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 Zp of p-adic integers. Recall that Zp = lim

R

←

Z/pnZ, R : Z/pn+1Z → Z/pnZ

Origins of Witt vectors: Teichm¨ uller

Consider the ring Zp of p-adic integers. Recall that Zp = lim

R

←

Z/pnZ, R : Z/pn+1Z → Z/pnZ and any a ∈ Zp is represented by a series a =

• i

aipi, with ai ∈ {0, . . . , p − 1}

Origins of Witt vectors: Teichm¨ uller

Consider the ring Zp of p-adic integers. Recall that Zp = lim

R

←

Z/pnZ, R : Z/pn+1Z → Z/pnZ and any a ∈ Zp is represented by a series a =

• i

aipi, with ai ∈ {0, . . . , p − 1}, or {1, . . . , p}

Origins of Witt vectors: Teichm¨ uller

Consider the ring Zp of p-adic integers. Recall that Zp = lim

R

←

Z/pnZ, R : Z/pn+1Z → Z/pnZ and any a ∈ Zp is represented by a series a =

• i

aipi, with ai ∈ {0, . . . , p − 1}, or {1, . . . , p}, or some other set of representatives of mod p residue classes x ∈ Fp = Z/pZ.

Origins of Witt vectors: Teichm¨ uller

Consider the ring Zp of p-adic integers. Recall that Zp = lim

R

←

Z/pnZ, R : Z/pn+1Z → Z/pnZ and any a ∈ Zp is represented by a series a =

• i

aipi, with ai ∈ {0, . . . , p − 1}, or {1, . . . , p}, or some other set of representatives of mod p residue classes x ∈ Fp = Z/pZ. Observation (Teichm¨ uller). There is a canonical choice of representatives [x] ∈ Zp for classes x ∈ Fp.

Origins of Witt vectors: Teichm¨ uller

Consider the ring Zp of p-adic integers. Recall that Zp = lim

R

←

Z/pnZ, R : Z/pn+1Z → Z/pnZ and any a ∈ Zp is represented by a series a =

• i

aipi, with ai ∈ {0, . . . , p − 1}, or {1, . . . , p}, or some other set of representatives of mod p residue classes x ∈ Fp = Z/pZ. Observation (Teichm¨ uller). There is a canonical choice of representatives [x] ∈ Zp for classes x ∈ Fp. Namely, for any n, the map (Z/pnZ)∗ → F∗

p admits a unique

splitting T : F∗

p → (Z/pnZ)∗. Set T(0) = 0 and [x] = T(x).

Origins of Witt vectors: Witt

We thus have a natural isomorphism of sets T q :

• i≥0

Fp ∼ = Zp, T q(x0, x1, . . . ) =

• i

[xi]pi Question: how to write down the ring operations?

Origins of Witt vectors: Witt

We thus have a natural isomorphism of sets T q :

• i≥0

Fp ∼ = Zp, T q(x0, x1, . . . ) =

• i

[xi]pi Question: how to write down the ring operations? For any n ≥ 1 and commutative ring A, denote by Wn(A) the set

• An. Let R : Wn+1(A) → Wn(A), V : Wn(A) → Wn+1(A) be the

maps R(a0, . . . , an) = a0, . . . , an−1, V (a0, . . . , an−1) = 0, a0, . . . , an−1

Origins of Witt vectors: Witt

We thus have a natural isomorphism of sets T q :

• i≥0

Fp ∼ = Zp, T q(x0, x1, . . . ) =

• i

[xi]pi Question: how to write down the ring operations? For any n ≥ 1 and commutative ring A, denote by Wn(A) the set

• An. Let R : Wn+1(A) → Wn(A), V : Wn(A) → Wn+1(A) be the

maps R(a0, . . . , an) = a0, . . . , an−1, V (a0, . . . , an−1) = 0, a0, . . . , an−1 Thm (Witt). There exists a unique set of functorial abelian groups structures on Wn(A), n ≥ 1 such that R, V , and T q : Wn(Fp) ∼ = Z/pnZ are additive.

Origins of Witt vectors: Witt

We thus have a natural isomorphism of sets T q :

• i≥0

Fp ∼ = Zp, T q(x0, x1, . . . ) =

• i

[xi]pi Question: how to write down the ring operations? For any n ≥ 1 and commutative ring A, denote by Wn(A) the set

• An. Let R : Wn+1(A) → Wn(A), V : Wn(A) → Wn+1(A) be the

maps R(a0, . . . , an) = a0, . . . , an−1, V (a0, . . . , an−1) = 0, a0, . . . , an−1 Thm (Witt). There exists a unique set of functorial abelian groups structures on Wn(A), n ≥ 1 such that R, V , and T q : Wn(Fp) ∼ = Z/pnZ are additive. Key idea of the theorem: functoriality.

First ingredient: ghost map

We cannot write down a formula for T : Fp → Z/pnZ, but we do have a formula for the composition Z/pn+1Z

Rn

− − − − → Fp

T

− − − − → Z/pn+1Z

First ingredient: ghost map

We cannot write down a formula for T : Fp → Z/pnZ, but we do have a formula for the composition Z/pn+1Z

Rn

− − − − → Fp

T

− − − − → Z/pn+1Z

• Lemma. x ∈ Z/pn+1Z is a Teichm¨

uller representative iff xpn = x. For any x ∈ Z/pn+1Z with residue Rn(x) ∈ Fp, we have xpn = [Rn(x)] = T(Rn(x)).

First ingredient: ghost map

We cannot write down a formula for T : Fp → Z/pnZ, but we do have a formula for the composition Z/pn+1Z

Rn

− − − − → Fp

T

− − − − → Z/pn+1Z

• Lemma. x ∈ Z/pn+1Z is a Teichm¨

uller representative iff xpn = x. For any x ∈ Z/pn+1Z with residue Rn(x) ∈ Fp, we have xpn = [Rn(x)] = T(Rn(x)). Introduce a ghost map wn : Wn+1(A) → A, wn(a0, . . . , an) = apn

0 + papn−1 1

+ · · · + pnan

First ingredient: ghost map

We cannot write down a formula for T : Fp → Z/pnZ, but we do have a formula for the composition Z/pn+1Z

Rn

− − − − → Fp

T

− − − − → Z/pn+1Z

• Lemma. x ∈ Z/pn+1Z is a Teichm¨

uller representative iff xpn = x. For any x ∈ Z/pn+1Z with residue Rn(x) ∈ Fp, we have xpn = [Rn(x)] = T(Rn(x)). Introduce a ghost map wn : Wn+1(A) → A, wn(a0, . . . , an) = apn

0 + papn−1 1

+ · · · + pnan

• Lemma. Assume given functorial abelian group structures on

Wn(A) s.t. wn are additive. Then T : Wn(Fp) ∼ = Z/pnZ is additive.

First ingredient: ghost map

We cannot write down a formula for T : Fp → Z/pnZ, but we do have a formula for the composition Z/pn+1Z

Rn

− − − − → Fp

T

− − − − → Z/pn+1Z

• Lemma. x ∈ Z/pn+1Z is a Teichm¨

uller representative iff xpn = x. For any x ∈ Z/pn+1Z with residue Rn(x) ∈ Fp, we have xpn = [Rn(x)] = T(Rn(x)). Introduce a ghost map wn : Wn+1(A) → A, wn(a0, . . . , an) = apn

0 + papn−1 1

+ · · · + pnan

• Lemma. Assume given functorial abelian group structures on

Wn(A) s.t. wn are additive. Then T : Wn(Fp) ∼ = Z/pnZ is additive.

• Pf. The map Wn(Z/pnZ) → Wn(Fp) is surjective, and wn for

A = Z/pnZ is the composition Wn(Z/pnZ) − − − − → Wn(Fp)

T

− − − − → Z/pnZ.

Second ingredient: recursive formula

Now, to construct Wn(A), use induction on n and the following

• bservation:

Second ingredient: recursive formula

Now, to construct Wn(A), use induction on n and the following

• bservation:
• Lemma. There exists a unique collection of universal polynomials

ci(−, −), i ≥ 1, s.t. for any n and commuting x0, x1, we have (*) (x0 + x1)pn = xpn

0 + xpn 1 + n

• i=1

pici(x0, x1)pn−i.

Second ingredient: recursive formula

Now, to construct Wn(A), use induction on n and the following

• bservation:
• Lemma. There exists a unique collection of universal polynomials

ci(−, −), i ≥ 1, s.t. for any n and commuting x0, x1, we have (*) (x0 + x1)pn = xpn

0 + xpn 1 + n

• i=1

pici(x0, x1)pn−i. Proof of Witt’s Theorem (sketch).

Second ingredient: recursive formula

Now, to construct Wn(A), use induction on n and the following

• bservation:
• Lemma. There exists a unique collection of universal polynomials

ci(−, −), i ≥ 1, s.t. for any n and commuting x0, x1, we have (*) (x0 + x1)pn = xpn

0 + xpn 1 + n

• i=1

pici(x0, x1)pn−i. Proof of Witt’s Theorem (sketch). We want to have a short exact sequence of abelian groups 0 − − − − → Wn(A)

V

− − − − → Wn+1(A)

Rn

− − − − → W1(A) = A − − − − → 0,

Second ingredient: recursive formula

Now, to construct Wn(A), use induction on n and the following

• bservation:
• Lemma. There exists a unique collection of universal polynomials

ci(−, −), i ≥ 1, s.t. for any n and commuting x0, x1, we have (*) (x0 + x1)pn = xpn

0 + xpn 1 + n

• i=1

pici(x0, x1)pn−i. Proof of Witt’s Theorem (sketch). We want to have a short exact sequence of abelian groups 0 − − − − → Wn(A)

V

− − − − → Wn+1(A)

Rn

− − − − → W1(A) = A − − − − → 0, and we have Wn+1 ∼ = A × Wn(A) as sets.

Second ingredient: recursive formula

Now, to construct Wn(A), use induction on n and the following

• bservation:
• Lemma. There exists a unique collection of universal polynomials

ci(−, −), i ≥ 1, s.t. for any n and commuting x0, x1, we have (*) (x0 + x1)pn = xpn

0 + xpn 1 + n

• i=1

pici(x0, x1)pn−i. Proof of Witt’s Theorem (sketch). We want to have a short exact sequence of abelian groups 0 − − − − → Wn(A)

V

− − − − → Wn+1(A)

Rn

− − − − → W1(A) = A − − − − → 0, and we have Wn+1 ∼ = A × Wn(A) as sets. Thus the abelian group structure must be a0, b0 + a1, b1 = a0 + a1, b0 + b1 + c q(a0, a1) for some cocycle c q.

Second ingredient: recursive formula

Now, to construct Wn(A), use induction on n and the following

• bservation:
• Lemma. There exists a unique collection of universal polynomials

ci(−, −), i ≥ 1, s.t. for any n and commuting x0, x1, we have (*) (x0 + x1)pn = xpn

0 + xpn 1 + n

• i=1

pici(x0, x1)pn−i. Proof of Witt’s Theorem (sketch). We want to have a short exact sequence of abelian groups 0 − − − − → Wn(A)

V

− − − − → Wn+1(A)

Rn

− − − − → W1(A) = A − − − − → 0, and we have Wn+1 ∼ = A × Wn(A) as sets. Thus the abelian group structure must be a0, b0 + a1, b1 = a0 + a1, b0 + b1 + c q(a0, a1) for some cocycle c q. Then wn is additive iff c q(−, −) satisfy (*).

Non-commutative recursive formula: naive version

• Question. Can we refine the recursive formula (*) to make

c q(−, −) non-commutative?

Non-commutative recursive formula: naive version

• Question. Can we refine the recursive formula (*) to make

c q(−, −) non-commutative? Ideally, we would like to have non-commutative polynomials ci(−, −), i ≥ 1 of degrees pi such that for any n ≥ 1 (x0 + x1)⊗pn = x⊗pn + x⊗pn

1

+

n

• i=1

pici(x0, x1)⊗pn−i ∈ T pn(x0, x1), where T pn(x0, x1) is the component of degree pn of the free associative Z-algebra T q(x0, x1) on variables x0, x1.

Non-commutative recursive formula: naive version

• Question. Can we refine the recursive formula (*) to make

c q(−, −) non-commutative? Ideally, we would like to have non-commutative polynomials ci(−, −), i ≥ 1 of degrees pi such that for any n ≥ 1 (x0 + x1)⊗pn = x⊗pn + x⊗pn

1

+

n

• i=1

pici(x0, x1)⊗pn−i ∈ T pn(x0, x1), where T pn(x0, x1) is the component of degree pn of the free associative Z-algebra T q(x0, x1) on variables x0, x1. This is not possible “as is”: already for n = 1, the non-commutative polynomial (x0 + x1)⊗p − x⊗p − x⊗p

1

is not divisible by p. Something has to be modified.

Non-commutative recursive formula: naive version

• Question. Can we refine the recursive formula (*) to make

c q(−, −) non-commutative? Ideally, we would like to have non-commutative polynomials ci(−, −), i ≥ 1 of degrees pi such that for any n ≥ 1 (x0 + x1)⊗pn = x⊗pn + x⊗pn

1

+

n

• i=1

pici(x0, x1)⊗pn−i ∈ T pn(x0, x1), where T pn(x0, x1) is the component of degree pn of the free associative Z-algebra T q(x0, x1) on variables x0, x1. This is not possible “as is”: already for n = 1, the non-commutative polynomial (x0 + x1)⊗p − x⊗p − x⊗p

1

is not divisible by p. Something has to be modified. This “something” turns out to be the coefficient pi.

Non-commutative recursive formula: correct version

• Notation. For any free Z-module N and integer n ≥ 1, denote by

σ : N⊗n → N⊗n the permutation of order n.

Non-commutative recursive formula: correct version

• Notation. For any free Z-module N and integer n ≥ 1, denote by

σ : N⊗n → N⊗n the permutation of order n. In particular, T n(x0, x1) = Z[x0, x1]⊗n, so σ acts on it.

Non-commutative recursive formula: correct version

• Notation. For any free Z-module N and integer n ≥ 1, denote by

σ : N⊗n → N⊗n the permutation of order n. In particular, T n(x0, x1) = Z[x0, x1]⊗n, so σ acts on it.

• Thm. There exist non-commutative polynomials ci(x0, x1), i ≥ 1 of

degrees pi s.t. for any n ≥ 1 (x0+x1)⊗pn = x⊗pn +x⊗pn

1

+

n

• i=1

(id +σ+· · ·+σpi−1)ci(x0, x1)⊗pn−i.

Non-commutative recursive formula: correct version

• Notation. For any free Z-module N and integer n ≥ 1, denote by

σ : N⊗n → N⊗n the permutation of order n. In particular, T n(x0, x1) = Z[x0, x1]⊗n, so σ acts on it.

• Thm. There exist non-commutative polynomials ci(x0, x1), i ≥ 1 of

degrees pi s.t. for any n ≥ 1 (x0+x1)⊗pn = x⊗pn +x⊗pn

1

+

n

• i=1

(id +σ+· · ·+σpi−1)ci(x0, x1)⊗pn−i. This has a direct combinatorial proof, due to Lars Hesselholt (rather non-trivial). Let me present a more conceptual proof.

The functors Qn

• Def. For a free Z-module N and integer n ≥ 1, let Qn(N) be the

cokernel of the map

• N⊗pn

σ id +σ+···+σpn−1

− − − − − − − − − − →

• N⊗pnσ

The functors Qn

• Def. For a free Z-module N and integer n ≥ 1, let Qn(N) be the

cokernel of the map

• N⊗pn

σ id +σ+···+σpn−1

− − − − − − − − − − →

• N⊗pnσ
• Remark. To construct cn(−, −), it suffices to prove that

(x0+x1)⊗pn = x⊗pn +x⊗pn

1

+

n−1

• i=1

(id +σ+· · ·+σpi−1)ci(x0, x1)⊗pn−i in the group Qn(Z[x0, x1]).

The functors Qn

• Def. For a free Z-module N and integer n ≥ 1, let Qn(N) be the

cokernel of the map

• N⊗pn

σ id +σ+···+σpn−1

− − − − − − − − − − →

• N⊗pnσ
• Remark. To construct cn(−, −), it suffices to prove that

(x0+x1)⊗pn = x⊗pn +x⊗pn

1

+

n−1

• i=1

(id +σ+· · ·+σpi−1)ci(x0, x1)⊗pn−i in the group Qn(Z[x0, x1]).

• Lemma. The correspondence x → x⊗pn factors as

N − − − − → N/pN

T

− − − − → Qn(N) for some functorial map T.

• Pf. Need to show that x⊗pn

− (x0 + px1)⊗pn lies in the image of (id +σ + · · · + σpn−1). Direct computation.

Polynomial Witt vectors

• Prop. There exist a unique functor Wn : Fp-Vect → Z-mod s.t.

Qn(N) ∼ = Wn(N/pN).

Polynomial Witt vectors

• Prop. There exist a unique functor Wn : Fp-Vect → Z-mod s.t.

Qn(N) ∼ = Wn(N/pN). (That is, Qn(N) only depends on N/pN.)

Polynomial Witt vectors

• Prop. There exist a unique functor Wn : Fp-Vect → Z-mod s.t.

Qn(N) ∼ = Wn(N/pN). (That is, Qn(N) only depends on N/pN.)

• Pf. Need to show that for two maps a0, a1 : N → N s.t. a0 = a1

mod p, we have Qn(a0) = Qn(a1).

Polynomial Witt vectors

• Prop. There exist a unique functor Wn : Fp-Vect → Z-mod s.t.

Qn(N) ∼ = Wn(N/pN). (That is, Qn(N) only depends on N/pN.)

• Pf. Need to show that for two maps a0, a1 : N → N s.t. a0 = a1

mod p, we have Qn(a0) = Qn(a1). But Qn(a) is induced by a⊗pn, so this immediately follows from the Lemma.

Polynomial Witt vectors

• Prop. There exist a unique functor Wn : Fp-Vect → Z-mod s.t.

Qn(N) ∼ = Wn(N/pN). (That is, Qn(N) only depends on N/pN.)

• Pf. Need to show that for two maps a0, a1 : N → N s.t. a0 = a1

mod p, we have Qn(a0) = Qn(a1). But Qn(a) is induced by a⊗pn, so this immediately follows from the Lemma.

• Def. Wn(M) are the groups of polynomial Witt vectors of an

Fp-vector space M.

Polynomial Witt vectors

• Prop. There exist a unique functor Wn : Fp-Vect → Z-mod s.t.

Qn(N) ∼ = Wn(N/pN). (That is, Qn(N) only depends on N/pN.)

• Pf. Need to show that for two maps a0, a1 : N → N s.t. a0 = a1

mod p, we have Qn(a0) = Qn(a1). But Qn(a) is induced by a⊗pn, so this immediately follows from the Lemma.

• Def. Wn(M) are the groups of polynomial Witt vectors of an

Fp-vector space M. Reasons for terminology: Wn(M) behave similarly to Wn(A).

Polynomial Witt vectors

• Prop. There exist a unique functor Wn : Fp-Vect → Z-mod s.t.

Qn(N) ∼ = Wn(N/pN). (That is, Qn(N) only depends on N/pN.)

• Pf. Need to show that for two maps a0, a1 : N → N s.t. a0 = a1

mod p, we have Qn(a0) = Qn(a1). But Qn(a) is induced by a⊗pn, so this immediately follows from the Lemma.

• Def. Wn(M) are the groups of polynomial Witt vectors of an

Fp-vector space M. Reasons for terminology: Wn(M) behave similarly to Wn(A). We have the Teichm¨ uller map T : M → Wn(M).

Polynomial Witt vectors

• Prop. There exist a unique functor Wn : Fp-Vect → Z-mod s.t.

Qn(N) ∼ = Wn(N/pN). (That is, Qn(N) only depends on N/pN.)

• Pf. Need to show that for two maps a0, a1 : N → N s.t. a0 = a1

mod p, we have Qn(a0) = Qn(a1). But Qn(a) is induced by a⊗pn, so this immediately follows from the Lemma.

• Def. Wn(M) are the groups of polynomial Witt vectors of an

Fp-vector space M. Reasons for terminology: Wn(M) behave similarly to Wn(A). We have the Teichm¨ uller map T : M → Wn(M). We also have V : Wn−1(M⊗p) → Wn(M) given by V = id +σ + · · · + σ⊗p−1

Polynomial Witt vectors

• Prop. There exist a unique functor Wn : Fp-Vect → Z-mod s.t.

Qn(N) ∼ = Wn(N/pN). (That is, Qn(N) only depends on N/pN.)

• Pf. Need to show that for two maps a0, a1 : N → N s.t. a0 = a1

mod p, we have Qn(a0) = Qn(a1). But Qn(a) is induced by a⊗pn, so this immediately follows from the Lemma.

• Def. Wn(M) are the groups of polynomial Witt vectors of an

Fp-vector space M. Reasons for terminology: Wn(M) behave similarly to Wn(A). We have the Teichm¨ uller map T : M → Wn(M). We also have V : Wn−1(M⊗p) → Wn(M) given by V = id +σ + · · · + σ⊗p−1 (σ is induced by σpn−1 on N⊗pn ∼ = (N⊗p)⊗pn−1, where M ∼ = N/pN.)

Restriction maps

• Def. For a free Z-module N and integer n ≥ 0, Q′

n(N) is the

cokernel of the map

• N⊗pn

σ id +σ+···+σpn+1−1

− − − − − − − − − − − →

• N⊗pnσ

Restriction maps

• Def. For a free Z-module N and integer n ≥ 0, Q′

n(N) is the

cokernel of the map

• N⊗pn

σ id +σ+···+σpn+1−1

− − − − − − − − − − − →

• N⊗pnσ

Then as before, we have a map T ′ : N/pN → Q′

n(N), and

Q′

n(N) = W ′ n(N/pN) for a functor W ′ n : Fp-Vect → Z-mod.

Restriction maps

• Def. For a free Z-module N and integer n ≥ 0, Q′

n(N) is the

cokernel of the map

• N⊗pn

σ id +σ+···+σpn+1−1

− − − − − − − − − − − →

• N⊗pnσ

Then as before, we have a map T ′ : N/pN → Q′

n(N), and

Q′

n(N) = W ′ n(N/pN) for a functor W ′ n : Fp-Vect → Z-mod.

Moreover, id +σ + · · · + σpn+1−1 = p(id +σ + · · · + σpn−1), so we have a natural map R : W ′

n(M) → Wn(M).

Restriction maps

• Def. For a free Z-module N and integer n ≥ 0, Q′

n(N) is the

cokernel of the map

• N⊗pn

σ id +σ+···+σpn+1−1

− − − − − − − − − − − →

• N⊗pnσ

Then as before, we have a map T ′ : N/pN → Q′

n(N), and

Q′

n(N) = W ′ n(N/pN) for a functor W ′ n : Fp-Vect → Z-mod.

Moreover, id +σ + · · · + σpn+1−1 = p(id +σ + · · · + σpn−1), so we have a natural map R : W ′

n(M) → Wn(M).

• Def. A map c : N → (N⊗p)σ is standard if c = T after projecting

to Q1(N) ∼ = N/pN.

Restriction maps

• Def. For a free Z-module N and integer n ≥ 0, Q′

n(N) is the

cokernel of the map

• N⊗pn

σ id +σ+···+σpn+1−1

− − − − − − − − − − − →

• N⊗pnσ

Then as before, we have a map T ′ : N/pN → Q′

n(N), and

Q′

n(N) = W ′ n(N/pN) for a functor W ′ n : Fp-Vect → Z-mod.

Moreover, id +σ + · · · + σpn+1−1 = p(id +σ + · · · + σpn−1), so we have a natural map R : W ′

n(M) → Wn(M).

• Def. A map c : N → (N⊗p)σ is standard if c = T after projecting

to Q1(N) ∼ = N/pN.

• Prop. For any standard map c and any n ≥ 1, the map

c⊗pn : Q′

n(N) → Qn+1(N)

is an isomorphism, it does not depend on c, and c⊗pn ◦ T ′ = T.

Exact sequences

As a corollary of the Proposition, we have W ′

n(M) ∼

= Wn+1(M) for any Fp-vector space M, so we get a functorial map R : Wn+1(M) → Wn(M).

Exact sequences

As a corollary of the Proposition, we have W ′

n(M) ∼

= Wn+1(M) for any Fp-vector space M, so we get a functorial map R : Wn+1(M) → Wn(M). One can show that it fits into an exact sequence M⊗pn

V n

− − − − → Wn+1(M)

R

− − − − → Wn(M) − − − − → 0.

Exact sequences

As a corollary of the Proposition, we have W ′

n(M) ∼

= Wn+1(M) for any Fp-vector space M, so we get a functorial map R : Wn+1(M) → Wn(M). One can show that it fits into an exact sequence M⊗pn

V n

− − − − → Wn+1(M)

R

− − − − → Wn(M) − − − − → 0. Unlike the commutative case, this sequence is not exact on the left! In fact, the kernel of R is the space M⊗pn

σ

(the pn-th cyclic power of the vector space M).

Exact sequences

As a corollary of the Proposition, we have W ′

n(M) ∼

= Wn+1(M) for any Fp-vector space M, so we get a functorial map R : Wn+1(M) → Wn(M). One can show that it fits into an exact sequence M⊗pn

V n

− − − − → Wn+1(M)

R

− − − − → Wn(M) − − − − → 0. Unlike the commutative case, this sequence is not exact on the left! In fact, the kernel of R is the space M⊗pn

σ

(the pn-th cyclic power of the vector space M). On the other hand, we have an exact sequence Wn(M⊗p)

V

− − − − → Wn+1(M)

Rn

− − − − → W1(M) = M − − − − → 0.

Exact sequences

As a corollary of the Proposition, we have W ′

n(M) ∼

= Wn+1(M) for any Fp-vector space M, so we get a functorial map R : Wn+1(M) → Wn(M). One can show that it fits into an exact sequence M⊗pn

V n

− − − − → Wn+1(M)

R

− − − − → Wn(M) − − − − → 0. Unlike the commutative case, this sequence is not exact on the left! In fact, the kernel of R is the space M⊗pn

σ

(the pn-th cyclic power of the vector space M). On the other hand, we have an exact sequence Wn(M⊗p)

V

− − − − → Wn+1(M)

Rn

− − − − → W1(M) = M − − − − → 0. (Again, the kernel is actually Wn(M⊗p)σ.)

Exact sequences

As a corollary of the Proposition, we have W ′

n(M) ∼

= Wn+1(M) for any Fp-vector space M, so we get a functorial map R : Wn+1(M) → Wn(M). One can show that it fits into an exact sequence M⊗pn

V n

− − − − → Wn+1(M)

R

− − − − → Wn(M) − − − − → 0. Unlike the commutative case, this sequence is not exact on the left! In fact, the kernel of R is the space M⊗pn

σ

(the pn-th cyclic power of the vector space M). On the other hand, we have an exact sequence Wn(M⊗p)

V

− − − − → Wn+1(M)

Rn

− − − − → W1(M) = M − − − − → 0. (Again, the kernel is actually Wn(M⊗p)σ.) A similar sequence exists for V i and Rn+1−i, 2 ≤ i ≤ n − 1.

Proof of the recursive formula

The upshot: we have a series of functors Wn : Fp-Vect → Z-mod that behave exactly like Wn(A): there are additive maps F, V , and the Teichm¨ uller splitting map T : M → Wn(M).

Proof of the recursive formula

The upshot: we have a series of functors Wn : Fp-Vect → Z-mod that behave exactly like Wn(A): there are additive maps F, V , and the Teichm¨ uller splitting map T : M → Wn(M). Proof of the Theorem.

Proof of the recursive formula

The upshot: we have a series of functors Wn : Fp-Vect → Z-mod that behave exactly like Wn(A): there are additive maps F, V , and the Teichm¨ uller splitting map T : M → Wn(M). Proof of the Theorem. Assume by induction that we have ci(−, −) for i ≤ n − 1. Then projecting the main equality for n − 1 to Q′

n−1(Z[x0, x1]), we see that

T ′(x0 + x1) = T ′(x0) + T ′(x1) +

n−1

• i=1

V i(T ′(ci(x0, x1))).

Proof of the recursive formula

The upshot: we have a series of functors Wn : Fp-Vect → Z-mod that behave exactly like Wn(A): there are additive maps F, V , and the Teichm¨ uller splitting map T : M → Wn(M). Proof of the Theorem. Assume by induction that we have ci(−, −) for i ≤ n − 1. Then projecting the main equality for n − 1 to Q′

n−1(Z[x0, x1]), we see that

T ′(x0 + x1) = T ′(x0) + T ′(x1) +

n−1

• i=1

V i(T ′(ci(x0, x1))). However, Q′

n−1(Z[x0, x1]) ∼

= Qn(Z[x0, x1]) by Proposition, and this implies that (x0+x1)⊗pn = x⊗pn +x⊗pn

1

+

n−1

• i=1

(id +σ+· · ·+σpi−1)ci(x0, x1)⊗pn−i in the group Qn(Z[x0, x1]).

Proof of the recursive formula

The upshot: we have a series of functors Wn : Fp-Vect → Z-mod that behave exactly like Wn(A): there are additive maps F, V , and the Teichm¨ uller splitting map T : M → Wn(M). Proof of the Theorem. Assume by induction that we have ci(−, −) for i ≤ n − 1. Then projecting the main equality for n − 1 to Q′

n−1(Z[x0, x1]), we see that

T ′(x0 + x1) = T ′(x0) + T ′(x1) +

n−1

• i=1

V i(T ′(ci(x0, x1))). However, Q′

n−1(Z[x0, x1]) ∼

= Qn(Z[x0, x1]) by Proposition, and this implies that (x0+x1)⊗pn = x⊗pn +x⊗pn

1

+

n−1

• i=1

(id +σ+· · ·+σpi−1)ci(x0, x1)⊗pn−i in the group Qn(Z[x0, x1]). As we have remarked, this implies the existence of cn(−, −) and proves the Theorem.

Algebras

Assume now given an associative unital Fp-algebra A. Can we define Witt vectors of A?

Algebras

Assume now given an associative unital Fp-algebra A. Can we define Witt vectors of A? We can treat A as a vector space and consider polynomial Witt vectors Wn(A), but if A is commutative, this is different from the classical Witt vector W cl

n (A).

Algebras

Assume now given an associative unital Fp-algebra A. Can we define Witt vectors of A? We can treat A as a vector space and consider polynomial Witt vectors Wn(A), but if A is commutative, this is different from the classical Witt vector W cl

n (A). E.g. W cl n+1(A) is an extension of

Wn(A) by A, while Wn+1(A) is an extension of Wn(A) by the cyclic power A⊗pn

σ

.

Algebras

Assume now given an associative unital Fp-algebra A. Can we define Witt vectors of A? We can treat A as a vector space and consider polynomial Witt vectors Wn(A), but if A is commutative, this is different from the classical Witt vector W cl

n (A). E.g. W cl n+1(A) is an extension of

Wn(A) by A, while Wn+1(A) is an extension of Wn(A) by the cyclic power A⊗pn

σ

. It turns out that the correct thing to generalize is not A itself but its Hochschild homology HH q(A).

Algebras

Assume now given an associative unital Fp-algebra A. Can we define Witt vectors of A? We can treat A as a vector space and consider polynomial Witt vectors Wn(A), but if A is commutative, this is different from the classical Witt vector W cl

n (A). E.g. W cl n+1(A) is an extension of

Wn(A) by A, while Wn+1(A) is an extension of Wn(A) by the cyclic power A⊗pn

σ

. It turns out that the correct thing to generalize is not A itself but its Hochschild homology HH q(A).

• Def. The Hochschild homology complex CH q(A) is the complex

. . .

b

− − − − → A⊗n

b

− − − − → . . .

b

− − − − → A⊗2

b

− − − − → A (we will not need the exact form of the differential b).

Hochschild-Witt homology

• Resume. One can define functorial differentials

bn : Wn(A⊗ q+1) → Wn(A⊗ q) giving Hochschild-Witt complexes WnCH q(A) with Hochschild-Witt homology WnHH q(A).

Hochschild-Witt homology

• Resume. One can define functorial differentials

bn : Wn(A⊗ q+1) → Wn(A⊗ q) giving Hochschild-Witt complexes WnCH q(A) with Hochschild-Witt homology WnHH q(A). There are maps R : Wn+1CH q(A) → WnCH q(A), V : WnCH q(A) → Wn+1CH q(A).

Hochschild-Witt homology

• Resume. One can define functorial differentials

bn : Wn(A⊗ q+1) → Wn(A⊗ q) giving Hochschild-Witt complexes WnCH q(A) with Hochschild-Witt homology WnHH q(A). There are maps R : Wn+1CH q(A) → WnCH q(A), V : WnCH q(A) → Wn+1CH q(A). W1CH q(A) is the Hochschild complex CH q(A); the rest are new.

Hochschild-Witt homology

• Resume. One can define functorial differentials

bn : Wn(A⊗ q+1) → Wn(A⊗ q) giving Hochschild-Witt complexes WnCH q(A) with Hochschild-Witt homology WnHH q(A). There are maps R : Wn+1CH q(A) → WnCH q(A), V : WnCH q(A) → Wn+1CH q(A). W1CH q(A) is the Hochschild complex CH q(A); the rest are new. The construction is somewhat technical (uses Connes’ cyclic category Λ and the so-called “trace functors”).

Hochschild-Witt homology

• Resume. One can define functorial differentials

bn : Wn(A⊗ q+1) → Wn(A⊗ q) giving Hochschild-Witt complexes WnCH q(A) with Hochschild-Witt homology WnHH q(A). There are maps R : Wn+1CH q(A) → WnCH q(A), V : WnCH q(A) → Wn+1CH q(A). W1CH q(A) is the Hochschild complex CH q(A); the rest are new. The construction is somewhat technical (uses Connes’ cyclic category Λ and the so-called “trace functors”). In degree 0, HH0(A) = A/[A, A] (the quotient by the subspace spanned by commutators).

Hochschild-Witt homology

• Resume. One can define functorial differentials

bn : Wn(A⊗ q+1) → Wn(A⊗ q) giving Hochschild-Witt complexes WnCH q(A) with Hochschild-Witt homology WnHH q(A). There are maps R : Wn+1CH q(A) → WnCH q(A), V : WnCH q(A) → Wn+1CH q(A). W1CH q(A) is the Hochschild complex CH q(A); the rest are new. The construction is somewhat technical (uses Connes’ cyclic category Λ and the so-called “trace functors”). In degree 0, HH0(A) = A/[A, A] (the quotient by the subspace spanned by commutators). Then WnHH0(A) fit into exact sequences HH0(A)

V n

− − − − → Wn+1HH0(A)

R

− − − − → WnHH0(A) − − − − → 0 and coincide with the groups W H

q (A) of “non-commutative Witt

vectors” discovered by Hesselholt.

The commutative case

As an illustration, consider the commutative case.

The commutative case

As an illustration, consider the commutative case. If A is commutative, then HH0(A) ∼ = A.

The commutative case

As an illustration, consider the commutative case. If A is commutative, then HH0(A) ∼ = A. If moreover X = SpecA is smooth, we have the Hochschild-Kostant-Rosenberg isomorphism HHi(A) ∼ = H0(X, Ωi

X)

(the spaces of degree-i differential forms).

The commutative case

As an illustration, consider the commutative case. If A is commutative, then HH0(A) ∼ = A. If moreover X = SpecA is smooth, we have the Hochschild-Kostant-Rosenberg isomorphism HHi(A) ∼ = H0(X, Ωi

X)

(the spaces of degree-i differential forms).

• Thm. In the assumptions above, we have natural isomorphisms

WnHHi(A) ∼ = H0(X, WnΩi

X),

where W qΩ q

X is the “de Rham-Witt complex” of X introduced by

Deligne and Illusie.

The commutative case

As an illustration, consider the commutative case. If A is commutative, then HH0(A) ∼ = A. If moreover X = SpecA is smooth, we have the Hochschild-Kostant-Rosenberg isomorphism HHi(A) ∼ = H0(X, Ωi

X)

(the spaces of degree-i differential forms).

• Thm. In the assumptions above, we have natural isomorphisms

WnHHi(A) ∼ = H0(X, WnΩi

X),

where W qΩ q

X is the “de Rham-Witt complex” of X introduced by

Deligne and Illusie. Since de Rham-Witt complex computes cristalline cohomology, our Hochschild-Witt Homology gives a non-commutative generalization

• f cristalline cohomology theory.

Thank you for your attention!