Rationally Isotropic Quadratic Spaces

Are Locally Isotropic: II

Ivan Panin and Konstantin Pimenov

Received: September 06, 2009 Revised: April 17, 2010

Abstract. The results of the present article extend the results of [Pa]. The main result of the article is Theorem 1.1 below. The proof is based on a moving lemma from [LM], a recent improvement due to O. Gabber of de Jong’s alteration theorem, and the main theorem of [PR]. A purity theorem for quadratic spaces is proved as well in the same generality as Theorem 1.1, provided that R is local. It generalizes the main purity result from [OP] and it is used to prove the main result in [ChP].

1 Introduction

Let A be a commutative ring and P be a finitely generated projective A-module. An element v ∈ P is called unimodular if the A-submodule vA of P splits off as a direct summand. If P = A^{n} and v = (a_{1}, a_{2}, …, a_{n}) then v is unimodular if and only if a_{1}A + a_{2}A + ⋯ + a_{n}A = A. Let \frac{1}{2} ∈ A. A quadratic space over A is a pair (P, α) consisting of a finitely generated projective A-module P and an A-isomorphism α : P → P^{∗} satisfying α = α^{∗}, where P^{∗} = Hom_{R}(P, R). Two spaces (P, α) and (Q, β) are isomorphic if there exists an A-isomorphism φ : P → Q such that α = φ^{∗} ∘ β ∘ φ. Let (P, φ) be a quadratic space over A. One says that it is isotropic over A, if there exists a unimodular v ∈ P with φ(v) = 0.

Theorem 1.1. Let R be a semi-local regular integral domain containing a field. Assume that all the residue fields of R are infinite and \frac{1}{2} ∈ R. Let K be the fraction field of R and (V, φ) a quadratic space over R. If (V, φ) ⊗_{R} K is isotropic over K, then (V, φ) is isotropic over R.

This Theorem is a consequence of the following result.

Theorem 1.2. Let k be an infinite perfect field of characteristic different from 2, B a k-smooth algebra. Let p_{1}, p_{2}, …, p_{n} be prime ideals of B, S = B−∪_{j=1}^{n}p_{j} and R := B_{S} be the localization of B with respect to S (note that B_{S} is a semilocal ring). Let K be the ring of fractions of R with respect to all non-zero divisors and (V, φ) be a quadratic space over R. If (V, φ) ⊗_{R} K is isotropic over K, then (V, φ) is isotropic over R.

For arbitrary discrete valuation rings, Theorem 1.1 holds trivially. It also holds for arbitrary regular local two-dimensional rings in which 2 is invertible, as proved by M. Ojanguren in [O]. To conclude the Introduction let us add a historical remark which might help the general reader. Let R be a regular local ring, G/R a reductive group scheme. The question whether a principal homogeneous space over R which admits a rational section actually admits a section goes back to the foundations of étale cohomology. It was raised by J.-P. Serre and A. Grothendieck (séminaire Chevalley “Anneaux de Chow”). In the geometric case, this question has essentially been solved, provided that G/R comes from a ground field k. Namely, J.-L. Colliot-Thélène and M. Ojanguren in [CT-O] deal with the case where the ground field k is infinite and perfect. There were later papers [Ra1] and [Ra2] by M.S. Raghunathan, which handled the case k infinite but not necessarily perfect. O. Gabber later announced a proof in the general case. One may then raise the question whether a similar result holds for homogeneous spaces. A specific instance is that of projective homogeneous spaces. An even more specific instance is that of smooth projective quadrics (question raised in [C-T], Montpellier 1977). This last case is handled in the present paper. Remark 3.5 deals with the semi-local case. The key point of the proof of Theorem 1.2 is the combination of the moving lemma in [LM] and Gabber’s improvement of the alteration theorem due to de Jong with the generalization of Springer’s result in [PR]. Theorem 1.1 is deduced from Theorem 1.2 using D. Popescu’s theorem.

2 Auxiliary results

Let k be a field. To prove Theorem 1 we need auxiliary results. We start recalling the notion of transversality as it is defined in [LM, Def.1.1.1].

Definition 2.1. Let f : X → Z, g : Y → Z be morphisms of k-smooth schemes. We say that f and g are transverse if

1. T or_{q}^{𝒪_{Z}} (𝒪_{Y} , 𝒪_{X}) = 0 for all q > 0.

2. The fibre product X ×_{Z} Y is a k-smooth scheme.

Lemma 2.2. Let f : X → Z and g : Y → Z be transverse, and pr_{Y} : Y ×_{Z} X → Y and h : T → Y be transverse, then f and g ∘ h are transverse.

This is just Lemma 1 from [Pa].

Since this moment and till Remark 2.6 (including that Remark) let k be an infinite perfect field of characteristic different from 2. Let U be a smooth irreducible quasi-projective variety over k and let j : u → U be a closed point of U. In particular, the field extension k(u)/k is finite. It is also separable since k is perfect. Thus u = Spec(k(u)) is a k-smooth variety.

Lemma 2.3. Let U be as above. Let Y be a k-smooth irreducible variety of the same dimension as U. Let v = {v_{1}, v_{2}, …, v_{s}} ⊂ U be a finite set of closed points. Let q : Y → U be a projective morphism such that q^{−1}(v) ≠ ∅. Assume q : Y → U and j_{v} : v ↪ U are transverse. Then q is finite étale over an affine neighborhood of the set v ⊂ U.

Proof. There is a v_{i} ∈ v such that q^{−1}(v_{i}) ≠ ∅. By [Pa, Lemma 2] q is finite étale over a neighborhood V_{i} of the point v_{i} ∈ U. This implies that V_{i} ⊂ q(Y). It follows that q(Y) = U, since q is projective and U is irreducible. Whence for each i = 1, 2, …, s one has q^{−1}(v_{i}) ≠ ∅. By [Pa, Lemma 2] for each m = 1, 2, …, s the morphism q is finite étale over a neighborhood V_{m} of the point v_{m} ∈ U. Since U is quasi-projective, q is finite étale over an affine neighborhood V of the set v ⊂ U.

Let U be as above. Let p : 𝒳 → U be a smooth projective k-morphism. Let X = p^{−1}(u) be the fibre of p over u. Since p is smooth the k(u)-scheme X is smooth. Since k(u)/k is separable X is smooth as a k-scheme. Thus for a morphism f : Y → 𝒳 of a k-smooth scheme Y it makes sense to say that f and the embedding i : X ↪ 𝒳 are transverse. So one can state the following

Lemma 2.4. Let p : 𝒳 → U be as above, let j_{v} : v ↪ U be as in Lemma 2.3 and let X = p^{−1}(v) be as above. Let Y be a k-smooth irreducible variety with dim(Y) = dim(U). Let f : Y → 𝒳 be a projective morphism such that f^{−1}(X) ≠ ∅. Suppose that f and the closed embedding i : X ↪ 𝒳 are transverse. Then the morphism q = p ∘ f : Y → U is finite étale over an affine neighborhood of the set v.

Proof. For each i = 1, 2, …, s the extension k(u)/k is finite. Since k is perfect, the scheme v is k-smooth. The morphism p : 𝒳 → U is smooth. Thus the morphism j_{v} and the morphism p are transverse. Morphisms j_{v} and q = p ∘ f are transverse by Lemma 2.2, since j_{v} and f are transverse. One has q^{−1}(v) = f^{−1}(X) ≠ ∅. Now Lemma 2.3 completes the proof of the Lemma.

For a k-smooth variety W let CH_{d}(W) be the group of dimension d algebraic cycles modulo rational equivalence on W (see [Fu]). The next lemma is a variant of the proposition [LM, Prop. 3.3.1] for the Chow groups Ch_{d} := CH_{d}/2CH_{d} of algebraic cycles modulo rational equivalence with ℤ/2ℤ-coefficients.

Lemma 2.5 (A moving lemma). Suppose that k is an infinite perfect field (the characteristic of k is different from 2 as above). Let W be a k-smooth scheme and let i : X ↪ W be a k-smooth closed subscheme. Then Cd_{d}(W) is generated by the elements of the form f_{∗}([Y]) where Y is an irreducible k-smooth variety of dimension d, [Y] ∈ Cd_{d}(Y) is the fundamental class of Y, f : Y → W is a projective morphism such that f and i are transverse and f_{∗} : Ch_{d}(Y) → Ch_{d}(W) is the push-forward.

Proof. The group Ch_{d}(W) is generated by cycles of the form [Z], where Z ⊂ W is a closed irreducible subvariety of dimension d. Since k is perfect of characteristic different from 2, applying a recent result due to Gabber [I, Thm. 1.3], one can find a k-smooth irreducible quasi-projective variety Z′ and a proper morphism π : Z′ → Z with k-smooth quasi-projective variety Z′ and such that the degree [k(Z′) : k(Z)] is odd. The morphism p is necessary projective, since the k-variety Z′ is quasi-projective and p is a proper morphism (see [Ha, Ch.II, Cor.4.8.e]). Write π′ for the composition Z′ → Z ↪ W. Clearly, π′_{∗}([Z′]) = [Z] ∈ Cd_{d}(W). The lemma is not proved yet, since π′ and i are not transverse. However to complete the proof it remains to repeat literally the proof of proposition [LM, Prop. 3.3.1]. The proof of that proposition does not use the resolution of singularities. Whence the lemma.

Remark 2.6. Note that at the end of the previous proof we actually used a Chow version of [LM, Prop. 3.3.1] instead of Prop. 3.3.1 itself.

The following theorem proved in [PR] is a generalization of a theorem of Springer. See [La, Chap.VII, Thm.2.3] for the original theorem by Springer.

Theorem 2.7. Let R be a local Noetherian domain which has an infinite residue field of characteristic different from 2. Let R ⊂ S be a finite R-algebra which is étale over R. Let (V, φ) be a quadratic space over R such that the space (V, φ) ⊗_{R} S contains an isotropic unimodular vector. If the degree [S : R] is odd then the space (V, φ) already contains a unimodular isotropic vector.

Remark 2.8. Theorem 2.7 is equivalent to the main result of [PR], since the R-algebra S from Theorem 2.7 one always has the form R[T]/(F (T)), where F (T) is a separable polynomial of degree [S : R] (see [AK, Chap.VI, Defn.6.11, Thm.6.12]).

Repeating verbatim the proof of Theorem 2.7 given in [PR] we get the following result.

Theorem 2.9. Let R be a semi-local Noetherian integral domain SUCH THAT ALL ITS residue fields ARE INFINITE of characteristic different from 2. Let R ⊂ S be a finite R-algebra which is étale over R. Let (V, φ) be a quadratic space over R such that the space (V, φ) ⊗_{R} S contains an isotropic unimodular vector. If the degree [S : R] is odd then the space (V, φ) already contains a unimodular isotropic vector.

3 Proofs of Theorems 1.2 and 1.1

Proof of Theorem 1.2. Let k be an infinite perfect field of characteristic different from 2. Let p_{1}, p_{2}, …, p_{n} be prime ideals of B, S = B − ∪_{j=1}^{n}p_{j} and R = B_{S} be the localization of B with respect to S. Clearly, it is sufficient to prove the theorem in the case when B is an integral domain. So, in the rest of the proof we will assume that B is an integral domain. We first reduce the proof to the localization at a set of maximal ideals. To do that we follow the arguments from [CT-O, page 101]. Clearly, there exist f ∈ S and a quadratic space (W, ψ) over B_{f} such that (W, ψ) ⊗_{B_{f}} B_{S} = (V, φ). For each index j let m_{j} be a maximal ideal of B containing p_{j} and such that f ∉ m_{j}. Let T = B − ∪_{j=1}^{n}m_{j} . Now B_{T} is a localization of B_{f} and one has B_{f} ⊂ B_{T} ⊂ B_{S} = R. Replace R by B_{T}. From now on and until the end of the proof of Theorem 1.2 we assume that R = 𝒪_{U,{u_{1},u_{2},...,u_{n}}} is the semi-local ring of a finite set of closed points u = {u_{1}, u_{2}, …, u_{n}} on a k- smooth d-dimensional irreducible affine variety U. Let 𝒳 ⊂ P_{R}(V) be a projective quadric given by the equation φ = 0 in the projective space P_{R}(V) = P roj(S^{∗}(V^{∨})). Let X = p^{−1}(u) be the schemetheoretic pre-image of u under the projection p : 𝒳 → Spec(R). Shrinking U we may assume that u is still in U and the quadratic space (V, φ) is defined over U. We still write 𝒳 for the projective quadric in P_{U}(V) given by the equation φ = 0 and still write p : 𝒳 → U for the projection. Let η : Spec(K) → U be the generic point of U and let 𝒳_{η} be the generic fibre of p : 𝒳 → U. Since the equation φ = 0 has a solution over K there exists a K-rational point y of 𝒳_{η}. Let Y ⊂ 𝒳 be its closure in 𝒳 and let [Y] ∈ Ch_{d}(𝒳) be the class of Y in the Chow groups with ℤ/2ℤ-coefficients. Since p is smooth the scheme X is k(u)-smooth. Since k(u)/k is a finite étale algebra X is smooth as a k-scheme. By Lemma 2.5 there exist a finite family of integers n_{r} ∈ ℤ and a finite family of projective morphisms f_{r} : Y_{r} → 𝒳 (with k-smooth irreducible Y_{r}’s of dimension dim(U)) which are transverse to the closed embedding i : X ↪ 𝒳 and such that ∑ n_{r}f_{r,∗}([Y_{r}]) = [Y] in Ch_{d}(𝒳). Shrinking U we may assume that for each index r one has f_{r}^{−1}(X) ≠ ∅. By Lemma 2.4 for any index r the morphism q_{r} = p ∘ f_{r} : Y_{r} → U is finite étale over an affine neighborhood U′ of the set u. Shrinking U we may assume that U′ = U. Let deg : Ch_{0}(𝒳_{η}) → ℤ/2ℤ be the degree map. Since deg(y) = 1 and ∑ n_{r}f_{r,∗}[Y_{r}] = [Y] ∈ Ch_{d}(𝒳) there exists an index r such that the degree of the finite étale morphism q_{r} : Y_{r} → U is odd. Without loss of generality we may assume that the degree of q_{1} is odd. The existence of the Y_{1}-point f_{1} : Y_{1} → 𝒳 of 𝒳 shows that we are under the hypotheses of Theorem 2.9. Hence shrinking U once more we see that there exists a section s : U → 𝒳 of the projection 𝒳 → U. Theorem 1.2 is proven.

Proof of Theorem 1.1. Let R be a regular semi-local integral domain containing a field. Let k be the prime field of R. By Popescu’s theorem R = \underrightarrow{lim} B_{α}, where the B_{α}’s are smooth k-algebras (see [P] or [Sw]). Let can_{α} : B_{α} → R be the canonical k-algebra homomorphism. We first observe that we may replace the direct system of the B_{α}’s by a system of essentially smooth semi-local k-algebras which are integral domains. In fact, if m_{j} is a maximal ideal of R, we can take p_{α,j} := can_{α}^{−1}(m_{j}), S_{α} := B_{α} − ∪_{j=1}^{n}p_{α,j} and replace each B_{α} by (B_{α})_{S_{α}}, Note that in this case the canonical morphisms can_{α} : B_{α} → R take maximal ideals to maximal ones and every B_{α} is a regular semi-local k-algebra. We claim that B_{α} is an integral domain. In fact, since B_{α} is a regular semilocal k-algebra it is a product ∏_{i=1}^{s} B_{α,i} of regular semi-local integral domains B_{α,i}. The ideal q_{α} := can_{α}^{−1}(0) ⊂ B_{α} is prime and is contained in each of the maximal ideals can_{α}^{−1}(m_{j}) of the ring B_{α}. The latter ideal runs over all the maximal ideals of B_{α}. Thus the prime ideal q_{α} is contained in all maximal ideals of B_{α} = ∏_{i=1}^{s} B_{α,i}. Since q_{α} is prime after reordering the indices it must be of the form q_{1} × ∏_{i=2}^{s} B_{α,i}. If s ≥ 2 then the latter ideal is not contained in a maximal ideal of the form ∏_{i=1}^{s−1} B_{α,i} × m for a maximal ideal m of B_{α,s}. Whence s = 1 and B_{α} is indeed an integral domain. There exists an index α and a quadratic space φ_{α} over B_{α} such that φ_{α}⊗_{B_{α}} R ≅ φ. For each index β ≥ α we will write φ_{β} for the B_{β}-space φ_{α} ⊗_{B_{α}} B_{β}. Clearly, φ_{β} ⊗_{B_{β}} R ≅ φ. The space φ_{K} is isotropic. Thus there exists an element f ∈ R such that the space (V_{f}, φ_{f}) is isotropic. There exists an index β ≥ α and a non-zero element f_{β} ∈ B_{β} such that can_{β}(f_{β}) = f and the space φ_{β} localized at f_{β} is isotropic over the ring (B_{β})_{f_{β}}. If char(k) = 0 or if char(k) = p > 0 and the field k is infinite perfect, then by Theorem 1.2 the space φ_{β} is isotropic. Whence the space φ is isotropic too. If char(k) = p > 0 and the field k is finite, then choose a prime number l different from 2 and from p and take the field k_{l} which is the composite of all l-primary finite extensions k′ of k in a fixed algebraic closure k̄ of k. Note that for each field k′′ which is between k and k_{l} and is finite over k the degree [k′′ : k] is a power of l. In particular, it is odd. Note as well that k_{l} is a perfect infinite field. Take the k_{l}-algebra k_{l} ⊗_{k} B_{β}. It is a semi-local essentially k_{l}-smooth algebra, which is not an integral domain in general. The element 1 ⊗ f_{β} is not a zero divisor. In fact, k_{l} is a flat k-algebra and the element f is not a zero divisor in B_{β}. The quadratic space k_{l}⊗_{k}φ_{β} localized at 1⊗f_{β} is isotropic over (k_{l}⊗_{k}B_{β})_{1⊗f_{β}} = k_{l} ⊗_{k} (B_{β})_{f_{β}} and 1 ⊗ f_{β} is not a zero divisor in k_{l} ⊗_{k} B_{β}. By Theorem 1.2 the space k_{l} ⊗_{k} φ_{β} is isotropic over k_{l} ⊗_{k} B_{β}. Whence there exists a finite extension k ⊂ k′ ⊂ k_{l} of k such that the space k′ ⊗_{k} φ_{β} is isotropic over k′ ⊗_{k} B_{β}. Thus the space k′⊗_{k} φ is isotropic over k′⊗_{k} R. Now k′⊗_{k} R is a finite étale extension of R of odd degree. All residue fields of R are infinite. By Theorem 2.9 the space φ is isotropic over R.

To state the first corollary of Theorem 1.1 we need to recall the notion of unramified spaces. Let R be a Noetherian integral domain and K be its fraction field. Recall that a quadratic space (W, ψ) over K is unramified if for every height one prime ideal ℘ of R there exists a quadratic space (V_{𝔭}, φ_{℘}) over R_{℘} such that the spaces (V_{℘}, φ_{℘}) ⊗_{R_{℘}} K and (W, ψ) are isomorphic.

Corollary 3.1 (A purity theorem). Let R be a regular local ring containing a field of characteristic different from 2 and such that the residue field of R is infinite. Let K be the field of fractions of R. Let (W, ψ) be a quadratic space over K which is unramified over R. Then there exists a quadratic space (V, φ) over R extending the space (W, ψ), that is the spaces (V, φ) ⊗_{R} K and (W, ψ) are isomorphic.

Proof. By the purity theorem [OP, Theorem A] there exists a quadratic space (V, φ) over R and an integer n ≥ 0 such that (V, φ)⊗_{R}K ≅ (W, ψ) ⊥ ℍ_{K}^{n}, where ℍ_{K} is a hyperbolic plane. If n > 0 then the space (V, φ) ⊗_{R} K is isotropic. By Theorem 1.1 the space (V, φ) is isotropic too. Thus (V, φ) ≅ (V′, φ′) ⊥ ℍ_{R} for a quadratic space (V′, φ′) over R. Now Witt’s Cancellation theorem over a field [La, Chap.I, Thm.4.2] shows that (V′, φ′) ⊗_{R} K ≅ (W, ψ) ⊥ ℍ_{K}^{n−1}. Repeating this procedure several times we may assume that n = 0, which means that (V, φ) ⊗_{R} K ≅ (W, ψ).

Remark 3.2. Corollary 3.1 is used in the proof of the main result in [ChP]. The main result in [ChP] holds now in the case of a local regular ring R containing a field provided that the residue field of R is infinite and \frac{1}{2} ∈ R.

Corollary 3.3. Let R be a semi-local regular integral domain containing a field. Assume that all the residue fields of R are infinite and \frac{1}{2} ∈ R. Let K be the fraction field of R. Let (V, φ) be a quadratic space over R and let u ∈ R^{×} be a unit. Suppose the equation φ = u has a solution over K then it has a solution over R, that is there exists a vector v ∈ V with φ(v) = u (clearly the vector v is unimodular).

Proof. It is very standard. However for the completeness of the exposition let us recall the arguments from [C-T, Proof of Prop.1.2]. Let (R, −u) be the rank one quadratic space over R corresponding to the unit −u. The space (V, φ)_{K} ⊥ (K, −u) is isotropic thus the space (V, φ) ⊥ (R, −u) is isotropic by Theorem 1.1. By the lemma below there exists a vector v ∈ V with φ(v) = u. Clearly v is unimodular.

Lemma 3.4. Let (V, φ) be as above. Let (W, ψ) = (V, φ) ⊥ (R, −u). The space (W, ψ) is isotropic if and only if there exists a vector v ∈ V with φ(v) = u.

Proof. It is standard. See [C-T, the proof of Proposition 1.2.].

Remark 3.5. It would be nice to extend the result of Corollary 3.1 to the semilocal case. The difficulty is to extend the purity theorem [OP, Theorem A] to that semi-local case.

4 Acknowledgements

The work was supported by the Presidium of the RAS Program “Fundamental Research in modern mathematics”, the joint DFG-RFBR grant 09-01-91333 NNIO-a, and the RFBR-grant 10-01-00551. The Authors thanks very much J.-L. Colliot-Thélène for explaining Gabber’s modification of the de Jong alteration theorem, M. Levine for his comments concerning the proof of proposition [LM, Prop. 3.3.1] and N. Durov for his stimulating interest in the present work.

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Ivan Panin Steklov Institute of Mathematics St.Petersburg Russia paniniv@gmail.com Konstantin Pimenov St.Petersburg State University St.Petersburg Russia kip302002@yahoo.com 524

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