On SK 1 of Iwasawa algebras joint work with Peter Schneider Otmar - PowerPoint PPT Presentation

An integral version of a result of Kostant on Lie algebras Uniform pro- p -groups and their Lie algebras Iwasawa algebras and SK 1 Application On SK 1 of Iwasawa algebras joint work with Peter Schneider Otmar Venjakob Mathematisches Institut Universit¨ at Heidelberg Cartagena, 14.02.2012 Otmar Venjakob On SK 1 of Iwasawa algebras

An integral version of a result of Kostant on Lie algebras The problem Uniform pro- p -groups and their Lie algebras A counterexample Iwasawa algebras and SK 1 Chevalley orders Application Main result on Lie algebras The setup R commutative ring, L a R -Lie algebra, finitely generated free as R -module [ , ] : L ∧ L → L � L := < x ∧ y | [ x , y ] L = 0 > R ⊆ ker[ , ] When does � L = ker[ , ] hold? Question: Otmar Venjakob On SK 1 of Iwasawa algebras

An integral version of a result of Kostant on Lie algebras The problem Uniform pro- p -groups and their Lie algebras A counterexample Iwasawa algebras and SK 1 Chevalley orders Application Main result on Lie algebras A counterexample Assume that 2 ǫ R × . V := R 4 with standard basis e 1 , . . . , e 4 and W := � 2 V / R ( e 1 ∧ e 2 + e 3 ∧ e 4 ) (rank 5) 2 � pr ∂ : V − − → W . Note that ker ∂ does not contain any nonzero vector of the form a ∧ b . L ′ := V ⊕ W with bracket 2 2 ∂ ⊆ � pr � L ′ → L ′ [ , ] : − − → − → W − − V makes L ′ into a 2-step nilpotent Lie algebra over R with center Z ( L ′ ) = [ L ′ , L ′ ] = W and e 1 ∧ e 2 + e 3 ∧ e 4 ǫ ker[ , ] \ � L ′ . Otmar Venjakob On SK 1 of Iwasawa algebras

An integral version of a result of Kostant on Lie algebras The problem Uniform pro- p -groups and their Lie algebras A counterexample Iwasawa algebras and SK 1 Chevalley orders Application Main result on Lie algebras Chevalley orders F field of characteristic zero g a F -split reductive Lie algebra over F with center z , Cartan subalgebra h and root system Φ, [ X α , X − α ] = − H α Q ∨ := � α ǫ Φ Z H α ⊆ h coroot lattice P ∨ := { h ǫ � α ǫ Φ Q H α : β ( h ) ǫ Z for any β ǫ Φ } ⊆ h coweight lattice of the root system Φ h Z ⊆ h Z -lattice such that Q ∨ ⊆ h Z ⊆ P ∨ ⊕ z , � g Z := h Z + Z X α ⊆ g . α ǫ Φ g Z is a Z -Lie subalgebra (Chevalley order) of g . g R := R ⊗ Z g Z is a R -Lie algebra. Otmar Venjakob On SK 1 of Iwasawa algebras

An integral version of a result of Kostant on Lie algebras The problem Uniform pro- p -groups and their Lie algebras A counterexample Iwasawa algebras and SK 1 Chevalley orders Application Main result on Lie algebras Theorem If 2 and 3 are invertible in R then ker[ , ] = � g R . Kostant had proved the case R = C by different methods. This is an integral version of his result. Otmar Venjakob On SK 1 of Iwasawa algebras

An integral version of a result of Kostant on Lie algebras Uniform pro- p -groups and their Lie algebras Uniform pro- p -groups Iwasawa algebras and SK 1 The associated Lie algebra Application Uniform pro- p -groups G (topologically) finitely generated pro- p group with 1 [ G , G ] ⊆ G p , · p � G i +1 / G i +2 for all i ≥ 1, G i / G i +1 2 ∼ = where G 1 := G and G i := [ G , G i ] G p is the lower p -central series, is called uniform pro- p group. Facts: · pi − 1 � G i is homeomorphic (but not homomorphic in G 1 ∼ = general). 2 (Lazard) A pro-finite group G is a p -adic Lie group ⇐ ⇒ G has an open characteristic subgroup which is uniform. Otmar Venjakob On SK 1 of Iwasawa algebras

An integral version of a result of Kostant on Lie algebras Uniform pro- p -groups and their Lie algebras Uniform pro- p -groups Iwasawa algebras and SK 1 The associated Lie algebra Application The associated Lie algebra The operations 1 n →∞ ( x p n y p n ) x + y := lim pn 1 n →∞ [ x p n , y p n ] ( x , y ) := lim p 2 n make G into a Z p Lie algebra, denoted L := L ( G ), and we have an equivalence of categories { G uniform } ← → {L with ( L , L ) ⊆ p L ,i.e., powerful } Otmar Venjakob On SK 1 of Iwasawa algebras

An integral version of a result of Kostant on Lie algebras Uniform pro- p -groups and their Lie algebras Iwasawa algebras Iwasawa algebras and SK 1 Vanishing of SK 1 Application Iwasawa algebras G pro-finite group Λ( G ) := lim Z p [ G / U ] Iwasawa algebra ← − U ⊳ G open Λ ∞ ( G ) := lim Q p [ G / U ] ← − U ⊳ G open � � SK 1 ( Z p [ G / U ]) := ker K 1 ( Z p [ G / U ]) − → K 1 ( Q p [ G / U ]) is known to be finite! � ∼ � SK 1 (Λ( G )) := ker K 1 (Λ( G )) − → K 1 (Λ ∞ ( G )) = lim − SK 1 ( Z p [ G / U ]) ← by a result of Fukaya and Kato. Otmar Venjakob On SK 1 of Iwasawa algebras

An integral version of a result of Kostant on Lie algebras Uniform pro- p -groups and their Lie algebras Iwasawa algebras Iwasawa algebras and SK 1 Vanishing of SK 1 Application A homological description Oliver: for H finite we have ⊕ A ⊆ H H 2 ( A , Z ) − → H 2 ( H , Z ) − → SK 1 ( Z p [ H ]) − → 0 where A runs trough all abelian subgroups of H . Dualizing with − ∨ := Hom cts ( − , Q p / Z p ) and taking limits gives: SK 1 (Λ( G )) ∨ = ker H 2 ( G , Q p / Z p ) − � H 2 ( A , Q p / Z p ) � � → lim . − → N A ⊆ G / N Otmar Venjakob On SK 1 of Iwasawa algebras

An integral version of a result of Kostant on Lie algebras Uniform pro- p -groups and their Lie algebras Iwasawa algebras Iwasawa algebras and SK 1 Vanishing of SK 1 Application A cohomological criterion If G has no torsion, then SK 1 = 0 ⇐ ⇒ δ � → H 1 ( G , Q p / Z p ) / p → H 2 ( G , F p ) res H 2 ( A , F p ) 0 − − − − → A ⊆ G is exact. As a consequence of Whiteheads Lemma and a result of Lazard we obtain Corollary If G is a compact p-adic Lie group such that L ( G ) := Q p ⊗ Z p L ( G ) is semi-simple, then SK 1 (Λ( G )) is finite. Otmar Venjakob On SK 1 of Iwasawa algebras

An integral version of a result of Kostant on Lie algebras Uniform pro- p -groups and their Lie algebras Iwasawa algebras Iwasawa algebras and SK 1 Vanishing of SK 1 Application The uniform case H ∗ ( G , F p ) = � H 1 ( G , F p ) Lazard: V := G / G p . Then SK 1 = 0 ⇐ ⇒ 2 δ ∨ ⊆ � � → G ab [ p ] − 0 − → − − → − − → 0 , V V is exact Otmar Venjakob On SK 1 of Iwasawa algebras

An integral version of a result of Kostant on Lie algebras Uniform pro- p -groups and their Lie algebras Iwasawa algebras Iwasawa algebras and SK 1 Vanishing of SK 1 Application The uniform case H ∗ ( G , F p ) = � H 1 ( G , F p ) Lazard: V := G / G p . Then SK 1 = 0 ⇐ ⇒ 2 δ ∨ ⊆ � � → G ab [ p ] − 0 − → − − → − − → 0 , V V is exact ⇐ ⇒ � V = ker ∂ → ( G p / [ G p , G ])[ p ] where ∂ : V ∧ V − gG p ∧ hG p �− → [ g , h ] mod [ G p , G ] Otmar Venjakob On SK 1 of Iwasawa algebras

An integral version of a result of Kostant on Lie algebras Uniform pro- p -groups and their Lie algebras Iwasawa algebras Iwasawa algebras and SK 1 Vanishing of SK 1 Application A Lie criterion � SK 1 = 0 ⇐ ⇒ L = ker[ , ] Otmar Venjakob On SK 1 of Iwasawa algebras

An integral version of a result of Kostant on Lie algebras Uniform pro- p -groups and their Lie algebras Iwasawa algebras Iwasawa algebras and SK 1 Vanishing of SK 1 Application Vanishing of SK 1 R = Z p for p � = 2 , 3 g a Q p -split reductive Lie algebra g Z ⊆ g a Chevalley order. Then, for any n ≥ 1 , p n g Z p corresponds to unique uniform p -adic Lie group G ( p n ) with Z p -Lie algebra L ( G ( p n )) = p n g Z p . Theorem In the above setting we have SK 1 (Λ( G ( p n ))) = 0 . Otmar Venjakob On SK 1 of Iwasawa algebras

An integral version of a result of Kostant on Lie algebras Uniform pro- p -groups and their Lie algebras Iwasawa algebras Iwasawa algebras and SK 1 Vanishing of SK 1 Application Examples G a split reductive group scheme over Z G ( p n ) := ker G ( Z p ) → G ( Z / p n ) � � satisfies conditions of the theorem, e.g. for m ≥ 1 ker ( SL d ( Z p ) → SL d ( Z p / p m )) . Otmar Venjakob On SK 1 of Iwasawa algebras

� � An integral version of a result of Kostant on Lie algebras Uniform pro- p -groups and their Lie algebras Iwasawa algebras and SK 1 Application Iwasawa Main Conjecture Uniqueness-statements in Main Conjectures of Iwasawa theory: � [ X E ] L , L ′ ✤ SK 1 (Λ( G )) � � ∂ � K 1 (Λ( G ) S ) � � K 0 ( S − tor ) K 1 (Λ( G )) DET � DET × ) � Maps ( Irr ( G ) , Q p ∪ {∞} ) Maps ( Irr ( G ) , Z p That is, if SK 1 (Λ( G )) = 1 and if L is induced form Λ( G ) ∩ Λ( G ) × S (no poles), then L is unique with 1 ∂ L = [ X E ], 2 DET ( L ) satisfies some interpolation property. Otmar Venjakob On SK 1 of Iwasawa algebras