The Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is quasi-split and R is unramified. Some of the techniques that allow us to overcome obstacles that have so far kept the mixed characteristic case out of reach include a version of Noether normalization over discrete valuation rings, as well as a suitable presentation lemma for smooth relative curves in mixed characteristic that facilitates passage to the relative affine line via excision and patching.

Although there is no natural internal product for hermitian forms over an algebra with involution of the first kind, we describe how to multiply two $\varepsilon $ -hermitian forms to obtain a quadratic form over the base field. This allows to define a commutative graded ring structure by taking together bilinear forms and $\varepsilon $ -hermitian forms, which we call the mixed Witt ring of an algebra with involution. We also describe a less powerful version of this construction for unitary involutions, which still defines a ring, but with a grading over $\mathbb {Z}$ instead of the Klein group.

We first describe a general framework for defining graded rings out of monoidal functors from monoidal categories with strong symmetry properties to categories of modules. We then give a description of such a strongly symmetric category $\mathbf {Br}_h(K,\iota )$ which encodes the usual hermitian Morita theory of algebras with involutions over a field K.

We can therefore apply the general framework to $\mathbf {Br}_h(K,\iota )$ and the Witt group functors to define our mixed Witt rings, and derive their basic properties, including explicit formulas for products of diagonal forms in terms of involution trace forms, explicit computations for the case of quaternion algebras, and reciprocity formulas relative to scalar extensions.

We intend to describe in future articles further properties of those rings, such as a $\lambda $ -ring structure, and relations with the Milnor conjecture and the theory of signatures of hermitian forms.

The main purpose of this article is to define a quadratic analogue of the Chern character, the so-called Borel character, that identifies rational higher Grothendieck-Witt groups with a sum of rational Milnor-Witt (MW)-motivic cohomologies and rational motivic cohomologies. We also discuss the notion of ternary laws due to Walter, a quadratic analogue of formal group laws, and compute what we call the additive ternary laws, associated with MW-motivic cohomology. Finally, we provide an application of the Borel character by showing that the Milnor-Witt K-theory of a field F embeds into suitable higher Grothendieck-Witt groups of F modulo explicit torsion.

We show how the techniques of Voevodsky’s proof of the Milnor conjecture and the Voevodsky–Rost proof of its generalization the Bloch–Kato conjecture can be used to study counterexamples to the classical Lüroth problem. By generalizing a method due to Peyre, we produce for any prime number $\ell $ and any integer $n\geq 2$, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree $n$ unramified étale cohomology class with $\ell $-torsion coefficients. When $\ell = 2$, the varieties that are constructed are furthermore unirational and non-rationality cannot be detected by a torsion unramified étale cohomology class of lower degree.

We challenge the classical belief that products of degenerate quadratic forms must remain degenerate and we show that this fails in general, e.g. over tensor triangulated categories with duality. This opens new ways of constructing non-degenerate quadratic forms and hence classes in Witt groups. In addition, we encapsulate in a Leibniz-type formula the behavior of the product with respect to the symmetric cone construction. We illustrate these ideas by computing the total Witt group of regular projective spaces.

Let $X$ be a regular separated scheme of finite Krull dimension and let $U^{n}_{X} \subset A^{n}_{X}$ be the punctured affine $n$-space over $X$. We show that the total graded Witt ring of $U^{n}_{X}$ is a free graded module over the total graded Witt ring of $X$ with two generators $1$ and $\epsilon$. The second generator satisfies the equation $\epsilon^{2} = 1$ when $n = 1$ and $\epsilon^{2} = 0$ when $n \geq 2$.

To every pair of algebraic number fields with isomorphic Witt rings one can associate a number, called the minimum number of wild primes. Earlier investigations have established lower bounds for this number. In this paper an analysis is presented that expresses the minimum number of wild primes in terms of the number of wild dyadic primes. This formula not only gives immediate upper bounds, but can be considered to be an exact formula for the minimum number of wild primes.

Let $A$ be a finite-dimensional division algebra containing a base field $k$ in its center $F$. $A$ is defined over a subfield $F_0$ if there exists an $F_0$-algebra $A_0$ such that $A = A_0 \bigotimes_{F_0} F$. The following are shown. (i) In many cases $A$ can be defined over a rational extension of $k$. (ii) If $A$ has odd degree $n \geq 5$, then $A$ is defined over a field $F_0$ of transcendence degree $\leq {\frac{1}{2}}(n-1)(n-2)$ over $k$. (iii) If $A$ is a $\mathbb{Z}/m \times \mathbb{Z}/2$-crossed product for some $m \geq 2$ (and in particular, if $A$ is any algebra of degree 4) then $A$ is Brauer equivalent to a tensor product of two symbol algebras. Consequently, ${\rm M}_m(A)$ can be defined over a field $F_0$ such that ${\rm trdeg}_k(F_0) \leq 4$. (iv) If $A$ has degree 4 then the trace form of $A$ can be defined over a field $F_0$ of transcendence degree $\leq 4$. (In (i), (iii) and (iv) it is assumed that the center of $A$ contains certain roots of unity.)

To each field $F$ of characteristic not 2, one can associate a certain Galois group ${{\mathcal{G}}_{F}}$, the so-called $\text{W}$-group of $F$, which carries essentially the same information as the Witt ring $W(F)$ of $F$. In this paper we investigate the connection between ${{\mathcal{G}}_{F}}$ and ${{\mathcal{G}}_{F(\sqrt{a})}}$, where $F(\sqrt{a})$ is a proper quadratic extension of $F$. We obtain a precise description in the case when $F$ is a pythagorean formally real field and $a=-1$, and show that the $\text{W}$-group of a proper field extension $K/F$ is a subgroup of the $\text{W}$-group of $F$ if and only if $F$ is a formally real pythagorean field and $K=F(\sqrt{-1)}$. This theorem can be viewed as an analogue of the classical Artin-Schreier’s theorem describing fields fixed by finite subgroups of absolute Galois groups. We also obtain precise results in the case when $a$ is a double-rigid element in $F$. Some of these results carry over to the general setting.

The abstract Witt rings which are Gorenstein have been classified when the dimension is one and the classification problem for those of dimension zero has been reduced to the case of socle degree three. Here we classify the Gorenstein Witt rings of fields with dimension zero and socle degree three. They are of elementary type.

To each field F of characteristic not 2, one can associate a certain Galois group 𝒢F, the so-called W-group of F, which carries essentially the same information as the Witt ring W(F) of F. In this paper we show that direct products of Witt rings correspond to free products of these Galois groups (in the appropriate category), group ring construction of Witt rings corresponds to semidirect products of W-groups, and the basic part of W(F) is related to the center of 𝒢F. In an appendix we provide a complete list of Witt rings and corresponding w-groups for fields F with |Ḟ/Ḟ2| ≤ 16.

An abstract Witt ring (R, G) of positive characteristic is known to be a group ring S[Δ] with ﹛1﹜ ≠ Δ ⊆ G if and only if it contains a form〈1,x〉, x ≠1, which represents only the two elements 1 and x. Carson and Marshall have characterized all Witt rings of characteristic 2 which contain binary forms representing exactly four elements. Such results which show R is isomorphic to a product of smaller rings are helpful in settling the conjecture that every finitely generated Witt ring is of elementary type. Here, some special situations are considered. In particular if char(R) = 8, |D〈l, 1〉| = 4, and R contains no rigid elements, then R is isomorphic to the Witt ring of the 2-adic numbers. If char(R) = 4, |D〈l,a〉| = 4 where a ∈ D〈1, 1〉, and R contains no rigid elements, then R is either a ring of order 8 or is the specified product of two Witt rings at least one of which is a group ring. In several cases R is realized by a field.