# The Elman-Lam-Krüskemper Theorem

Authors: Karim Johannes Becher; David B. Leep
Journal: ISRN Algebra
 (2011)

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## Abstract

For a (formally) real fieldK, the vanishing of a certain power of the fundamental ideal in the Witt ring of K(-1) implies that the same power of the fundamental ideal in the Witt ring of K is torsion free. The proof of this statement involves a fact on the structure of the torsion part of powers of the fundamental ideal in the Witt ring of K. This fact is very difficult to prove in general, but has an elementary proof under an assumption on the stability index of K. We present an exposition of these results.


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## Body

## 1. Introduction

In the study of sums of squares and quadratic forms over fields, it has often been fruitful to consider the quadratic extension obtained by adjoining-1. Consider, for example, an extension K/R of transcendence degree n of a real closed field R. Tsen-Lang theory yields that quadratic forms over K(-1) of dimension greater than 2n are isotropic. From this, one can derive that in K any totally positive element is a sum of 2n squares and that torsion quadratic forms over K of dimension greater than 2n+2 are isotropic. The proof of the statement on sums of squares in K uses the observation that a certain field property (An+1) goes down a quadratic extension.We assume that the reader is familiar with the classical theory of quadratic forms over fields as presented in [1]. We use standard notation and terminology. When we speak about quadratic forms we always assume them to be regular, and we often just call them forms.LetK be a field of characteristic different from 2. We denote by K× the multiplicative group of K and by ∑K2 the subgroup of the nonzero sums of squares in K. By the Artin-Schreier Theorem, K admits a field ordering if and only if -1∉∑K2; in this case, we say that K is real, otherwise nonreal.LetWK denote the Witt ring of K and IK its fundamental ideal. Let WtK be the torsion part of WK. If K is real, then WtK⊆IK, otherwise WtK=WK.Letn be a positive integer. We put InK=(IK)n and ItnK=WtK∩InK. Following [2] and [1, Chapter XI, Section 4], we denote by (An) the condition that every torsion n-fold Pfister form over a given field is hyperbolic. Recall that InK is generated (as a group) by the n-fold Pfister forms over K. Hence, if K is a nonreal field, then K satisfies (An) if and only if InK=0. It was conjectured in [2] that, without assuming that K is nonreal, the following holds.Conjecture 1.1.
The fieldK satisfies  (An) if and only if  ItnK=0.This is obviously implied by the following stronger conjecture, which was proved in [3, (2.8)] as a consequence of Voevodsky's proof of the Milnor Conjecture and the deep results in [4].Conjecture 1.2.
One hasItnK=ItK·In-1K.Nevertheless, one may wish to have more elementary proofs of these conjectures, even for special cases. Forn≤3, such a proof for Conjecture 1.1 can be found in [1, Chapter XI, (4.1) and Chapter XII, (3.1)]. For an elementary proof of Conjecture 1.2 for n=2, see [1, Chapter XI, (4.2)] or Proposition 3.2 below. For arbitrary n, Krüskemper showed in [5, Theorem 3] by elementary arguments that a sum of up to three scaled n-fold Pfister forms lies in ItnK if and only if it lies in ItK·In-1K; in Theorem 3.8, we give a self-contained exposition of this result.There is another situation where an elementary proof of Conjecture1.2 is possible, depending on the stability index of the field K. Following [6], the (reduced) stability index ofK is defined as
(1.1)st(K)=min⁡{r∈N∣Ir+1K=2IrK+Itr+1K}∈N∪{∞}.
Note that st(K)=0 if and only if K is either nonreal or uniquely ordered.In [7], Krüskemper outlined an elementary proof of Conjecture 1.2 for the case where n≥st(K), using several facts stated in [8] without a detailed proof.Our efforts to understand the mentioned results from [5, 7] led us to a more detailed and structured exposition, which we want to make available to a wider audience, though we do not claim any originality of the ideas. Shortly after completing this note, we learned that another exposition of Krüskemper's proof that Conjectures 1.2 and 1.1 hold for n≥st(K) appeared in [9, Section 35.B]. Here, following [7], we work within the more general framework of preorderings and the reduced theory of quadratic forms.Let us point out that having Conjecture1.2 for n>st(K) leads to a proof of the following statement, as observed in [5, Proposition 1].Theorem 1.3 (Elman-Lam-Krüskemper).
Forn≥1, the following statements are equivalent:
(i)InK(-1)=0.
(ii)ItnK=0 and InK=2In-1K.
(iii)ItnK=0 and st(K)<n.Assuming thatItnK=0, it is clear that InK=2In-1K holds if and only if st(K)<n. Therefore, (ii) and (iii) are equivalent. Elman and Lam showed in [2, (4.4), (4.5), (4.7)] that InK(-1)=0 if and only if K satisfies (An) and InK=2In-1K (see also [1, Chapter XI, (4.7), (4.14), (4.18)]); this fact was rediscovered by Elman and Prestel in [10, (3.3)]. Hence, to complete the proof of Theorem 1.3,  it is sufficient to know that Conjecture 1.1 holds when n>st(K).Finally, note that the equivalent conditions in Theorem1.3  hold in the case where K is an extension of a real closed field of transcendence degree strictly smaller than n. In fact, since any quadratic form over K(-1) of dimension exceeding 2n-1 is isotropic, it is clear that InK(-1)=0, and [2, (6.1)] gives a more direct way to show that ItnK=0 in this case.


## 2. Preorderings

We recall some facts from real algebra and the reduced theory of quadratic forms modulo a preordering. For details, we refer to [11]. Let X(K) denote the set of all field orderings of K. For a quadratic form φ over K and P∈X(K), we denote by signP(φ) the signature of φ at P.Apreordering ofK is a proper subset T⊊K that contains the squares in K and that is closed under addition and multiplication in K. Note that these conditions entail that {0}∪∑K2⊆T and -1∉T, in particular K is real. Conversely, if K is real, then K has preorderings and the smallest one is ∑K2∪{0}.LetT be a fixed preordering of K. We denote T×=T∩K×=T∖{0}. We denote by XT(K) the set of field orderings of K that contain T. By [1, Chapter VIII, (9.6)], we have T=⋂P∈XT(K)P. The set XT(K) is considered as a topological space with the Harrison topology, which is generated by the sets H(a)={P∈XT(K)∣a∈P} where a∈K×. For r∈ℕ and a1,…,ar∈K×, we put
(2.1)H(a1,…,ar)=H(a1)∩⋯∩H(ar)={P∈X(K)∣a1,…,ar∈P};
the sets of this shape are called Harrison sets, and they form a base of the Harrison topology on XT(K). For a nontrivial Harrison set H of XT(K), the smallest r∈ℕ such that H=H(a1,…,ar) with a1,…,ar∈K× is called the degree ofH; the trivial Harrison sets ∅ and XT(K) are given the degree 0.We write(2.2)signT:WK⟶ZXT(K),[φ]⟼(signP(φ))P∈XT(K)
and denote the kernel of this homomorphism by ITK. (Note that in [11] this notation is used for what is IK/ITK in our notation: the fundamental ideal in the reduced Witt ring WTK=WK/ITK.) For n∈ℕ, let ITnK=InK∩ITK.Letφ be a quadratic form over K. We say that φ is T-positive if DK(φ)⊆T. We say that φ is T-isotropic if ϑ⊗φ is isotropic for some T-positive form ϑ over K; by [11, (1.20)], this is equivalent to having signT(φ)=signT(ψ) for some form ψ over K with dim(ψ)<dim(φ). We write DT(φ) for the set of elements a∈K× such that a∈DK(ϑ⊗φ) for some T-positive form ϑ over K. By [11, (1.16)], signT(φ)=0 if and only if there is a T-positive Pfister form ϑ over K such that ϑ⊗φ is hyperbolic. Moreover, the ideal  ITK is generated by the binary forms 〈1,-t〉 with t∈T×, by [11, (1.26)]. A form ψ over K with dim(ψ)=dim(φ) and signT(ψ)=signT(φ) is said to be T-isometric toφ; by [11, (1.19)] this implies that DT(ψ)=DT(φ). These facts will be used in the next section.Thestability index ofT is defined as
(2.3)st(T)=sup⁡{deg⁡(H)∣HisaHarrisonsetofXT(K)}∈N∪{∞}.
This is reconcilable with the above definition of the stability index of a field. In fact, if K is real then for the preordering S=∑K2∪{0}, we have st(K)=st(S) by [11, (13.1)], and furthermore ISK=WtK and ISnK=ItnK for n≥1.


## 3. Krüskemper's Theorem

LetT be a preordering on K and n∈ℕ. A more general version of Conjecture 1.2 is the following conjecture.Conjecture 3.1.
One hasITnK=ITK·In-1K.This conjecture is proved in [3, (2.7)], but the proof of the nontrivial inclusion ITnK⊆ITK·In-1K uses the deep results in [4]. In [7], Krüskemper sketches an elementary proof of Conjecture 3.1 for n≥st(T). We want to provide a complete exposition of Krüskemper's result, which we achieve with Theorem 3.10. On the way, we shall obtain with Theorem 3.8 that, for any n, a sum of up to three scaled n-fold Pfister forms over K belongs to ITnK if and only if it belongs to ITK·In-1K.We begin with the elementary proof of Conjecture3.1 for n=2, adapted from [1, Chapter XI, (4.1)].Proposition 3.2.
One hasIT2K=ITK·IK.Proof.
Fora∈K× and t∈T×, one has 〈a,-at〉≡〈1,-t〉modITK·IK. For t,t′∈T×, one has 〈1,-t〉⊥〈1,-t′〉≡〈1,-tt′〉modITK·IK. As the ideal ITK is generated by the forms 〈1,-t〉 with t∈T×, any element of IT2K is given by a form φ=〈a1,-a1t1〉⊥⋯⊥〈ar,-artr〉, where r∈ℕ, a1,…,ar∈K×, and t1,…,tr∈T× with t1⋯tr∈K×2. Using induction on r and the above congruences shows that φ∈ITK·IK.Lemma 3.3.
Letψ and ρ be Pfister forms over K, and let ρ′ be the form over K such that ρ=〈1〉⊥ρ′. Then, for b∈DT(ψ⊗ρ′), there exists a Pfister form μ over K with signT(ψ⊗ρ)=signT(ψ⊗〈1,b〉⊗μ).Proof.
There exists aT-positive Pfister form τ such that b∈DK(τ⊗ψ⊗ρ′). By [1, Chapter X, (1.10)], we have τ⊗ψ⊗ρ=τ⊗ψ⊗〈1,b〉⊗μ for a Pfister form μ, and then signT(ψ⊗ρ)=signT(ψ⊗〈1,b〉⊗μ) holds.The next two statements are contained without proof in [8, Section 5].Lemma 3.4 (Marshall).
Letπ be an n-fold Pfister form over K where n≥1. If a∈DT(π′), then there is an (n-1)-fold Pfister form ρ over K such that π≡〈1,a〉⊗ρmodITK·In-1K. If b∈DT(π), then bπ≡πmodITK·InK.Proof.
We use induction onn. For n=1, the first part is trivial. In the induction step, we will use both parts of the statement for n-1 to prove the first part for n. Before we do so, we explain how, for given π, the second part of the statement follows from the first part. This together will give the proof.
Letb∈DT(π). Then, b∈DT(〈1,a〉) for some a∈DT(π′). By Proposition 3.2, we have 〈1,-b〉⊗〈1,a〉∈ITK·IK. If now π≡〈1,a〉⊗ρmodITK·In-1K for an (n-1)-fold Pfister form ρ, then 〈1,-b〉⊗π≡〈1,-b〉⊗〈1,a〉⊗ρ≡0modITK·InK and thus bπ≡πmodITK·InK.
For the induction step, assume now thatn>1 and write π=〈1,c〉⊗σ with c∈K× and an (n-1)-fold Pfister form σ. Let a∈DT(π′). Since π′=σ′⊥cσ, there exist u∈DT(σ′) and v∈DT(σ) such that a∈DK(〈u,cv〉) and thus 〈u,cv〉=〈a,acuv〉. By the induction hypothesis, we then have σ≡〈1,u〉⊗τmodITK·In-2K for some (n-2)-fold Pfister form τ, and σ≡vσmodITK·In-1K. We obtain
(3.1)π≡〈1,cv〉⊗σ≡〈1,u,cv,cuv〉⊗τ≡〈1,a,acuv,cuv〉⊗τmod⁡ITK⋅In-1K
and then choose ρ=〈1,cuv〉⊗τ.Lemma 3.5 (Marshall).
Assume thatn≥1, and let π1 and π2 be two n-fold Pfister forms over K. If signT(π1)=signT(π2), then π1≡π2modITK·In-1K.Proof.
The proof is by induction onn. The claim is trivial for n=1. Write π1=〈1,a〉⊗σ1 with a∈K× and an (n-1)-fold Pfister form σ1. As π1′ and π2′ are T-isometric, we have a∈DT(π1′)=DT(π2′), and thus Lemma 3.4 yields that π2≡〈1,a〉⊗σ2modITK·In-1K for some (n-1)-fold Pfister form σ2. If a∈-T, then πi≡〈1,a〉⊗σi≡0modITK·In-1K for i=1,2. Assume now that a∉-T. Then T′=T+aT is a preordering. For any r∈T′×, there are s,t∈T× with r∈DK(〈s,at〉), and then
(3.2)〈1,a〉⊗〈1,-r〉≡〈1,at,-r,-art〉≡〈1,-s,arst,-art〉≡0mod⁡ITK⋅IK.
This shows that 〈1,a〉·IT′K⊆ITK·IK. Since signT′(σ1)=signT′(σ2), we have σ1≡σ2modIT′K·In-2K by the induction hypothesis. Multiplying with 〈1,a〉, we now obtain that π1≡π2modITK·In-1K.Corollary 3.6.
Letπ be an n-fold Pfister form over K, and ψ a form over K of even dimension. Then, ψ⊗π∈ITn+1K if and only if ψ⊗π∈ITK·InK.Proof.
One implication is clear as triviallyITK·InK⊆ITn+1K. Assume that ψ⊗π∈ITn+1K. Then, there exists a T-positive Pfister form τ such that ψ⊗τ⊗π is hyperbolic. For s∈DK(τ⊗π), the (n+1)-fold Pfister form 〈1,-s〉⊗π lies in the annihilator of τ, thus signT(〈1,-s〉⊗π)=0, and by Lemma 3.5 then 〈1,-s〉⊗π∈ITK·InK. By [1, Chapter XI, (3.1)], the binary forms 〈1,-s〉 with s∈DK(τ⊗π) generate the subideal of IK annihilated by τ⊗π, and since ψ belongs to this ideal, we conclude that ψ⊗π∈ITK·InK.Lemma 3.7.
Assume thatn≥1, and let π1,π2,π3 be n-fold Pfister forms over K and b∈K× such that signT(π1)=signT(π2)+signT(bπ3). Then, there exists an (n-1)-fold Pfister form ψ over K and c1,c2,c3∈K× with c3=c1c2 such that πi≡〈1,-ci〉⊗ψmodITK·In-1K for i=1,2,3.Proof.
Choose anm-fold Pfister form ψ and (n-m)-fold Pfister forms ρ1,ρ2 such that signT(πi)=signT(ψ⊗ρi) for i=1,2, where 0≤m≤n-1 and where m is as large as possible in this range.
We claim thatm=n-1. Suppose on the contrary that m<n-1. Then, the form ϑ=ψ⊗(ρ1′⊥-ρ2′) is T-isotropic, given that signT(ϑ)=signT(bπ3) and dim(ϑ)=2n+1-2m+1>2n=dim(bπ3). So, there exists an element b∈DT(ψ⊗ρ1′)∩DT(ψ⊗ρ2′), and Lemma 3.3 now gives a contradiction to the maximality of m. Hence, ρ1 and ρ2 are 1-fold Pfister forms.
We choosec1,c2∈K× such that ρi=〈1,-ci〉 for i=1,2. Then, we have signT(πi)=signT(〈1,-ci〉⊗ψ) for i=1,2. Let c3=c1c2. It follows that
(3.3)signT(π3)=signT(bπ1⊥-bπ2)=signT(bc2〈1,-c3〉⊗ψ).
Since the signatures of the Pfister forms π3 and 〈1,-c3〉⊗ψ lie in {0,2n}, we thus have signT(πi)=signT(〈1,-ci〉⊗ψ) for i=1,2,3. Now, Lemma 3.5 yields that πi≡〈1,-ci〉⊗ψmodITK·In-1K for i=1,2,3.The following result appears in [5, Theorem 3].Theorem 3.8 (Krüskemper).
Letn≥1. Let φ=a1π1⊥a2π2⊥a3π3 for n-fold Pfister forms π1,π2,π3 over K and a1,a2,a3∈K×. Then, signT(φ)=0 if and only if φ∈ITK·In-1K.Proof.
Ifφ∈ITK·In-1K, then obviously signT(φ)=0. Assume now that signT(φ)=0. Then, signT(a1π1⊥a2π2)=signT(-a3π3), and comparing dimensions we conclude that a1π1⊥a2π2 is T-isotropic, so there is an element a∈DT(a1π1)∩DT(-a2π2). Using Lemma 3.4, we obtain that (-1)i-1aaiπi≡πimodITK·InK for i=1,2. Hence, with d=a3, we have
(3.4)φ≡aπ1⊥-aπ2⊥dπ3mod⁡ITK⋅In-1K.
By Lemma 3.7 applied with b=-ad, there is an (n-1)-fold Pfister form ψ over K and c1,c2,c3∈K× with c3=c1c2 such that πi≡〈1,-ci〉⊗ψmodITK·In-1K for i=1,2,3. It follows that
(3.5)φ≡d〈1,-c1c2,-adc1,adc2〉⊗ψmod⁡ITK⋅In-1K.
As signT(φ)=0, we obtain that signT(〈1,-c1c2,-adc1,adc2〉⊗ψ)=0. By Lemma 3.5, then 〈1,-c1c2,-adc1,adc2〉⊗ψ∈ITK·InK, so φ∈ITK·In-1K.Given a Pfister formπ over K, we put H(π)={P∈XT(K)∣πisP-positive}. Hence, for r≥1 and a1,…,ar∈K×, we have H(a1,…,ar)=H(π) with π=〈1,a1〉⊗⋯⊗〈1,ar〉.Proposition 3.9.
Assume thatn≥st(T). Every class of InK/ITK·In-1K is given by a form ψ1⊗π1⊥⋯⊥ψr⊗πr where r∈ℕ, ψ1,…,ψr are forms over K, and π1,…,πr are n-fold Pfister forms over K such that H(π1),…,H(πr) are nonempty and pairwise disjoint.Proof.
Considera∈K× and an n-fold Pfister form π1 over K. As st(T)≤n, there exist n-fold Pfister forms π2 and π3 over K such that
(3.6)H(π2)=H(π1)∩H(a),H(π3)=H(π1)∩H(-a).
It follows that signT(π1)=signT(π2)+signT(π3). Applying Theorem 3.8 with a1=1 and a2=a3=-1 yields that
(3.7)π1≡π2⊥π3mod⁡ITK⋅In-1K.
Consider now a finite setF⊆K×. For S⊆F, let HS denote the Harrison set consisting of the orderings P∈XT(K) with S⊆P and F∖S⊆-P. Let 𝒫=𝒫(F), the power set of F. Then, (HS)S∈𝒫 is a finite partition of XT(K). (However, HS can be empty for many S∈𝒫.) Induction on the cardinality of F, using the above argument in the induction step, yields that, given an arbitrary n-fold Pfister form ρ over K, there exists a family of n-fold Pfister forms (ρS)S∈𝒫 such that H(ρS)=H(ρ)∩HS for every S∈𝒫 and
(3.8)ρ≡∑S∈PρSmod⁡ITK⋅In-1K.
Moreover, if ρ=〈1,a1〉⊗⋯⊗〈1,an〉 for a1,…,an∈F and S∈𝒫, then H(ρS)=HS in case a1,…,an all belong to S and H(ρS)=∅ otherwise.
An arbitrary element ofInK is given by a form
(3.9)a1ρ1⊥⋯⊥amρm,
where m≥1, a1,…,am∈K× and where ρ1,…,ρm are n-fold Pfister forms over K. Choose a finite set F⊆K× such that for 1≤i≤m there exist ci1,…,cin∈F such that ρi=〈1,ci1〉⊗⋯⊗〈1,cin〉. Put 𝒫=𝒫(F). Using that n≥st(T), we choose a family of n-fold Pfister forms (πS)S∈𝒫 such that H(πS)=HS for S∈𝒫.
By the above argument, we choose for1≤i≤m a family of n-fold Pfister forms (ρi,S)S∈𝒫 such that
(3.10)ρi≡∑S∈Pρi,Smod⁡ITK⋅In-1K
and such that H(ρi,S) is either empty or equal to HS for any S∈𝒫. Using Lemma 3.5, for 1≤i≤m and S∈𝒫, it follows that ρi,S≡0modITK·In-1K if H(ρi,S)=∅ and ρi,S≡πSmodITK·In-1K otherwise. Put 𝒫′={S∈𝒫∣HS≠∅}. For S∈𝒫′, let ψS be the orthogonal sum of those 〈ai〉 with 1≤i≤m such that H(ρi,S)=HS. We conclude that
(3.11)a1ρ1⊥⋯⊥amρm≡∑S∈P'ψS⊗πSmod⁡ITK⋅In-1K.Theorem 3.10 (Krüskemper).
Ifn≥st(T), then ITnK=ITK·In-1K.Proof.
We trivially have thatITK·In-1K⊆ITnK. To prove the converse inclusion, we assume that n≥st(T) and φ∈ITnK. By Proposition 3.9, we have
(3.12)φ≡ψ1⊗π1⊥⋯⊥ψr⊗πrmod⁡ITK⋅In-1K,
where r∈ℕ, ψ1,…,ψr are forms and where π1,…,πr are n-fold Pfister forms such that H(π1),…,H(πr) are nonempty and pairwise disjoint. Using this and the fact that signT(φ)=0, we obtain for 1≤i≤r that signT(ψi⊗πi)=0 and that ψi∈IK because signT(πi)≠0, so that ψi⊗πi∈ITK·InK by Corollary 3.6. Therefore, φ∈ITK·In-1K.



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