Cornell 2017

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Function-field symmetric functions:

In search of an F

^q

[T ]-combinatorics

Darij Grinberg (UMN)

27 February 2017, Cornell

slides:

http://www.cip.ifi.lmu.de/~grinberg/algebra/

cornell-feb17.pdf

preprint (WIP, and currently a mess):

http:

//www.cip.ifi.lmu.de/~grinberg/algebra/schur-ore.pdf

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Symmetric functions and Witt vectors

The connection between symmetric functions and (big) Witt vectors is due to Cartier around 1970 (vaguely; made explicit by Reutenauer in 1995), and can be used to the benefit of either.

Modern references: e.g., Hazewinkel’sWitt vectors, part 1 (arXiv:0804.3888v1, see alsoerrata), and works of James Borger (mainlyarXiv:0801.1691v6, as well as

arXiv:math/0407227v1 joint with Wieland).

Let N+={1,2,3, . . .}. The(big) Witt vector functor is a functor W :CRing→CRing, sending any commutative ring A to a new commutative ringW(A) with some extra

structure.

Note thatW(A) is a ring, not anA-algebra.

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Symmetric functions and Witt vectors

The connection between symmetric functions and (big) Witt vectors is due to Cartier around 1970 (vaguely; made explicit by Reutenauer in 1995), and can be used to the benefit of either.

Modern references: e.g., Hazewinkel’sWitt vectors, part 1 (arXiv:0804.3888v1, see alsoerrata), and works of James Borger (mainlyarXiv:0801.1691v6, as well as

arXiv:math/0407227v1 joint with Wieland).

Let N+={1,2,3, . . .}. The(big) Witt vector functor is a functor W :CRing→CRing, sending any commutative ring A to a new commutative ringW(A) with some extra

structure.

Note thatW(A) is a ring, not anA-algebra.

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Definition of Witt vectors, 1: ghost maps

Let Abe a commutative ring.

We abbreviate a family (ak)_k∈_N₊ ∈A^N⁺ asa. Similarly for other letters.

For eachn∈N+, define a map wn:A^N⁺ →A by wn(a) =X

d|n

da^n/d_d .

The mapwnis called the n-th ghost projection.

Examples:

w1 =a1.

Ifp is a prime, thenw_p=a₁^p+pa_p. w₆ =a⁶₁+ 2a³₂+ 3a²₃+ 6a₆.

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We abbreviate a family (a_k)_k∈

N+ ∈A^N⁺ asa. Similarly for other letters.

For eachn∈N+, define a map wn:A^N⁺ →A by w_n(a) =X

d|n

da^n/d_d .

Let w :A^N⁺ →A^N⁺ be the map given by w(a) = (wn(a))_n∈_N₊. We callw theghost map.

This ghost map w is not linear and in general not injective or surjective. However, its image turns out to be a subring of A^N⁺. It is called thering of ghost-Witt vectors.

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We abbreviate a family (a_k)_k∈

N+ ∈A^N⁺ asa. Similarly for other letters.

For eachn∈N+, define a map wn:A^N⁺ →A by w_n(a) =X

d|n

da^n/d_d .

Let w :A^N⁺ →A^N⁺ be the map given by w(a) = (wn(a))_n∈_N₊. We callw theghost map.

This ghost map w is not linear and in general not injective or surjective. However, its image turns out to be a subring of

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Definition of Witt vectors, 2: addition

For example, for anya,b∈A^N⁺, we have

w(a) +w(b) =w(c) for somec∈A^N⁺. How to compute thisc ?

Good news:

w is injective if Ais torsionfree (asZ-module).

w is bijective if Ais a Q-vector space.

Hence, we can compute cback fromw(c) by recursion (coordinate by coordinate). Miraculously, the denominators vanish.

Examples:

w1(c) =w1(a) +w1(b) ⇐⇒ c1=a1+b1. w₂(c) =w₂(a) +w₂(b) ⇐⇒

c₁²+ 2c2 = a₁²+ 2a2

+ b₁²+ 2b2

naturality

⇐⇒ c2 =a2+b2+1

2

a²₁+b²₁−(a1+b1)²

, and the RHS is indeed a Z-polynomial.

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Good news:

Examples:

w1(c) =w1(a) +w1(b) ⇐⇒c1=a1+b1. w₂(c) =w₂(a) +w₂(b) ⇐⇒

c₁²+ 2c2 = a₁²+ 2a2

+ b₁²+ 2b2

naturality

⇐⇒

c2 =a2+b2+1

a²+b²−(a1+b1)²

, and the RHS is

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Good news:

Examples:

w1(c) =w1(a) +w1(b) ⇐⇒c1=a1+b1. w₂(c) =w₂(a) +w₂(b) ⇐⇒

c₁²+ 2c2 = a₁²+ 2a2

+ b₁²+ 2b2

naturality

⇐⇒

c2 =a2+b2+1 2

a²₁+b²₁−(a1+b1)²

, and the RHS is indeed a Z-polynomial.

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Definition of Witt vectors, 3: W(A)

Let’s make a new ring out of this: We defineW(A) to be the ring that equals A^N⁺ as a set, but whose ring structure is such that W :CRing→CRing is a functor, andw is a natural (in A) ring homomorphism fromW(A) toA^N⁺.

This looks abstract and confusing, but the underlying idea is simple: Define addition on W(A) so that

w_n(a+b) =w_n(a) +w_n(b) for all n.

Thus, a+b is the cfrom last page.

Functoriality is needed, because there might be several choices for a given A (ifA is not torsionfree), but only one consistent choice for all rings A. Functoriality forces us to pick the consistent choice.

Ifa∈W(A), then the an are called the Witt coordinates of a, while the w_n(a) are called the ghost coordinates ofa.

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Ifa∈W(A), then the an are called the Witt coordinates of a, while the w_n(a) are called the ghost coordinates of a.

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Ifa∈W(A), then the an are called the Witt coordinates of a, while the w (a) are called the ghost coordinates of a.

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Definition of Witt vectors, 4: coda

The ring W(A) is called thering of (big) Witt vectors over A.

The functorCRing →CRing, A7→W(A) is called the(big) Witt vector functor.

For any given primep, there is a canonical quotient Wp(A) of W(A) called thering of p-typical Witt vectors of A. Number theorists usually care about the latter ring. For example, Wp(Fp) =Zp (thep-adics). We have nothing to say about it here.

W(A) comes with more structure: Frobenius and

Verschiebung endomorphisms, a comonad comultiplication map W(A)→W(W(A)), etc.

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Definition of Witt vectors, 4: coda

The ring W(A) is called thering of (big) Witt vectors over A.

The functorCRing →CRing, A7→W(A) is called the(big) Witt vector functor.

For any given primep, there is a canonical quotient Wp(A) of W(A) called thering of p-typical Witt vectors of A. Number theorists usually care about the latter ring. For example, Wp(Fp) =Zp (thep-adics). We have nothing to say about it here.

W(A) comes with more structure: Frobenius and

Verschiebung endomorphisms, a comonad comultiplication map W(A)→W(W(A)), etc.

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Avatars of Witt vectors, 1: Power series

There are some equivalent ways to define W(A). Let me show two.

One is the Grothendieck construction using power series (see, again, Hazewinkel, or Rabinoff’s arXiv:1409.7445):

Let Λ(A) be the topological ring defined as follows:

As topological spaces, Λ(A) = 1 +tA[[t]] = {power series with constant term 1}.

Addition + in Λ(A) is multiplication of power series.b Multiplicationb·in Λ(A) is given by

(1−at)b·(1−bt) = 1−abt

(and distributivity and continuity, and naturality inA).

Canonical ring isomorphism

W(A)→Λ(A), a7→

∞

Y

n=1

(1−a_ntⁿ).

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Avatars of Witt vectors, 1: Power series

There are some equivalent ways to define W(A). Let me show two.

One is the Grothendieck construction using power series (see, again, Hazewinkel, or Rabinoff’s arXiv:1409.7445):

Let Λ(A) be the topological ring defined as follows:

As topological spaces, Λ(A) = 1 +tA[[t]] = {power series with constant term 1}.

Addition + in Λ(A) is multiplication of power series.b Multiplicationb·in Λ(A) is given by

(1−at)b·(1−bt) = 1−abt

(and distributivity and continuity, and naturality inA).

Canonical ring isomorphism

W(A)→Λ(A), a7→

∞

Y(1−a tⁿ).

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Avatars of Witt vectors, 2: Characters of Λ (virtual alphabets)

Here is another: Let Λ be the Hopf algebra of symmetric functions over Z. (No direct relation to Λ(A); just traditional notations clashing.)

Define ring Alg(Λ,A) as follows:

As set, Alg(Λ,A) ={algebra homomorphisms Λ→A}.

Addition = convolution.

Multiplication = convolution using the second

comultiplicationon Λ (= Kronecker comultiplication = Hall dual of Kronecker multiplication).

The elements of Alg(Λ,A) are known ascharacters of Λ (as in Aguiar-Bergeron-Sottile) or virtual alphabets (to the Lascoux school) or as specializations of symmetric functions (as in Stanley’s EC2).

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The elements of Alg(Λ,A) are known ascharacters of Λ (as in Aguiar-Bergeron-Sottile) or virtual alphabets (to the Lascoux school) or as specializations of symmetric functions (as in Stanley’s EC2).

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There is a unique family (w_n)_n∈

N+ of symmetric functions satisfying pn =P

d|ndw_d^n/d for all n∈N+. (Equivalently, it is determined byhn=P

λ`nw_λ, wherew_λ =w_λ₁w_λ₂· · ·.) These are called the Witt coordinates.

We have a ring isomorphism

Alg(Λ,A)→W(A), f 7→(f (w_n))_n∈

N+.

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Avatars of Witt vectors, 2: Characters of Λ, cont’d There is a unique family (w_n)_n∈

d|ndw_d^n/d for all n∈N+. (Equivalently, it is determined byh_n=P

λ`nw_λ, wherew_λ =w_λ₁w_λ₂· · ·.) These generate Λ as a ring, are called the Witt coordinates, and were first introduced in 1995 by Reutenauer.

Alg(Λ,A)→W(A), f 7→(f (wn))_n∈_N₊.

We also have a ring homomorphism (isomorphism whenAis a Q-algebra)

Alg(Λ,A)→A^N⁺, f 7→(f (pn))_n∈_N₊. These form a commutative diagram

Alg(Λ,A) ^∼⁼ ^//W(A)

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Avatars of Witt vectors, 2: Characters of Λ, cont’d There is a unique family (w_n)_n∈

d|ndw_d^n/d for all n∈N+. (Equivalently, it is determined byh_n=P

λ`nw_λ, wherew_λ =w_λ₁w_λ₂· · ·.) These generate Λ as a ring, are called the Witt coordinates, and were first introduced in 1995 by Reutenauer.

Alg(Λ,A)→W(A), f 7→(f (wn))_n∈_N₊.

We also have a ring homomorphism (isomorphism whenAis a Q-algebra)

Alg(Λ,A)→A^N⁺, f 7→(f (pn))_n∈_N₊. These form a commutative diagram

Alg(Λ,A) ^∼⁼ ^//

%%

W(A)

w

A^N⁺

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Reconstructing Λ fromW = Alg(Λ,−)

This also works in reverse: We can reconstruct Λ from the functor W, as its representing object. Namely:

The functor Forget◦W :CRing →Set determines Λ as a ring (by Yoneda).

The functor Forget◦W :CRing →Ab (additive group of W(A)) determines Λ as a Hopf algebra.

The functorW :CRing→CRing determines Λ as a Hopf algebra equipped with a second comultiplication.

The comonad structure onW additionally determines plethysm on Λ.

Thus, if symmetric functions hadn’t been around, Witt vectors would have let us rediscover them.

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Reconstructing Λ fromW = Alg(Λ,−)

This also works in reverse: We can reconstruct Λ from the functor W, as its representing object. Namely:

The functor Forget◦W :CRing →Set determines Λ as a ring (by Yoneda).

The functor Forget◦W :CRing →Ab (additive group of W(A)) determines Λ as a Hopf algebra.

The functorW :CRing→CRing determines Λ as a Hopf algebra equipped with a second comultiplication.

The comonad structure onW additionally determines plethysm on Λ.

Thus, if symmetric functions hadn’t been around, Witt vectors would have let us rediscover them.

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The ghost-Witt integrality theorem (aka Dwork lemma), 1

Assume you don’t know about Λ(A) or Λ. How would you go about proving that the Witt vector functorW exists?

In other words, why do the denominators (e.g., in the computation of csatisfying w(a) +w(b) =w(c))

“miraculously” vanish?

This is a consequence of theghost-Witt integrality theorem, also known (in parts) asDwork’s lemma. I shall state a (more or less) maximalist version of it; only the C ⇐⇒ E part is actually needed.

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Ghost-Witt integrality theorem.

Let Abe a commutative ring. For everyn∈N+, let

ϕn:A→A be an endomorphism of the ring A. Assume that:

We haveϕp(a)≡a^pmodpA for every a∈Aand every primep.

We haveϕ₁ = id, and we have ϕ_n◦ϕ_m=ϕ_nm for every n,m∈N+. (Thus,n 7→ϕn is an action of the

multiplicative monoidN+ onA by ring endomorphisms.) [For a stupid example, let A=Z andϕn= id.

For an example that is actually useful to Witt vectors, let A be a polynomial ring over Z, and let ϕ_n send each

indeterminate to its n-th power.]

Let b= (b_n)_n∈

N+ ∈A^N⁺ be a sequence of elements ofA. Then, the following assertions are equivalent: [continued on next page]

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Let b= (b_n)_n∈

N+ ∈A^N⁺ be a sequence of elements ofA.

Then, the following assertions are equivalent: [continued on

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Let b= (b_n)_n∈

N+ ∈A^N⁺ be a sequence of elements ofA.

Then, the following assertions are equivalent: [continued on next page]

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Ghost-Witt integrality theorem, continued.

The following are equivalent:

C: Everyn∈N+ and every prime divisorp of n satisfy ϕp b_n/p

≡bnmodp^v^p⁽ⁿ⁾A

(where vp(n) is the multiplicity ofp in the factorization of n).

D: There exists a sequencex= (x_n)_n∈

N+ ∈A^N⁺ of elements of Asuch that

b_n =X

d|n

dx_d^n/d =w_n(x) for everyn∈N+.

In other words,x belongs to the image of the ghost map w.

E: There exists a sequencey= (yn)_n∈_N₊ ∈A^N⁺ of elements of Asuch that

bn=X

d|n

dϕ_n/d(yd) for everyn ∈N+.

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b_n =X

d|n

bn=X

d|n

dϕ_n/d(yd) for everyn ∈N+.

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b_n =X

d|n

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F: Everyn∈N+ satisfies X

d|n

µ(d)ϕd b_n/d

∈nA.

G: Everyn∈N+ satisfies X

d|n

φ(d)ϕ_d b_n/d

∈nA.

J: There exists a ring homomorphism from the ring Λ toA which sends p_n (the n-th power sum symmetric function) to b_n for everyn ∈N+.

Note that this theorem has various neat consequences, like the famous necklace divisibility n|P

d|n

µ(d)q^n/d forn ∈N+ and q ∈Z. (And various generalizations.)

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d|n

µ(d)ϕd b_n/d

∈nA.

d|n

φ(d)ϕ_d b_n/d

∈nA.

d|n

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d|n

µ(d)ϕd b_n/d

∈nA.

d|n

φ(d)ϕ_d b_n/d

∈nA.

d|n

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d|n

µ(d)ϕd b_n/d

∈nA.

d|n

φ(d)ϕ_d b_n/d

∈nA.

µ(d)q^n/d forn ∈N+ and

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Z and Fq[T]: a tale of two rings

Now to something completely different...

Fix a prime power q.

There is a famous analogy between the elements ofZ and the elements ofFq[T]. (This is related toq-enumeration, the lore of the field with 1 element, etc.)

All that matters to us is that

positive integers inZcorrespond to monicpolynomials in Fq[T];

primes inZcorrespond toirreducible monic polynomials in Fq[T].

Let Fq[T]₊ be the set of all monicpolynomials inFq[T].

Let’s define an analogue of (big) Witt vectors for Fq[T] instead of Z.

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Let’s define an analogue of (big) Witt vectors forFq[T] instead ofZ.

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Let’s define an analogue of (big) Witt vectors forFq[T] instead ofZ.

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Definition of Fq[T]-Witt vectors, 1: ghost maps

Let Abe a commutative Fq[T]-algebra.

We abbreviate a family (a_N)_N∈

Fq[T]₊ ∈A^F^q^[T]⁺ as a.

For eachN ∈Fq[T]₊, define a map w_N :A^F^q^[T^]⁺ →Aby

w_N(a) =X

D|N

Da^q_D^deg(N/D),

where the sum is over all monicdivisorsD of N.

Let w :A^F^q^[T^]⁺ →A^F^q^[T]⁺ be the map given by w(a) = (w_N(a))_N∈

Fq[T]₊.

This “ghost map” w is Fq-linear, but notFq[T]-linear.

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Definition of Fq[T]-Witt vectors, 1: ghost maps

Let Abe a commutative Fq[T]-algebra.

For eachN ∈Fq[T]₊, define a map w_N :A^F^q^[T^]⁺ →Aby

w_N(a) =X

D|N

Da^q_D^deg(N/D),

Fq[T]₊.

This “ghost map” w is Fq-linear, but notFq[T]-linear.

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Definition of Fq[T]-Witt vectors, 2: Wq(A)

Let’s make a newFq[T]-algebra out of this: We define W_q(A) to be theFq[T]-algebra

that equalsA^F^q^[T]⁺ as a set, but

which is functorial in A(that is, we are really defining a functor Wq :CRing_F_q_[T_]→CRing_F_q_[T], whereCRing_R is the category of commutativeR-algebras), and

whoseFq[T]-algebra structure is such thatw is a natural (inA) homomorphism ofFq[T]-algebras from Wq(A) to A^F^q^[T]⁺.

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Definition of Fq[T]-Witt vectors, 2: Wq(A), cont’d

Example: The addition inWq(A) is the same as in A^F^q^[T]⁺ (sincew isFq-linear, and soWq(A) =A^F^q^[T]⁺ as

Fq-modules), so this would be boring. Instead, let’s set c=Tain W_q(A), and computew_π(c) for an irreducibleπ.

Start with c1=Ta1, which is easy to check. wπ(c) =Twπ(a)

⇐⇒ c₁^q^deg^π+πcπ =Ta^q₁^deg^π+Tπaπ c1=Ta1

⇐⇒ (Ta1)^q^degπ+πcπ =Ta^q₁^deg^π+Tπaπ

⇐⇒ πcπ =Tπaπ−

T^q^deg^π−T

a^q₁^deg^π

naturality

⇐⇒ c_π =Ta_π− T^q^deg^π−T π a^q₁^deg^π.

The fraction on the RHS is a polynomial due to a known fact from Galois theory (namely:

T^q^k−T = Q

γ∈Fq[T]₊ irreducible; degγ|k

γ).

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Start with c1=Ta1, which is easy to check.

wπ(c) =Twπ(a)

⇐⇒ (Ta1)^q^degπ+πcπ =Ta^q₁^deg^π+Tπaπ

T^q^deg^π−T

a^q₁^deg^π

naturality

T^q^k−T = Q

γ).

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wπ(c) =Twπ(a)

⇐⇒ (Ta1)^q^deg^π+πcπ =Ta^q₁^deg^π+Tπaπ

T^q^deg^π−T

a^q₁^deg^π

naturality

T^q^k−T = Q

γ).

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wπ(c) =Twπ(a)

⇐⇒ (Ta1)^q^deg^π+πcπ =Ta^q₁^deg^π+Tπaπ

T^q^deg^π−T

a^q₁^deg^π

naturality

T^q^k−T = Q

γ).

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Definition of Fq[T]-Witt vectors, 3: coda

There is also a second construction ofW_q(A), using Carlitz polynomials, yielding an isomorphicFq[T]-algebra. (See the preprint.)

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Avatars of Fq[T]-Witt vectors?

Can we find anything similar to the two avatars of W(A) ? Power series? This appears to require a notion of power series where the exponents are polynomials inFq[T]. Product ill-defined due to lack of actual “positivity”. Seems too much to wish...

Alg(Λ,A)? Well, we can try brute force: Remember how Λ was reconstructed from W, and do something similar to

“reconstruct” a representing object fromWq. We’ll come back to this shortly.

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Surprise: F-modules, 1

First, a surprise...

We aren’t using the whole Fq[T]-algebra structure on A! (This is unlike the Z-case, where it seems that we use the commutative ring Ain full.)

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First, a surprise...

We aren’t using the whole Fq[T]-algebra structure on A! (This is unlike the Z-case, where it seems that we use the commutative ring Ain full.)

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Let F be the noncommutative ring

FqhF,T |FT =T^qFi. This is an Fq-vector space with basis TⁱF^j

(i,j)∈N², and is an Ore polynomial ring. It shares many properties of usual univariate polynomials (see papers of Ore).

Actually,

F ∼=

Fq[T] [X]_q−lin,+,◦ ,

whereFq[T] [X]_q−lin are the polynomials inX overFq[T] whereX occurs only with exponentsq^k, and where ◦is composition of polynomials.

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Actually,

F ∼=

Fq[T] [X]_q−lin,+,◦ ,

whereFq[T] [X]_q−lin are the polynomials inX overFq[T] whereX occurs only with exponentsq^k, and where ◦is composition of polynomials.

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What matters to us:

Each commutative Fq[T]-algebra canonically becomes a (left) F-module by having

T act as multiplication byT, and

F act as the Frobenius (i.e., takingq-th powers).

Thus, we have a functor CRing_F_q_[T_]→ModF.

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What matters to us:

Each commutative Fq[T]-algebra canonically becomes a (left) F-module by having

T act as multiplication byT, and

F act as the Frobenius (i.e., takingq-th powers).

Thus, we have a functor CRing_F_q_[T_]→ModF.

There are other sources ofF-modules too (cf. Jacobson on

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Fq[T]-Witt vectors of an F-module Let Abe a (left)F-module.

For eachN ∈Fq[T]₊, define a map w_N :A^F^q^[T^]⁺ →Aby w_N(a) =X

D|N

DF^deg(N/D)a_D,

Fq[T]₊. We define Wq(A) to be theF-module

which is functorial in A(that is, we are really defining a functor W_q :ModF →ModF), and

whose F-module structure is such that w is a natural (in A) homomorphism ofF-modules fromWq(A) toA^F^q^[T^]⁺.

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Fq[T]-Witt vectors of an F-module Let Abe a (left)F-module.

For eachN ∈Fq[T]₊, define a map w_N :A^F^q^[T^]⁺ →Aby w_N(a) =X

D|N

DF^deg(N/D)a_D,

Fq[T]₊. We define Wq(A) to be theF-module

which is functorial in A(that is, we are really defining a functor W_q :ModF →ModF), and

F-module structure is such that

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An F-ghost-Witt integrality theorem, 1

Again, there is a “ghost-Witt integrality theorem” that helps prove the existence of the Wq functors.

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F-ghost-Witt integrality theorem.

Let Abe a (left)F-module. For everyP ∈Fq[T]₊, let ϕ_P :A→Abe an endomorphism of the F-module A.

Assume that:

We haveϕπ(a)≡F^deg^πamodπAfor every a∈Aand every monic irreducibleπ ∈Fq[T]₊.

We haveϕ1= id, and we have ϕN◦ϕM =ϕNM for every N,M ∈Fq[T]₊. (Thus,N 7→ϕ_N is an action of the multiplicative monoidFq[T]₊ onAbyF-module endomorphisms.)

Let b= (b_n)_n∈

N+ ∈A^F^q^[T]⁺ be a family of elements of A.

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F-ghost-Witt integrality theorem.

Let Abe a (left)F-module. For everyP ∈Fq[T]₊, let ϕ_P :A→Abe an endomorphism of the F-module A.

Assume that:

We haveϕπ(a)≡F^deg^πamodπAfor every a∈Aand every monic irreducibleπ ∈Fq[T]₊.

We haveϕ1= id, and we have ϕN◦ϕM =ϕNM for every N,M ∈Fq[T]₊. (Thus,N 7→ϕ_N is an action of the multiplicative monoidFq[T]₊ onAbyF-module endomorphisms.)

Let b= (b_n)_n∈

N+ ∈A^F^q^[T]⁺ be a family of elements of A.

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C: EveryN ∈Fq[T]₊ and every monic irreducible divisor π of N satisfy

ϕ_π b_N/π

≡b_Nmodπ^v^π^(N)A.

D₂: There exists a family x= (x_N)_N∈

Fq[T]₊ ∈A^F^q^[T]⁺ of elements ofA such that

b_N =X

D|N

DF^deg(N/D)x_D =w_N(x) for everyN∈Fq[T]₊.

E: There exists a family y= (yN)_N∈_F_q_[T_]

+ ∈A^F^q^[T^]⁺ of elements ofA such that

b_N =X

D|N

Dϕ_N/D(y_D) for everyN ∈Fq[T]₊.

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ϕ_π b_N/π

b_N =X

D|N

+ ∈A^F^q^[T^]⁺ of elements ofA such that

b_N =X

D|N

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ϕ_π b_N/π

D₁: There exists a family x= (x_N)_N∈

b_N =X

D|N

DN

D [T +F]x_D for everyN∈Fq[T]₊. [This is mainly interesting due to the connection to Carlitz polynomials.]

E: There exists a family y= (y_N)_N∈

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ϕ_π b_N/π

b_N =X

D|N

+ ∈A^F^q^[T]⁺ of elements ofA such that

b_N =X

D|N

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F: EveryN ∈Fq[T]₊ satisfies X

D|N

µ(D)ϕ_D b_N/D

∈NA.

Here, µis anFq[T]-version of the M¨obius function, defined as the usual one (i.e., squarefree7→ number of distinct irreducible factors; non-squarefree7→ 0).

G: EveryN ∈Fq[T]₊ satisfies X

D|N

φ(D)ϕD b_N/D

∈NA,

whereφis one of two reasonableFq[T]-versions of the Euler totient function.

J: ???

To stateJ, we need an Fq[T]-analogue of the symmetric functions.

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D|N

µ(D)ϕ_D b_N/D

∈NA.

D|N

φ(D)ϕD b_N/D

∈NA,

J: ???

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D|N

µ(D)ϕ_D b_N/D

∈NA.

D|N

φ(D)ϕD b_N/D

∈NA,

J: ???

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D|N

µ(D)ϕ_D b_N/D

∈NA.

D|N

φ(D)ϕD b_N/D

∈NA,

J: ???

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