[

[

Asher Auel Department of Mathematics & CS
Emory University
400 Dowman Drive NE W401
Atlanta, GA 30322
auel@mathcs.emory.edu
Abstract

The Milnor conjecture has been a driving force in the theory of quadratic forms over fields, guiding the development of the theory of cohomological invariants, ushering in the theory of motivic cohomology, and touching on questions ranging from sums of squares to the structure of absolute Galois groups. Here, we survey some recent work on generalizations of the Milnor conjecture to the context of schemes (mostly smooth varieties over fields of characteristic ). Surprisingly, a version of the Milnor conjecture fails to hold for certain smooth complete -adic curves with no rational theta characteristic (this is the work of Parimala, Scharlau, and Sridharan). We explain how these examples fit into the larger context of the unramified Milnor question, offer a new approach to the question, and discuss new results in the case of curves over local fields and surfaces over finite fields.

\articleinfo\SelectTips

cm10 Remarks on the Milnor conjecture over schemes] Remarks on the Milnor conjecture over schemes \subjclass[2010]Primary 11-02, 19-02; Secondary 11Exx, 19G12

The first cases of the (as of yet unnamed) Milnor conjecture were studied in Pfister’s Habilitationsschrift [85] in the mid 1960s. As Pfister [86, p. 3] himself points out, “[the Milnor conjecture] stimulated research for quite some time.” Indeed, it can be seen as one of the driving forces in the theory of quadratic forms since Milnor’s original formulation [66] in the early 1970s.

The classical cohomological invariants of quadratic forms (rank, discriminant, and Clifford–Hasse–Witt invariant) have a deep connection with the history and development of the subject. In particular, they are used in the classification (Hasse–Minkowski local-global theorem) of quadratic forms over local and global fields. The first “higher invariant” was described in Arason’s thesis [1], [3]. The celebrated results of Merkurjev [62] and Merkurjev–Suslin [64] settled special cases of the Milnor conjecture in the early 1980s, and served as a starting point for Voevodsky’s development of the theory of motivic cohomology. Other special cases were settled by Arason–Elman–Jacob [5] and Jacob–Rost [46]. Voevodsky’s motivic cohomology techniques [99] ultimately led to a complete solution of the Milnor conjecture, for which he was awarded the 2002 Fields Medal.

The consideration of quadratic forms over rings (more general than fields) has its roots in the number theoretic study of lattices (i.e. quadratic forms over ) by Gauss as well as the algebraic study of division algebras and hermitian forms (i.e. quadratic forms over algebras with involution) by Albert. A general framework for the study of quadratic forms over rings was established by Bass [17], with the case of (semi)local rings treated in depth by Baeza [9]. Bilinear forms over Dedekind domains (i.e. unimodular lattices) were studied in a number theoretic context by Fröhlich [35], while the consideration of quadratic forms over algebraic curves (and their function fields) was initiated by Geyer, Harder, Knebusch, Scharlau [44], [39], [53], [55]. The theory of quadratic (and bilinear) forms over schemes was developed by Knebusch [54], [52], and utilized by Arason [2], Dietel [27], Parimala [76], [83], Fernández-Carmena [32], Sujatha [94], [82], Arason–Elman–Jacob [6], [7], and others. A theory of symmetric bilinear forms in additive and abelian categories was developed by Quebbemann–Scharlau–Schulte [87], [88]. Further enrichment came eventually from the triangulated category techniques of Balmer [10], [11], [12], and Walter [101]. This article will focus on progress in generalizing the Milnor conjecture to these contexts.

These remarks grew out of a lecture at the RIMS-Camp-Style seminar “Galois-theoretic Arithmetic Geometry” held October 19-23, 2010, in Kyoto, Japan. The author would like to thank the organizers for their wonderful hospitality during that time. He would also like to thank Stefan Gille, Moritz Kerz, R. Parimala, and V. Suresh for many useful conversations. The author acknowledges the generous support of the Max Plank Institute for Mathematics in Bonn, Germany where this article was written under excellent working conditions. This author is also partially supported by National Science Foundation grant MSPRF DMS-0903039.

Conventions.

A graded abelian group or ring will be denoted by . If is a decreasing filtration of a ring by ideals, denote by the associated graded ring. Denote by the elements of order 2 in an abelian group . All abelian groups will be written additively.

1 The Milnor conjecture over fields

Let be a field of characteristic . The total Milnor -ring was introduced in [66]. The total Galois cohomology ring is canonically isomorphic, under our hypothesis on the characteristic of , to the total Galois cohomology ring with coefficients in the trivial Galois module . The Witt ring of nondegenerate quadratic forms modulo hyperbolic forms has an decreasing filtration generated by powers of the fundamental ideal of even rank forms. The Milnor conjecture relates these three objects: Milnor -theory, Galois cohomology, and quadratic forms.

The quotient map induces a graded ring homomorphism called the norm residue symbol by Bass–Tate [16]. The Pfister form map given by induces a group homomorphism (see Scharlau [91, 2 Lemma 12.10]), which extends to a surjective graded ring homomorphism , see Milnor [66, Thm. 4.1].

Theorem 1 (Milnor conjecture).

Let be a field of characteristic . There exists a graded ring homomorphism called the higher invariants of quadratic forms, which fits into the following diagram

of isomorphisms of graded rings.

Many excellent introductions to the Milnor conjecture and its proof exist in the literature. For example, see the surveys of Kahn [47], Friedlander–Rapoport–Suslin [34], Friedlander [33], Pfister [84], and Morel [68].

The conjecture breaks up naturally into three parts: the conjecture for the norm residue symbol , the conjecture for the Pfister form map , and the conjecture for the higher invariants . Milnor [66, Question 4.3, §6] originally made the conjecture for and , which was already known for finite, local, global, and real closed fields, see [66, Lemma 6.2]. For general fields, the conjecture for follows from Hilbert’s theorem 90, and for and by elementary arguments. The conjecture for is easy, see Pfister [85]. Merkurjev [62] proved the conjecture for (hence for as well), with alternate proofs given by Arason [4], Merkurjev [63], and Wadsworth [100]. The conjecture for was settled by Merkurjev–Suslin [64] (and independently by Rost [89]). The conjecture for can be divided into two parts: to show the existence of maps (which are a priori only defined on generators, the Pfister forms), and then to show they are surjective. The existence of was proved by Arason [1], [3]. The existence of was proved by Jacob–Rost [46] and independently Szyjewski [93]. Voevodsky [99] proved the conjecture for . Orlov–Vishik–Voevodsky [73] proved the conjecture for , with different proofs given by Morel [69] and Kahn–Sujatha [48].

1.1 Classical invariants of quadratic forms

The theory of quadratic forms over a general field has its genesis in Witt’s famous paper [102]. Because of the assumption of characteristic , we do not distinguish between quadratic and symmetric bilinear forms. The orthogonal sum and tensor product give a semiring structure on the set of isometry classes of symmetric bilinear forms over . The hyperbolic plane is the symmetric bilinear form , where and . The Witt ring of symmetric bilinear forms is the quotient of the Grothendieck ring of nondegenerate symmetric bilinear forms over with respect to and , modulo the ideal generated by the hyperbolic plane, see Scharlau [91, Ch. 2].

The rank of a bilinear form is the -vector space dimension of . Since the hyperbolic plane has rank 2, the rank modulo 2 is a well defined invariant of an element of the Witt ring, and gives rise to a surjective ring homomorphism

whose kernel is the fundamental ideal .

The signed discriminant of a non-degenerate bilinear form is defined as follows. Choosing an -vector space basis of , we consider the Gram matrix of , i.e. the matrix given by . Then is given by the formula , where . The Gram matrix of , with respect to a different basis for with change of basis matrix , is . Thus , which depends on the choice of basis, is only well-defined up to squares. For , denote by its class in the abelian group . The signed discriminant of is defined as . Introducing the sign into the signed discriminant ensures its vanishing on the ideal of hyperbolic forms, hence it descents to the Witt group. While the signed discriminant is not additive on , its restriction to gives rise to a group homomorphism

which is easily seen to be surjective. It’s then not difficult to check that its kernel coincides with the square of the fundamental ideal. See Scharlau [91, §2.2] for more details.

The Clifford invariant of a non-degenerate symmetric bilinear form is defined in terms of its Clifford algebra. The Clifford algebra of is the quotient of the tensor algebra by the two-sided ideal generated by . If has rank , then inherits the structure of a -graded semisimple -algebra of -dimension , see Scharlau [91, §9.2]. By the structure theory of the Clifford algebra, or is a central simple -algebra depending on whether has even or odd rank, respectively. The Clifford invariant is then the Brauer class of or , respectively. Since the Clifford algebra and its even subalgebra carry canonical involutions of the first kind, their respective classes in the Brauer group are of order 2, see Knus [57, § IV.7.8]. While the Clifford invariant is not additive on , its restriction to gives rise to a group homomorphism

see Knus [57, IV Prop. 8.1.1].

Any symmetric bilinear form over a field of characteristic can be diagonalized, i.e. a basis can be chosen for so that the Gram matrix is diagonal. For , we write for the standard symmetric bilinear form with associated diagonal Gram matrix. For , denote by the (quaternion) -algebra generated by symbols and subject to the relations , , and . For example, is Hamilton’s ring of quaternions. Then the discriminant and Clifford invariant can be conveniently calculated in terms of a diagonalization. For , we have

and

(1)

where

see Lam [59], Scharlau [91, II.12.7] or Esnault–Kahn–Levine–Viehweg [30, §1].

2 Globalization of cohomology theories

Generalizations (what we will call globalizations) of the Milnor conjecture to the context of rings and schemes have emerged from many sources, see Parimala [75], Colliot-Thélène–Parimala [21], Parimala–Sridharan [78], Monnier [67], Pardon [74], Elbaz-Vincent–Müller-Stach [28], Gille [41], and Kerz [49]. To begin with, one must ask for appropriate globalizations of the objects in the conjecture: Milnor -theory, Galois cohomology theory, and the Witt group with its fundamental filtration. While there are many possible choices of such globalizations, we will focus on two types: global and unramified.

2.1 Global globalization

Let be a field of characteristic . Let (resp. ) be the category of fields (resp. commutative unital rings) with an -algebra structure together with -algebra homomorphisms. Let be the category of -schemes, and the category of smooth -schemes. We will denote, by the same names, the associated (large) Zariski sites. Let (resp. ) be the category of abelian groups (resp. graded abelian groups), we will always consider as embedded in in degree 0.

Let be a functor. A globalization of to rings (resp. schemes) is a functor (resp. contravariant functor ) extending . If is a globalization of to rings, then we can define a globalization to schemes by taking the sheaf associated to the presheaf on (always considered with the Zariski topology).

“Naïve” Milnor -theory.

For a commutative unital ring , mimicking Milnor’s tensorial construction (with the additional relation that , which is automatic for fields) yields a graded ring , which should be referred to as “naïve” Milnor -theory. This already appears in Guin [43, §3] and later studied by Elbaz-Vincent–Müller-Stach [28]. Naïve Milnor -theory has some bad properties when has small finite residue fields, see Kerz [50] who also provides a improved Milnor -theory repairing these defects. Thomason [96] has shown that there exists no globalization of Milnor -theory to smooth schemes which satisfies -homotopy invariance and has a functorial homomorphism to algebraic -theory.

Étale cohomology.

Étale cohomology provides a natural globalization of Galois cohomology to schemes. We will thus consider the functor on .

Global Witt group.

For a scheme , the global Witt group of regular symmetric bilinear forms introduced by Knebusch [52] provides a natural globalization of the Witt group to schemes. Other possible globalizations are obtained from the Witt groups of triangulated category with duality introduced by Balmer [10], [11], [12], [13]. These include: the derived Witt group of the bounded derived category of coherent locally free -modules; the coherent Witt group of the bounded derived category of quasicoherent -modules with coherent cohomology (assuming has a dualizing complex, see Gille [40, §2.5], [41, §2]); the perfect Witt group of the derived category of perfect complexes of -modules. The global and derived Witt groups are canonically isomorphic by Balmer [12, Thm. 4.3]. All of the above Witt groups are isomorphic (though not necessarily canonically) if is assumed to be regular.

Fundamental filtration and the classical invariants.

Globalizations of the classical invariants of quadratic forms are defined as follows. Let be a regular symmetric bilinear form of rank on .

The rank (modulo 2) of gives rise to a functorial homomorphism

whose kernel is called the fundamental ideal of .

The signed discriminant form gives rise to a functorial homomorphism

see Knus [57, III §4.2] and Milne [65, III §4]. Alternatively, the center of the (even) Clifford -algebra of defines a class in called the Arf invariant, which coincides with the signed discriminant under the canonical morphism (see Knus [57, IV Prop. 4.6.3] or Parimala–Srinivas [81, §2.2]). Denote the kernel of by , which is an ideal of . Note that may not be the square of the ideal .

The Clifford -algebra gives rise to a functorial homomorphism

called the Clifford invariant, see Knus–Ojanguren [58] and Parimala–Srinivas [81, §2]. Denote the kernel of by , which is an ideal of .

As Parimala–Srinivas [81, p. 223] point out, there is no functorial map lifting the Clifford invariant. Instead, we can work with Grothendieck–Witt groups. The rank (modulo 2) gives rise to a functorial homomorphism

with kernel denoted by . The signed discriminant gives rise to a functorial homomorphism

with kernel denoted by . The class of the Clifford -algebra, together with it’s canonical involution (via the “involutive” Brauer group construction of Parimala–Srinivas [81, §2]), gives rise to a functorial homomorphism

also see Knus–Parimala–Sridharan [56]. Denote the kernel of by , which is an ideal of .

Lemma 2.1.

Let be a scheme satisfying . Then under the quotient map , the image of the ideal is precisely the ideal , for .

Proof.

For this is a consequence of the following diagram with exact rows and columns

which is commutative since hyperbolic spaces have even rank and trivial signed discriminant. Here, is the Grothendieck group of locally free -modules of finite rank and is the hyperbolic map .

For , we have the formula due to Esnault–Kahn–Viehweg [29, Prop. 5.5] (combined with (1)). Here is the 1st Chern class modulo 2, defined as the first coboundary map in the long-exact sequence in étale cohomology

arising from the Kummer exact sequence

see Grothendieck [42]. The claim then follows by a diagram chase through

where the right vertical column arises from the Kummer sequence, and is the subgroup of generated by locally free -modules whose determinant is a square. ∎

Remark 2.1.

The hypothesis that is satisfied if is a quasi-compact quasi-separated scheme admitting an ample invertible sheaf by de Jong’s extension [26] (see also [61, Th. 2.2.2.1]) of a result of Gabber [36].

The existence of global globalizations of the higher invariants (e.g. a globalization of the Arason invariant) remains a mystery. Esnault–Kahn–Levine–Viehweg [30] have shown that for a regular symmetric bilinear form that represents a class in , the obstruction to having an Arason invariant in is precisely the 2nd Chern class in the Chow group modulo 2 (note that the invariant of [30] is trivial if represents a class in ). They also provide examples where this obstruction does not vanish. On the other hand, higher cohomological invariants always exist in unramified cohomology.

2.2 Unramified globalization

A functorial framework for the notion of “unramified element” is established in Colliot-Thélène [20, §2]. See also the survey by Zainoulline [103, §3]. Rost [90, Rem. 5.2] gives a different perspective in terms of cycle modules, also see Morel [69, §2]. Assume that has finite Krull dimension and is equidimensional over a field . For simplicity of exposition, assume that is integral. Denote by its set of codimension points.

Denote by the category of local -algebras together with local -algebra morphisms. Given a functor , call

the group of unramified elements of over . Then is a globalization of to schemes.

Given a functor , there is a natural map . If this map is injective, surjective, or bijective we say that the injectivity, weak purity, or purity property hold, respectively. Whether these properties hold for various functors and schemes is the subject of many conjectures and open problems, see Colliot-Thélène [20, §2.2] for examples.

Unramified Milnor -theory.

Define the unramified Milnor -theory (resp. modulo 2) of to be the graded ring of unramified elements (resp. ) of the “naïve” Milnor -theory (resp. modulo 2) functor (resp. ) restricted to , see §2.1. Let be the Zariski sheaf on associated to “naïve” Milnor -theory and the associate sheaf quotient, which is the Zariski sheaf associated to the presheaf , see Morel [69, Lemma 2.7]. Then and when is smooth over an infinite field (compare with the remark in §2.1) by the Bloch–Ogus–Gabber theorem for Milnor -theory, see Colliot-Thélène–Hoobler–Kahn [23, Cor. 5.1.11, §7.3(5)]. Also, see Kerz [49]. Note that the long exact sequence in Zariski cohomology yields a short exact sequence

still assuming is smooth over an infinite field.

Unramified cohomology.

Define the unramified étale cohomology (modulo 2) of to be the graded ring of unramified elements of the functor . Letting be the Zariski sheaf on associated to the functor , then when is smooth over a field of characteristic by the exactness of the Gersten complex for étale cohomology, see Bloch–Ogus [19, §2.1, Thm. 4.2].

Unramified fundamental filtration of the Witt group.

Define the unramified Witt group of to be the abelian group of unramified elements of the global Witt group functor . Letting be the Zariski sheaf associated to the global Witt group functor, then when is regular over a field of characteristic by Ojanguren–Panin [72] (also see Morel [69, Thm. 2.2]). Writing , then the functors are Zariski sheaves (still assuming is regular), denoted by , which form a filtration of , see Morel [69, Thm. 2.3].

Note that the long exact sequence in Zariski cohomology yields short exact sequences

where and we are still assuming is regular over a field of characteristic . If the obstruction group is nontrivial, then not every element of is represented by a quadratic form on . If is the spectrum of a regular local ring, then , see Morel [69, Thm. 2.12].

Remark 2.2.

As before, the notation does not necessarily mean the th power of . This is true, however, when is the spectrum of a regular local ring containing an infinite field of characteristic , see Kerz–Müller-Stach [51, Cor. 0.5].

2.3 Gersten complexes

Gersten complexes (Cousin complexes) exists in a very general framework, but for our purposes, we will only need the Gersten complex for Milnor -theory, étale cohomology, and Witt groups.

Gersten complex for Milnor -theory.

Let be a regular excellent integral -scheme. Let denote the Gersten complex for Milnor -theory

where is the “tame symbol” homomorphism defined in Milnor [66, Lemma 2.1]. We have that . See Rost [90, §1] or Fasel [31, Ch. 2] for more details. We will also consider the Gersten complex for Milnor -theory modulo 2, for which we have that .

Gersten complex for étale cohomology.

Let be a smooth integral -scheme, with of characteristic . Let denote the Gersten complex for étale cohomology

where and is the homomorphism induced from the spectral sequence associated to the coniveau filtration, see Bloch–Ogus [19]. Then we have that is a resolution of .

Gersten complex for Witt groups.

Let be a regular integral -scheme of finite Krull dimension. Let denote the Gersten–Witt complex

where is the homomorphism induced from the second residue map for a set of choices of local parameters, see Balmer–Walter [15]. Because of these choices, is only defined up to isomorphism, though there is a canonical complex defined in terms of Witt groups of finite length modules over the local rings of points. We have that .

Fundamental filtration.

The filtration of the Gersten complex for Witt groups induced by the fundamental filtration was first studied methodically by Arason–Elman–Jacob [5], see also Parimala–Sridharan [78], Gille [41], and Fasel [31, §9].

The differentials of the Gersten complex for Witt groups respect the fundamental filtration as follows:

see Fasel [31, Thm. 9.2.4] and Gille [41]. Thus for all we have complexes

which provide a filtration of in the category of complexes of abelian groups. Here we write for . We have that .

The canonical inclusion respects the differentials, and so defines a cokernel complex

see Fasel [31, Déf. 9.2.10], where for a field . We have that

Unramified norm residue symbol.

The norm residue symbol for fields provides a morphism of complexes , where the map on terms of degree is . By the Milnor conjecture for fields, this is an isomorphism of complexes. Upon restriction, we have an isomorphism . Upon further restriction, we have an injection .

Unramified Pfister form map.

The Pfister form map for fields provides a morphism of complexes , where the map on terms of degree is . Upon restriction, we have a homomorphism . See Fasel [31, Thm. 10.2.6].

Unramified higher cohomological invariants.

By the Milnor conjecture for fields, there exists a higher cohomological invariant morphism of complexes , where the map on terms of degree is . Upon restriction, we have homomorphisms factoring through to .

Furthermore, on the level of complexes, the higher cohomological invariant morphisms factors through to a morphism of complexes , which by the Milnor conjecture over fields, is an isomorphism. Upon restriction, we have isomorphisms . Also see Morel [69, §2.3].

2.4 Motivic globalization

There is another important globalization of Milnor -theory and Galois cohomology, but we only briefly mention it here. Conjectured to exist by Beĭlinson [18] and Lichtenbaum [60], and then constructed by Voevodsky [99], motivic complexes modulo 2 give rise to Zariski and étale motivic cohomology groups modulo 2 and .

For a field , Nesterenko–Suslin [70] and Totaro [97] establish a canonical isomorphism while the work of Bloch, Gabber, and Suslin (see the survey by Geisser [37, §1.3.1]) establishes an isomorphism (for of characteristic ). The natural pullback map

induced from the change of site is then identified with the norm residue homomorphism. Thus and provide motivic globalizations of the mod 2 Milnor -theory and Galois cohomology functors, respectively. On the other hand, there does not seem to exist a direct motivic globalization of the Witt group or its fundamental filtration.

3 Globalization of the Milnor conjecture

Unramified.

Let be a field of characteristic . Summarizing the results cited in §2.2–2.3, we have a commutative diagram of isomorphisms

of sheaves of graded abelian groups on . In particular, we have such a commutative diagram of isomorphisms on the level of global sections. What we will consider as a globalization of the Milnor conjecture — the unramified Milnor question — is a refinement of this.

  • Let be a smooth scheme over a field of characteristic . Then restricting gives rise to a homomorphism .

Question 3.1 (Unramified Milnor question).

Let be a smooth scheme over a field of characteristic . Consider the following diagram:

  1. Is the inclusion surjective?

  2. Is the inclusion surjective?

  3. Does the restriction of to have image contained in ? If so, is it an isomorphism?

Note that in degree , Questions 3.1 (1), (2), and (3) can be rephrased in terms of the obstruction groups, respectively: does vanish; does vanish; and does the restriction of yield a map and is it an isomorphism?

From now on we shall focus mainly on the unramified Milnor question for quadratic forms (i.e. Question 3.1(1)), which was already explicitly asked by Parimala–Sridharan [78, Question Q].

Global Grothendieck–Witt.

We mention a global globalization of the Milnor conjecture for quadratic forms. Because of the conditional definition of the global cohomological invariants, we restrict ourselves to the classical invariants on Grothendieck–Witt groups defined in §2.1.

Question 3.2 (Global Merkurjev question).

Let be a regular scheme with 2 invertible. For , consider the homomorphisms,

induced from the (classical) cohomological invariants on Grothendieck–Witt groups. Are they surjective?

This can be viewed as a globalization of Merkurjev’s theorem. Indeed, first note that the cases of Question 3.2 are easy. Next, a consequence of Lemma 2.1 is that is surjective if and only is surjective. This, in turn, is a consequence of a positive answer to Question 3.1(1) for any satisfying weak purity for the Witt group (see §3.1 for examples).

Motivic.

The globalization of the Milnor conjecture for Milnor -theory using Zariski and étale motivic cohomology modulo 2 (see §2.4) is the modulo 2 case of the Beĭlinson–Lichtenbaum conjecture: for a smooth variety over a field, the canonical map is an isomorphism for . The combined work of Suslin–Voevodsky [95] and Geisser–Levine [38] show the Beĭlinson–Lichtenbaum conjecture to be a consequence of the Bloch–Kato conjecture, a proof of which has been announced by Rost and Voevodsky.

3.1 Some purity results

In this section we review some of the purity results (see §2.2) relating the global and unramified Witt groups and cohomology.

Purity results for Witt groups.

For a survey on purity results for Witt groups, see Zainoulline [103]. Purity for the global Witt group means that the natural map is an isomorphism.

Theorem 2.

Let be a regular integral noetherian scheme with 2 invertible. Then purity holds for the global Witt group functor under the following hypotheses:

  1. is dimension ,

  2. is the spectrum of a regular local ring of dimension ,

  3. is the spectrum of a regular local ring containing a field.

For part (1), the case of dimension is due to Colliot-Thélène–Sansuc [22, Cor. 2.5], the case of dimension and affine is due to Ojanguren–Parimala–Sridharan–Suresh [71], and for the general case (as well as (2)) see Balmer–Walter [15]. For (3), see Ojanguren–Panin [72].

As a consequence, for any scheme over which purity for the Witt group holds, the unramified Milnor question for (with ) is equivalent to the analogous global Milnor question. This is especially useful for the case of curves.

Purity results for étale cohomology.

For geometrically locally factorial and integral, the purity property holds for unramified cohomology in degree , i.e.

see Colliot-Thélène–Sansuc [24, Cor. 3.2, Prop. 4.1].

For smooth over a field of characteristic , we have a canonical identification by Bloch–Ogus [19] such that the canonical map is the map arising from the Kummer exact sequence already considered in the proof of Lemma 2.1.

3.2 Positive results

We now survey some of the known positive cases of the unramified Milnor question in the literature.

Theorem 3 (Kerz–Müller-Stach [51, Cor. 0.8], Kerz [49, Thm. 1.2]).

Let be a local ring with infinite residue field of characteristic . Then the unramified Milnor question (all parts of Question 3.1) has a positive answer over .

Hoobler [45] had already proved this in degree .

The following result was communicated to us by Stefan Gille (who was inspired by Totaro [98]).

Theorem 4.

Let be a proper smooth integral variety over a field of characteristic . If is -rational then the unramified Milnor question (all parts of Question 3.1) has a positive answer over .

Proof.

The groups , , and are birational invariants of smooth proper -varieties. To see this, one can use Colliot-Thélène [20, Prop. 2.1.8e] and the fact that the these functors satisfy weak purity for regular local rings (see Theorem 2). Another proof uses the fact that the complexes , , and are cycle modules in the sense of Rost, see [90, Cor. 12.10]. In any case, by Colliot-Thélène [20, Prop. 2.1.9] the pullback induces isomorphisms (first proved by Milnor [66, Thm. 2.3] for ), , and for all and . In particular, and , and the theorem follows from the Milnor conjecture over fields. ∎

The following positive results are known for low dimensional schemes. Recall the notion of cohomological dimension of a field (see Serre [92, I §3.1]), virtual cohomological 2-dimension and their 2-primary versions. Denoting by any of these notions of dimension, note that if and then .

Theorem 5 (Parimala–Sridharan [78], Monnier [67]).

Let be a smooth integral curve over a field of characteristic . Then the unramified Milnor question for quadratic forms (Question 3.1(1)) has a positive answer over in the following cases:

  1. ,

  2. ,

  3. and is affine,

  4. and is affine.

Proof.

For (1), this follows from Parimala–Sridharan [78, Lemma 4.1] and the fact that is always surjective. For (2), the case (i.e.  is real closed) is contained in Monnier [67, Cor. 3.2] and the case follows from a straightforward generalization to real closed fields of the results in [78, §5] for the real numbers. For (3), see [78, Lemma 4.2]. For (4), the statement follows from a generalization of [78, Thm. 6.1]. ∎

We wonder whether can be replaced by in Theorem 5. Parimala–Sridharan [78, Rem. 4] ask whether there exist affine curves (over a well-chosen field) over which the unramified Milnor question has a negative answer.

For surfaces, there are positive results are in the case of . If is algebraically closed, then the unramified Milnor question for quadratic forms (Question 3.1(1)) has a positive answer by a direct computation, see Fernández-Carmena [32]. If is real closed, one has the following result.

Theorem 6 (Monnier [67, Thm. 4.5]).

Let be smooth integral surface over a real closed field . If the number of connected components of is (i.e. in particular if ), then the unramified Milnor question for quadratic forms (Question 3.1(1)) has a positive answer over .

Examples of surfaces with many connected components over a real closed field, and over which the unramified Milnor question still has a positive answer, are also given in Monnier [67].

Finally, as a consequence of [8, Cor. 3.4], the unramified Milnor question for quadratic forms (Question 3.1(1)) has a positive answer over any scheme satisfying: is generated by quaternion Azumaya algebras (i.e. indexperiod for 2-torsion classes); or is generated by Azumaya algebras of degree dividing 4 (i.e. indexperiod for 2-torsion classes) and is 2-divisible. In particular, this recovers the known cases of curves over finite fields (by class field theory) and surfaces over algebraically closed fields (by de Jong [25]).

4 Negative results

Alex Hahn asked if there exists a ring over which the global Merkurjev question (Question 3.2) has a negative answer, i.e.  is not surjective. The results of Parimala, Scharlau, and Sridharan [77], [78], [79], show that there exist smooth complete curves (over -adic fields ) over which the unramified Milnor question (Question 3.1(1)) in degree 2 (and hence, by purity, the global Merkurjev question) has a negative answer.

Remark 4.1.

The assertion (in Gille [41, §10.7] and Pardon [74, §5]) that the unramified Milnor question (Question 3.1(1)) has a positive answer over any smooth scheme (over a field of characteristic ) is incorrect. In these texts, the distinction between the groups and is not made clear.

Definition 4.1 (Parimala–Sridharan [78]).

A scheme over a field has the extension property for quadratic forms if there exists such that every regular quadratic form on extends to .

Proposition 4.1 (Parimala–Sridharan [78, Lemma 4.3]).

Let be a field of characteristic and with and a smooth integral -curve. Then the unramified Milnor question for quadratic forms (Question 3.1(1)) has a positive answer for if and only if has the extension property.

The extension property is guaranteed when a residue theorem holds for the Witt group. The reside theorem for is due to Milnor [66, §5]. For nonrational curves, the choice of local uniformizers inherent in defining the residue maps is eliminated by considering quadratic forms with values in the canonical bundle .

Definition 4.2.

Let be a scheme and an invertible -module. An (-valued) symmetric bilinear form on is a triple , where is a locally free -module of finite rank and is an -module morphism.

Theorem 7 (Geyer–Harder–Knebusch–Scharlau [39]).

Let be a smooth proper integral curve over a field of characteristic . Then there is a canonical complex (which is exact at the first two terms)

and thus in particular has a residue theorem.

Now any choice of isomorphism , induces a group isomorphism via . Thus in particular, if is a square in , then has the extension property. Conversely:

Theorem 8 (Parimala–Sridharan [79, Thm. 3]).

Let be a local field of characteristic and a smooth integral hyperelliptic -curve of genus with . Then the unramified Milnor question for quadratic forms (Question 3.1(1)) holds over if and only if is a square.

Example 4.1 (Parimala–Sridharan [79, Rem. 3]).

Let be the smooth proper hyperelliptic curve over with affine model . One can show using [77, Thm. 2.4] that is not a square. The point defines a -rational point of . Hence by Theorem 8, the unramified Milnor question has a negative answer over .

Note that possible counter examples which are surfaces could be extracted from the following result.

Theorem 9 (Monnier [67, Thm. 4.5]).

Let be a smooth integral surface over a real closed field . Then the unramified Milnor question for quadratic forms (Question 3.1(1)) has a positive answer over if and only if the cokernel of the mod 2 signature homomorphism is 4-torsion.

5 Line bundle-valued quadratic forms

Let be a smooth -scheme. Let be the Witt group of -valued symmetric bilinear forms on and