclic subgroup of order gcd (cid:0) w ( F ) , | k | + 1 (cid:1) ′ . Moreover the left homomorphism isgiven by [ a ] c (cid:16)(cid:18) − a
00 (1 − a ) − (cid:19) , (cid:18) a a − (cid:19)(cid:17) . Let F = Q equipped with the p -adic valuation v p . The residue field is F p and w ( Q ) = 24[22, Example 2.1.2, Chap. VI]. If p = 2 ,
3, then 24 | p − | K ind3 ( F p ) | . We havegcd( w ( Q ) , p + 1) ′ = gcd(3 , p + 1) = ( | p + 10 if 3 ∤ p + 1 . From the previous theorem and the fact that c F p is of order gcd(6 , ( p + 1) /
2) in P ( F p ) [4,Lemma 7.11], we obtain the following corollary. Corollary 5.8.
For any prime number p ≥ , we have the exact sequence → P ( F p ) (cid:2) (cid:3) → H (Γ ( Z ( p ) ) , Z (cid:2) (cid:3) ) → H (SL ( Z ( p ) ) , Z (cid:2) (cid:3) ) → , where P ( F p ) = P ( F p ) / h c F p i if | p + 1 and P ( F p ) = P ( F p ) if ∤ p + 1 . As a final application of the results above, we show:
Proposition 5.9.
Let p ≥ be a prime. Let ¯ H (SL ( Z ( p ) ) , Z ) denote the image ofthe natural homomorphism H (SL ( Z ( p ) ) , Z ) → H (SL ( Q ) , Z ) and let ¯ H (SL ( Z ( p ) ) , Z ) denote ker( ¯ H (SL ( Z ( p ) ) , Z ) → K ind3 ( Q )) . Then ¯ H (SL ( Z ( p ) ) , Z (cid:2) (cid:3) ) ≃ L q = p P ( F p ) (cid:2) (cid:3) , where the sum runs over all primes q not equal to p .Proof. We have a commutative diagram with exact rows, whose middle column is exact byTheorem 2.1:0 ¯ H (SL ( Z ( p ) ) , Z (cid:2) (cid:3) ) ¯ H (SL ( Z ( p ) ) , Z (cid:2) (cid:3) ) K ind3 ( Q ) (cid:2) (cid:3) H (SL ( Q ) , Z (cid:2) (cid:3) ) H (SL ( Q ) , Z (cid:2) (cid:3) ) K ind3 ( Q ) (cid:2) (cid:3) P ( F p ) (cid:2) (cid:3) P ( F p ) (cid:2) (cid:3) = δ p ∆ p = Thus ¯ H (SL ( Z ( p ) ) , Z (cid:2) (cid:3) ) ∼ = ker(∆ p : ¯ H (SL ( Q ) , Z (cid:2) (cid:3) ) → P ( F p ) (cid:2) (cid:3) ). But this is equalto ker(∆ ′ p : ¯ H (SL ( Q ) , Z (cid:2) (cid:3) ) → P ( F p ) (cid:2) (cid:3) ) by Proposition 4.3 above. However ker ∆ ′ p ≃ L q = p P ( F p ) (cid:2) (cid:3) by the main theorem of [10] (where ∆ ′ p is denoted S p ). (cid:3) References [1] Bass, H., Tate, J. The Milnor ring of a global field. Lecture Notes in Math., Vol. , 349–446 (1973)46[2] Brown, K. S.: Cohomology of Groups. Graduate Texts in Mathematics, 87. Springer-Verlag, New York(1994) 31[3] Dupont, J- L., Sah, C. Scissors congruences. II. J. Pure Appl. Algebra (1982), no. 2, 159–195. 1[4] Hutchinson, K.: A Bloch-Wigner complex for SL . J. K-Theory , no. 1, 15–68 (2013) 1, 4, 5, 32, 47[5] Hutchinson, K.: A refined Bloch group and the third homology of SL of a field. J. Pure Appl. Algebra , 2003–2035 (2013) 5, 7, 9, 11[6] Hutchinson, K.: On the low-dimensional homology of SL ( k [ t, t − ]). J. Algebra , 324–366 (2015)40[7] Hutchinson, K.: The second homology of SL of S -integers. J. Number Theory , 223–272 (2016) 2 HE HOMOLOGY OF SL OF DISCRETE VALUATION RINGS 49 [8] Hutchinson, K.: The third homology of SL of local rings. Journal of Homotopy and Related Structures , 931–970 (2017) 2, 5, 6, 7, 8, 9, 11, 12, 20, 30, 32, 38[9] Hutchinson, K.: The third homology of SL of fields with discrete valuation. J. Pure Appl. Algebra (5), 1076–1111 (2017) 3, 8, 10, 11, 28[10] Hutchinson, K.: The third homology of SL ( Q ). Preprint. arxiv.org/abs/1906.11650 13, 15, 16, 38, 48[11] Hutchinson, K., Tao, L.: The third homology of the special linear group of a field. J. Pure Appl.Algebra (2009), no. 9, 1665–1680. 6[12] Mazzoleni A.: A new proof of a theorem of Suslin, K-Theory (3-4) 199–211 (2005) 35[13] Mirzaii, B.: Bloch-Wigner theorem over rings with many units. Math. Z. , 329–346 (2011). Erratumto: Bloch-Wigner theorem over rings with many units. Math. Z. , 653–655 (2013) 4, 32[14] Mirzaii, B., Mokari, F. Y.: A Bloch-Wigner theorem over rings with many units II. Journal of Pureand Applied Algebra , 5078–5096 (2015) 4[15] Mirzaii, B.: A Bloch-Wigner exact sequence over local rings. Journal of Algebra , 459–493 (2017)4, 32[16] Schlichting, M.: Euler class groups and the homology of elementary and special linear groups. Adv.Math , 1-81, (2017) 2[17] Gille S., Scully S., Zhong, C.: Milnor-Witt K -groups of local rings. Adv. Math. , 729-753, (2016) 2[18] Serre, J.-P.: Local Fields, Springer-Verlag, Berlin (1979) 45[19] Serre, J.-P.: Trees, Springer-Verlag, Berlin (1980) 21, 22[20] Sherman, C.: K-theory of discrete valuation rings. Journal of Pure and Applied Algebra K of a field and the Bloch group. Proc. Steklov Inst. Math. , no. 4, 217–239 (1991)1, 4, 8, 32[22] Weibel, C. A.: The K -book: An introduction to algebraic K -theory. Graduate Studies in Mathematics,vol. 145. American Mathematical Society, Providence (2013) 20, 44, 45, 46, 47 School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4,Ireland
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