THE MULTIPLICATIVE TRANSFER FOR THE GROTHENDIECK WITT RING

MARKUS ROST

Introduction

In this text we collect some material related with the multiplicative norm for the Grothendieck-Witt ring. It is a first draft.

1. Polynomials on semi groups

Let A be an additive semi group. In other words, we are given a set A and a commutative and associative operation (x, y) ↦ x + y on A.

We do not assume that A contains a neutral element 0. In any case, there is the semi group with neutral element A_{0} = A ∪ {0} (with A_{0} = A if 0 ∈ A).

Let A → Ā be the group completion of A.

Let further B be a commutative group. Then, by the very definition of group completion, a semi group homomorphism P: A → B extends to a unique group homomorphism P̄: Ā → B.

The purpose of this section is to note that every polynomial map P: A → B extends to a unique polynomial map P̄: Ā → B. This is perhaps well known (indeed, see [2]); it becomes almost a triviality after relating group completions and polynomial maps with the corresponding (semi) group rings.

Let Z[A] be the semi group ring of A. As a group, Z[A] is the free abelian group on A. For x ∈ A we denote by [x] the corresponding generator of Z[A]. Then the multiplication in Z[A] is given by [x][y] = [x + y].

If 0 ∈ A, then [0] = 1 is the unit element of Z[A]. In general, Z[A] is an ideal of the unital ring Z[A_{0}].

We consider A as subset of Z[A] in the natural way, via x ↦ [x]. It is clear that any map P: A → B extends to a unique group homomorphism P̂: Z[A] → B.

The subset A of Z[A] is a multiplicative subset. It is tautological that the group ring of the group completion Ā is localization of Z[A] at A:

Z[Ā] = A^{−1}Z[A]

Definition. Let P: A → B be a map.

(1) P is called a polynomial map of degree ≤ −1 if P = 0.

(2) For n ≥ 0, P is called a polynomial map of degree ≤ n if for each x ∈ P the map

Δ_{x}P: A → B
Δ_{x}P (y) = P (y + x) − P (y)

is a polynomial map of degree ≤ n − 1.

Date: January 28, 2003.

(3) P is called a polynomial map if it is a polynomial map of degree ≤ n for some n.

We denote by

ϵ_{A}: Z[A_{0}] → Z
ϵ_{A}([x]) = 1 (x ∈ A_{0})

the augmentation homomorphism and by I_{A} = ker ϵ_{A} its kernel.

For α ∈ Z[A] one has

{ \atop Δ_{x}P}(α) = P̂(([x] − 1)α)

This implies easily by an induction argument the following

Lemma 1. A map P: A → B is a polynomial map of degree ≤ n if and only if

P̂(I_{A}^{n+1}Z[A]) = 0

Corollary 1. There is a bijection

P ↦ (α mod I_{A}^{n+1}Z[A] ↦ P̂(α))

from the group of polynomial maps A → B of degree ≤ n to

Hom(Z[A]/I_{A}^{n+1}Z[A], B)

Example. The polynomial maps on A = N_{0}^{r} are the linear combinations of the polynomials of the form

P (x_{1}, …, x_{r}) = ∏_{i=1}^{r} \binom{x_{i}}{n_{i}}

where the n_{i} are nonnegative integers. (Use Corollary 1.)

For x ∈ A one has 1 − [x] ∈ I_{A} and therefore [x] mod I_{A}^{n+1} is invertible. Thus

Z[Ā]/In+1 ; Ā = A^{−1}Z[A]/I_{A}^{n+1}Z[A] = Z[A]/I_{A}^{n+1}Z[A]

and we have

Corollary 2. A polynomial map P: A → B extends uniquely to a polynomial map P̄: Ā → B. If P is of degree ≤ n, then P̄ is of degree ≤ n.

Let P: A → B be a polynomial map of degree ≤ n. Its extension P̄ can be made explicit as follows. For x, y ∈ A we have

[x − y](1 − [y])^{n+1} = ∑_{i=0}^{n+1}(−1)^{i}\binom{n + 1}{i}[x + (i − 1)y]

On the other hand, one has [x − y](1 − [y])^{n+1} ∈ In+1 ; Ā and therefore

ˆP̄([x − y](1 − [y])^{n+1}) = 0

This gives

(1)  P̄(x − y) = ∑_{i=0}^{n}(−1)^{i}\binom{n + 1}{i + 1}P (x + iy)

expressing any value of P̄ in terms of values of P.

2. Multiplicative polynomials on Z

Multiplicative polynomial maps appear from multiplicative transfer maps on cohomology rings, Grothendieck-Witt rings of symmetric bilinear forms, etc. In those cases it is an obvious question to compute the restriction to the subring generated by 1.

Definition. Let A, B be commutative rings. A polynomial map P: A → B is called multiplicative if P (1) = 1 and if

P (xy) = P (x)P (y)

for x, y ∈ A.

Let P: A → B be a multiplicative polynomial map. Then P (0)^{2} = P (0). If P (0) = 1, then P is constant, P ≡ 1. If P (0) = 0, we call P proper. If B is connected, then P is either constant or proper.

A polynomial map P: Z → B of degree ≤ n can be written uniquely as

(2)  P (k) = a_{0} + a_{1}k + a_{2}\binom{k}{2} + ⋯ + a_{n}\binom{k}{n}

with a_{i} ∈ B. If P (0) = 0 and P (1) = 1, then a_{0} = 0 and a_{1} = 1 and

(3)  P (k) = k + a_{2}\binom{k}{2} + ⋯ + a_{n}\binom{k}{n}

Let X_{n} be the scheme over Spec Z representing the multiplicative polynomials on Z. The presentation (2) identifies X_{n} as a closed subscheme of the affine space A^{n+1} with coordinates a_{0}, …, a_{n}.

Note that a_{0} + a_{1} = 1 and a_{0}^{2} = a_{0} on X_{n}.

Let Y_{n} = X_{n} ∩ {a_{0} = 0, a_{1} = 1}. Then X_{n} is the disjoint union of Y_{n} and a copy of Spec Z given by (a_{0}, a_{1}, …, a_{n}) = (1, 0, …, 0). The latter represents P = 1 and we may write

X_{n} = Y_{n} ∪ {1}

The scheme Y_{n} represents the proper multiplicative polynomials on Z. The presentation (3) identifies Y_{n} as a closed subscheme of the affine space A^{n−1} with coordinates a_{2}, …, a_{n}.

Lemma 2. The scheme X_{n} is flat and finite over Spec Z of degree n + 1. Let T = Z[(n!)^{−1}]. Then (X_{n})_{T} is consists of n + 1 copies of Spec T.

Proof. If n! is invertible in B, one may parametrize polynomial maps P: Z → B of degree ≤ n as

(4)  P (k) = b_{0} + b_{1}k + b_{2}k^{2} + ⋯ + b_{n}k^{n}

with b_{i} ∈ B. It is easy to see that P is multiplicative if and only if the b_{i} are pairwise orthogonal idempotents. This proves the second claim.

For the general case observe that X_{n} is the spectrum of

R_{n} = Z[a_{0}, …, a_{n}]/I

where the ideal I is generated by a_{0} +a_{1} −1 and elements a_{i}a_{j} −l_{ij} (1 ≤ i ≤ j ≤ n) with the l_{ij} linear in the a_{k}. Hence R_{n} is generated by a_{0}, …, a_{n} as a Z-module. Since dim_{Q} R_{n} ⊗ Q = n + 1 by the first part of the proof, it follows that R_{n} a free Z-module of rank n + 1. □

Example. The case n = 2. The proper multiplicative quadratic polynomials Z → B are

P (k) = k + a\binom{k}{2}

where a is subject to a^{2} = 2a. This follows easily from

\binom{kl}{2} = 2\binom{k}{2}\binom{l}{2} + \binom{k}{2}l + \binom{l}{2}k

This describes Y_{2} completely.

Example. The case n = 3 with 2 = 0 in B. If 2B = 0, the proper multiplicative polynomials Z → B of degree ≤ 3 are

P (k) = k + a\binom{k}{2} + b\binom{k}{2}k

where a, b are subject to a^{2} = ab = b^{2} = 0. This describes (Y_{3})_{F_{2}} completely.

Exercise. Let L/F be a finite field extension and let N_{L/F} denote the multiplicative norm for the Grothendieck-Witt ring (to be defined later). Describe the polynomial k ↦ N_{L/F} (k), k ∈ Z.

It is clear by Serre that the coefficients of N_{L/F} |Z can be expressed in terms of the exterior powers of the trace form. Is there a nice formula for that? Does the Pfister form P (L/F) show up? I have not considered anything here.

Exercise. Let L be a finite field extension of the field of rational numbers Q of degree d. For k ∈ Z let P (k) be the signature of N_{L/Q}(k), where N_{L/Q} is the multiplicative norm for the Grothendieck-Witt ring (to be defined later).

Show that P (k) = k^{r+s} where r, s are the numbers of real, complex embeddings of L, respectively (d = r + 2s).

3. The norm

See also [1].

3.1. Notations. The symmetric group on i letters is denoted by S_{i}. Let

sgn: S_{i} → {±1}

be the signature homomorphism.

Let F be a commutative ring and let M be an F -module. Let

T^{i}M = T_{F}^{i} M = M^{⊗i} = \underbrace{M ⊗_{F} ⋯ ⊗_{F} M}_{i factors}

be the i-fold tensor power of M over F. The symmetric group S_{i} acts in a natural way on M^{⊗i} by

s(x_{1} ⊗ ⋯ ⊗ x_{i}) = x_{s^{−1}1} ⊗ ⋯ ⊗ x_{s^{−1}i}

We write

Λ^{i}M = Λ_{F}^{i} M = M^{⊗i}/ ∑_{s∈S_{i}}(1 − sgn(s)s)M^{⊗i}
S_{i}M = S_{i}^{F} M = (M^{⊗i})^{S_{i}} = { α ∈ M^{⊗i} | s(α) = α for s ∈ S_{i} }

for the i-th exterior power of M and the module of symmetric i-tensors, respectively.

For x ∈ M we use the notation

[x] = [x]_{i} = \underbrace{x ⊗ ⋯ ⊗ x}_{i factors} ∈ S_{i}M

We will also use the trace maps

T_{i,j} = ∑_{s∈S_{i+j}/S_{i}×S_{j}} s : S_{i}M ⊗ S_{j}M → S_{i+j}M

3.2. The norm. Let F be a commutative ring and let E be a locally free ring extension of F of rank n.

For discussion of geometric interpretations let us write X = Spec E and Y = Spec F. Thus we have a morphism of schemes X → Y, finite and flat of degree n.

The tensor power E^{⊗i} is a locally free ring extension of F of rank n^{i} containing S_{i}E as subring (all tensors are considered over F). The S_{i}E-module structure on E^{⊗i} induces an S_{i}E-module structure on Λ^{i}E and thus we have homomorphisms

ρ_{i}: S_{i}E → End_{F} (Λ^{i}E)

The F -module Λ^{n}E is invertible. Therefore we get a ring homomorphism

ν_{E/F} = ρ_{n}: S_{n}E → End_{F} (Λ^{n}E) = F

called the norm of E/F. Indeed, for x ∈ E one has

ν_{E/F} ([x]) = (Λ^{n}E ⟶^{Λ^{n}μ_{x}} Λ^{n}E) = det(μ_{x}) = N_{E/F} (x)

where μ_{x}: E → E is the multiplication map.

For a finitely generated locally free E-module M we put

ν_{E/F} (M) = S_{n}M ⊗_{S_{n}E} F = (ν_{E/F})_{∗}(S_{n}M)

Suppose E/F is separable. Then ν_{E/F} (M) is a locally free F -module. If M is of rank d as E-module, then ν_{E/F} (M) is of rank d^{n} as F -module. If e_{1}, …, e_{n} are orthogonal idempotents of E (necessarily of rank 1: e_{i}E = F), then

(5)  ν_{E/F} (M) = M_{1} ⊗ ⋯ ⊗ M_{n}

where M_{i} = e_{i}M.

Let b: M × M → E be a non-degenerate bilinear symmetric form. Then the form

b^{⊗n}: M^{⊗n} × M^{⊗n} → E^{⊗n}

restricts to a form

S_{n}b: S_{n}M × S_{n}M → S_{n}E

which we tensor with S_{n}E → F to get a form

ν_{E/F} (b): ν_{E/F} (M) × ν_{E/F} (M) → F

If E/F is separable, then ν_{E/F} (b) is again a non-degenerate bilinear symmetric form. In the situation of (5) we have

ν_{E/F} (b) = b_{1} ⊗ ⋯ ⊗ b_{n}

with b_{i} = b|(M_{i} × M_{i}).

In the next sections we study basic properties of the norm.

4. The extensions C_{i_{1},...,i_{r}}E

Let (i_{1}, …, i_{r}) be a sequence of integers i_{j} ≥ 0 with i_{1} + ⋯ + i_{r} = n. Such a sequence will be called a partition of n. (We allow i_{j} = 0 for convenience.)

Let

S_{i_{1},...,i_{r}}E = S_{i_{1}}E ⊗ ⋯ ⊗ S_{i_{r}}E

Note that S_{n}E is a subring of S_{i_{1},...,i_{r}}E.

We define

C_{i_{1},...,i_{r}}E = (ν_{E/F})_{∗}(S_{i_{1},...,i_{r}}E) = S_{i_{1},...,i_{r}}E ⊗_{S_{n}E} F

Let C^{i_{1},...,i_{r}}X = Spec C_{i_{1},...,i_{r}}E. Then there is a pull back diagram

C^{i_{1},...,i_{r}}X ⟶ S^{i_{1}}X × ⋯ × S^{i_{r}}X
↓  ↓
Y  ⟶^{ν}  S^{n}X

If X/Y is separable, then C^{i_{1},...,i_{r}}X represents the decompositions of the fibers into subsets of size i_{1}, …, i_{r}.

If E/F is separable, then C_{i_{1},...,i_{r}}E is a separable F -algebra of rank

\binom{n}{i_{1}, …, i_{r}} = \binom{i_{1} + ⋯ + i_{r}}{i_{1}}\binom{i_{2} + ⋯ + i_{r}}{i_{2}} ⋯ \binom{i_{r}}{i_{r}}

Example. If E/F is separable, then

Ē = C_{1,...,1}E = E^{⊗n} ⊗_{S_{n}E} F

is a separable F -algebra of rank n!. Moreover there is a natural S_{n}-action on Ē. If E/F is a field extension with maximal possible Galois group, then Ē is the Galois closure of E with the n embeddings E → Ē given by the maps

E ↦ E^{⊗n} x ↦ 1 ⊗ ⋯ ⊗ 1 ⊗ x ⊗ 1 ⊗ ⋯ 1

5. The extensions C_{i}E

This section contains a first attempt to understand the algebras C_{i_{1},...,i_{r}}E.

We study the case r = 2. For 0 ≤ i ≤ n let C_{i}E = C_{i,n−i}E.

First a general fact:

Lemma 3. One has S_{i}E ⊗ S_{j}E = (S_{i}E ⊗ 1)S_{i+j}E.

Proof. In this first draft, we leave the proof to the reader. □

Geometrically speaking, the lemma says that

S^{i}X × S^{j}X → S^{i}X × S^{i+j}X

is a closed embedding.

Corollary 3. The homomorphism

ν_{i}: S_{i}E ⟶^{⊗1} S_{i}E ⊗ S_{n−i}E → C_{i}E

is surjective.

Geometrically speaking, this means that

C^{i}X → S^{i}X × S^{n−i}X ⟶^{projection} S^{i}X

is a closed embedding.

Note that we have a canonical isomorphism τ: C_{i}E → C_{n−i}E by switching factors of S_{i}E ⊗ S_{n−i}E (so that τ ∘ τ(α) = α when defined). If n is even, we get an involution τ: C_{n/2} → C_{n/2}.

Example. If E/F is separable, then C_{i}E is a separable F -algebra of rank \binom{n}{i}.

Example. If E/F is separable of degree 4, then (C_{2}E)^{τ} is the “cubic resolvent” of E/F.

Example. Consider the “most degenerate” extension of degree 4, E = F [x, y, z], x^{2} = y^{2} = z^{2} = xy = xz = yz = 0. In this case one finds that S_{2}E → C_{2}E is an isomorphism. Hence C_{2}E is of rank 10 in this case, in contrast to the case of separable quartic extensions where C_{2}E is of rank 6. See [1] for closely related examples.

Recall the homomorphisms ρ_{i}: S_{i}E → End_{F} (Λ^{i}E). Composing ρ_{n−i} with duality

End_{F} (Λ^{n−i}E) = End_{F} (Λ^{n}E ⊗ (Λ^{i}E)^{∗}) = End_{F} ((Λ^{i}E)^{∗}) = End_{F} (Λ^{i}E)

we get a homomorphism

ρ̄_{n−i}: S_{n−i}E → End_{F} (Λ^{i}E)

Lemma 4. The homomorphisms ρ_{i}, ρ̄_{n−i} commute, i.e.,

ρ_{i}(α)ρ̄_{n−i}(β) = ρ̄_{n−i}(β)ρ_{i}(α)

The resulting homomorphism

ρ_{i} ⊗ ρ̄_{n−i}: S_{i}E ⊗ S_{n−i}E → End_{F} (Λ^{i}E)

factors through a homomorphism

ρ̂_{i}: C_{i}E → End_{F} (Λ^{i}E)

Proof. In this first draft, we leave the proof to the reader. Note that we don’t assume any separability condition here.

One may check this first in the split separable case, which gives the general separable case, and then use specialization to the degenerate case. (I don’t like this way, rather I prefer a tensor game, which shouldn’t be very technical.) □

Let us define

trace_{i} = trace_{End_{F} (Λ^{i}E)} ∘ ρ̂_{i}: C_{i}E → F

In the separable case, ρ̂_{i} is the embedding of a maximal commutative subalgebra of End_{F} (Λ^{i}E). In this case trace_{i} is the usual trace map for C_{i}E/F.

Example. Consider the “most degenerate” extension of degree 4, see above. In this case one finds that the image of ρ̂_{2} has rank 4.

Exercise. Show that if E is a subalgebra of an Azumaya algebra A, then there is a natural homomorphism C_{i}E → λ^{i}A (see [3] for the definition of the exterior powers of Azumaya algebras). If E is locally a maximal commutative separable subalgebra of A, then C_{i}E locally a maximal commutative separable subalgebra of λ^{i}A. (I have not done this exercise.)

Proposition 1.

(1) C_{0}E = C_{n}E = F

(2) The map ν_{1} : E → C_{1}E is an isomorphism.

(3) The map

# : S_{n−1}E ⟶^{ν_{n−1}} C_{n−1}E ⟶^{τ} C_{1} ⟶^{ν_{1}^{−1}} E

is the adjoint map ([x]^{#}x = N(x)).

Proof. In this first draft, we leave the proof to the reader. Note that we don’t assume any separability condition here.

One may check this easily in the split separable case, which gives the general separable case.

As for (2) in the general case: We know from Corollary 3 that ν_{1} is surjective. On the other hand, one has

ρ̂_{1} ∘ ν_{1}(x) = μ_{x}

for x ∈ E. This shows the injectivity of ν_{1}. □

Lemma 5. The trace maps T_{i,j}: S_{i}E ⊗ S_{j}E → S_{i+j}E induce trace maps

T̄_{i,j}: C_{i}E ⊗ C_{j}E → C_{i+j}E

which are C_{i+j}E-linear. One has

T̄_{i,n−i}(α ⊗ β) = trace_{i}(ατ(β))

Proof. No proof. □

We formulate a basic formula.

Proposition 2. Let x, y ∈ E. Then

ν_{k}([x + y]) = ∑_{i=0}^{k} T_{i,k−i}(ν_{i}([x]) ⊗ ν_{k−i}([y]))

In particular,

N_{E/F} (x + y) = ∑_{i=0}^{n} trace_{i}(ν_{i}([x])τ(ν_{n−i}([y])))

Proof. In this first draft, we leave the proof to the reader. Note that we don’t assume any separability condition here.

□

It follows that the trace_{i}(ν_{i}([x])) are the coefficients of the characteristic polynomial of x.

6. The tensorial norm

Let F be a commutative ring and let E be a locally free ring extension of F of rank n. We assume that E/F is separable (=etale).

Let M_{1}, …, M_{r} be finitely generated locally free E-modules and let (i_{1}, …, i_{r}) be a partition of n.

We put

S_{i_{1},...,i_{r}}(M_{1}, …, M_{r}) = S_{i_{1}}M_{1} ⊗ ⋯ ⊗ S_{i_{r}}M_{r}

and

C_{i_{1},...,i_{r}}(M_{1}, …, M_{r}) = (ν_{E/F})_{∗}(S_{i_{1},...,i_{r}}(M_{1}, … M_{r}))
= S_{i_{1},...,i_{r}}(M_{1}, …, M_{r}) ⊗_{S_{i_{1},...,ir} E} C_{i_{1},...,i_{r}}E

Lemma 6. Let f_{r}: C_{i_{1},...,i_{r}}E → C_{r,i_{1}−r,...,i_{r}}E be the natural homomorphisms, induced from S_{i_{1}}E → S_{r}E ⊗ S_{i_{1}−r}E. Then

C_{i_{1},...,i_{r}}(M ⊕ N, M_{2}, … M_{r}) = ⨁_{r=0}^{i_{1}} f_{r}^{∗}(C_{r,i_{1}−r,...,i_{r}}(M, N, M_{2}, … M_{r}))

Proof. Easy. □

Corollary 4. There exist a unique multiplicative polynomial map

N_{E/F} : K_{0}^{⊕}E → K_{0}^{⊕}F

of degree n such that

N_{E/F} ([M]) = [ν_{E/F} (M)]

Proof. Use Lemma 6 and Corollary 2... □

7. The norm for the Grothendieck-Witt ring

We denote by GW (F) the Grothendieck-Witt ring of F of symmetric bilinear forms (let us say in this draft that F is a field).

Let E/F be a separable field extension of degree n. Using an obvious extension of section 6, one gets

Corollary 5. There exist a unique multiplicative polynomial map

N_{E/F} : GW (E) → GW (F)

of degree n such that

N_{E/F} ([b]) = [ν_{E/F} (b)]

I guess one should also define polynomial maps

ν_{i}: GW (E) → GW (C_{i}E)

of degree i. For instance, for n = 4 this would give a quadratic map

GW (E) ⟶^{ν_{2}} GW (C_{2}E) ⟶^{trace} GW ((C_{2}E)^{τ} )

to the Grothendieck-Witt ring of the cubic resolvent…

8. Computations

I haven’t done much besides for the case of a quadratic field extensions in characteristic not 2. See the text “A Pfister form invariant for etale algebras”.

References

[1] D. Ferrand, Un foncteur norme, Bull. Soc. Math. France 126 (1998), no. 1, 1–49.

[2] S. Joukhovitski, K-theory of the Weil transfer functor, K-Theory 20 (2000), no. 1, 1–21, Special issues dedicated to Daniel Quillen on the occasion of his sixtieth birthday, Part I.

[3] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions, American Mathematical Society, Providence, RI, 1998, With a preface in French by J. Tits.

Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH 43210, USA

   E-mail address: rost@math.ohio-state.edu
   URL: http://www.math.ohio-state.edu/~rost