Witt vectors, commutative and non-commutative Dmitry Kaledin - PowerPoint PPT Presentation

Witt vectors, commutative and non-commutative Dmitry Kaledin Steklov Math Institute & NRU HSE, Moscow

Origins of Witt vectors: Teichm¨ uller Consider the ring Z p of p -adic integers. Recall that Z / p n Z , R : Z / p n +1 Z → Z / p n Z Z p = lim R ←

Origins of Witt vectors: Teichm¨ uller Consider the ring Z p of p -adic integers. Recall that Z / p n Z , R : Z / p n +1 Z → Z / p n Z Z p = lim R ← and any a ∈ Z p is represented by a series � a i p i , a = i with a i ∈ { 0 , . . . , p − 1 }

Origins of Witt vectors: Teichm¨ uller Consider the ring Z p of p -adic integers. Recall that Z / p n Z , R : Z / p n +1 Z → Z / p n Z Z p = lim R ← and any a ∈ Z p is represented by a series � a i p i , a = i with a i ∈ { 0 , . . . , p − 1 } , or { 1 , . . . , p }

Origins of Witt vectors: Teichm¨ uller Consider the ring Z p of p -adic integers. Recall that Z / p n Z , R : Z / p n +1 Z → Z / p n Z Z p = lim R ← and any a ∈ Z p is represented by a series � a i p i , a = i with a i ∈ { 0 , . . . , p − 1 } , or { 1 , . . . , p } , or some other set of representatives of mod p residue classes x ∈ F p = Z / p Z .

Origins of Witt vectors: Teichm¨ uller Consider the ring Z p of p -adic integers. Recall that Z / p n Z , R : Z / p n +1 Z → Z / p n Z Z p = lim R ← and any a ∈ Z p is represented by a series � a i p i , a = i with a i ∈ { 0 , . . . , p − 1 } , or { 1 , . . . , p } , or some other set of representatives of mod p residue classes x ∈ F p = Z / p Z . Observation (Teichm¨ uller). There is a canonical choice of representatives [ x ] ∈ Z p for classes x ∈ F p .

Origins of Witt vectors: Teichm¨ uller Consider the ring Z p of p -adic integers. Recall that Z / p n Z , R : Z / p n +1 Z → Z / p n Z Z p = lim R ← and any a ∈ Z p is represented by a series � a i p i , a = i with a i ∈ { 0 , . . . , p − 1 } , or { 1 , . . . , p } , or some other set of representatives of mod p residue classes x ∈ F p = Z / p Z . Observation (Teichm¨ uller). There is a canonical choice of representatives [ x ] ∈ Z p for classes x ∈ F p . Namely, for any n , the map ( Z / p n Z ) ∗ → F ∗ p admits a unique p → ( Z / p n Z ) ∗ . Set T (0) = 0 and [ x ] = T ( x ). splitting T : F ∗

Origins of Witt vectors: Witt We thus have a natural isomorphism of sets F p ∼ � � [ x i ] p i T q : = Z p , T q ( � x 0 , x 1 , . . . � ) = i ≥ 0 i Question: how to write down the ring operations?

Origins of Witt vectors: Witt We thus have a natural isomorphism of sets F p ∼ � � [ x i ] p i T q : = Z p , T q ( � x 0 , x 1 , . . . � ) = i ≥ 0 i Question: how to write down the ring operations? For any n ≥ 1 and commutative ring A , denote by W n ( A ) the set A n . Let R : W n +1 ( A ) → W n ( A ), V : W n ( A ) → W n +1 ( A ) be the maps R ( � a 0 , . . . , a n � ) = � a 0 , . . . , a n − 1 � , V ( � a 0 , . . . , a n − 1 � ) = � 0 , a 0 , . . . , a n − 1 �

Origins of Witt vectors: Witt We thus have a natural isomorphism of sets F p ∼ � � [ x i ] p i T q : = Z p , T q ( � x 0 , x 1 , . . . � ) = i ≥ 0 i Question: how to write down the ring operations? For any n ≥ 1 and commutative ring A , denote by W n ( A ) the set A n . Let R : W n +1 ( A ) → W n ( A ), V : W n ( A ) → W n +1 ( A ) be the maps R ( � a 0 , . . . , a n � ) = � a 0 , . . . , a n − 1 � , V ( � a 0 , . . . , a n − 1 � ) = � 0 , a 0 , . . . , a n − 1 � Thm (Witt). There exists a unique set of functorial abelian groups structures on W n ( A ), n ≥ 1 such that R , V , and T q : W n ( F p ) ∼ = Z / p n Z are additive.

Origins of Witt vectors: Witt We thus have a natural isomorphism of sets F p ∼ � � [ x i ] p i T q : = Z p , T q ( � x 0 , x 1 , . . . � ) = i ≥ 0 i Question: how to write down the ring operations? For any n ≥ 1 and commutative ring A , denote by W n ( A ) the set A n . Let R : W n +1 ( A ) → W n ( A ), V : W n ( A ) → W n +1 ( A ) be the maps R ( � a 0 , . . . , a n � ) = � a 0 , . . . , a n − 1 � , V ( � a 0 , . . . , a n − 1 � ) = � 0 , a 0 , . . . , a n − 1 � Thm (Witt). There exists a unique set of functorial abelian groups structures on W n ( A ), n ≥ 1 such that R , V , and T q : W n ( F p ) ∼ = Z / p n Z are additive. Key idea of the theorem: functoriality.

First ingredient: ghost map We cannot write down a formula for T : F p → Z / p n Z , but we do have a formula for the composition R n T Z / p n +1 Z → Z / p n +1 Z − − − − → F p − − − −

First ingredient: ghost map We cannot write down a formula for T : F p → Z / p n Z , but we do have a formula for the composition R n T Z / p n +1 Z → Z / p n +1 Z − − − − → F p − − − − uller representative iff x p n = x . Lemma. x ∈ Z / p n +1 Z is a Teichm¨ For any x ∈ Z / p n +1 Z with residue R n ( x ) ∈ F p , we have x p n = [ R n ( x )] = T ( R n ( x )).

First ingredient: ghost map We cannot write down a formula for T : F p → Z / p n Z , but we do have a formula for the composition R n T Z / p n +1 Z → Z / p n +1 Z − − − − → F p − − − − uller representative iff x p n = x . Lemma. x ∈ Z / p n +1 Z is a Teichm¨ For any x ∈ Z / p n +1 Z with residue R n ( x ) ∈ F p , we have x p n = [ R n ( x )] = T ( R n ( x )). Introduce a ghost map w n : W n +1 ( A ) → A , w n ( a 0 , . . . , a n ) = a p n 0 + pa p n − 1 + · · · + p n a n 1

First ingredient: ghost map We cannot write down a formula for T : F p → Z / p n Z , but we do have a formula for the composition R n T Z / p n +1 Z → Z / p n +1 Z − − − − → F p − − − − uller representative iff x p n = x . Lemma. x ∈ Z / p n +1 Z is a Teichm¨ For any x ∈ Z / p n +1 Z with residue R n ( x ) ∈ F p , we have x p n = [ R n ( x )] = T ( R n ( x )). Introduce a ghost map w n : W n +1 ( A ) → A , w n ( a 0 , . . . , a n ) = a p n 0 + pa p n − 1 + · · · + p n a n 1 Lemma. Assume given functorial abelian group structures on W n ( A ) s.t. w n are additive. Then T : W n ( F p ) ∼ = Z / p n Z is additive.

First ingredient: ghost map We cannot write down a formula for T : F p → Z / p n Z , but we do have a formula for the composition R n T Z / p n +1 Z → Z / p n +1 Z − − − − → F p − − − − uller representative iff x p n = x . Lemma. x ∈ Z / p n +1 Z is a Teichm¨ For any x ∈ Z / p n +1 Z with residue R n ( x ) ∈ F p , we have x p n = [ R n ( x )] = T ( R n ( x )). Introduce a ghost map w n : W n +1 ( A ) → A , w n ( a 0 , . . . , a n ) = a p n 0 + pa p n − 1 + · · · + p n a n 1 Lemma. Assume given functorial abelian group structures on W n ( A ) s.t. w n are additive. Then T : W n ( F p ) ∼ = Z / p n Z is additive. Pf. The map W n ( Z / p n Z ) → W n ( F p ) is surjective, and w n for A = Z / p n Z is the composition T W n ( Z / p n Z ) − → Z / p n Z . − − − → W n ( F p ) − − − −

Second ingredient: recursive formula Now, to construct W n ( A ), use induction on n and the following observation:

Second ingredient: recursive formula Now, to construct W n ( A ), use induction on n and the following observation: Lemma. There exists a unique collection of universal polynomials c i ( − , − ), i ≥ 1, s.t. for any n and commuting x 0 , x 1 , we have n ( x 0 + x 1 ) p n = x p n 0 + x p n p i c i ( x 0 , x 1 ) p n − i . � (*) 1 + i =1

Second ingredient: recursive formula Now, to construct W n ( A ), use induction on n and the following observation: Lemma. There exists a unique collection of universal polynomials c i ( − , − ), i ≥ 1, s.t. for any n and commuting x 0 , x 1 , we have n ( x 0 + x 1 ) p n = x p n 0 + x p n p i c i ( x 0 , x 1 ) p n − i . � (*) 1 + i =1 Proof of Witt’s Theorem (sketch).

Second ingredient: recursive formula Now, to construct W n ( A ), use induction on n and the following observation: Lemma. There exists a unique collection of universal polynomials c i ( − , − ), i ≥ 1, s.t. for any n and commuting x 0 , x 1 , we have n ( x 0 + x 1 ) p n = x p n 0 + x p n p i c i ( x 0 , x 1 ) p n − i . � (*) 1 + i =1 Proof of Witt’s Theorem (sketch). We want to have a short exact sequence of abelian groups R n V 0 − − − − → W n ( A ) − − − − → W n +1 ( A ) − − − − → W 1 ( A ) = A − − − − → 0 ,

Second ingredient: recursive formula Now, to construct W n ( A ), use induction on n and the following observation: Lemma. There exists a unique collection of universal polynomials c i ( − , − ), i ≥ 1, s.t. for any n and commuting x 0 , x 1 , we have n ( x 0 + x 1 ) p n = x p n 0 + x p n p i c i ( x 0 , x 1 ) p n − i . � (*) 1 + i =1 Proof of Witt’s Theorem (sketch). We want to have a short exact sequence of abelian groups R n V 0 − − − − → W n ( A ) − − − − → W n +1 ( A ) − − − − → W 1 ( A ) = A − − − − → 0 , and we have W n +1 ∼ = A × W n ( A ) as sets.