Grothendieck Witt Groups Research Articles

Grothendieck–Witt groups of henselian valuation rings

Grothendieck–Witt groups of henselian valuation rings

Hermitian K-theory for stable infty -categories I: Foundations

This paper is the first in a series in which we offer a new framework for hermitian {text {K}}-theory in the realm of stable infty -categories. Our perspective yields solutions to a variety of classical problems involving Grothendieck-Witt groups of rings and clarifies the behaviour of these invariants when 2 is not invertible. In the present article we lay the foundations of our approach by considering Lurie’s notion of a Poincaré infty -category, which permits an abstract counterpart of unimodular forms called Poincaré objects. We analyse the special cases of hyperbolic and metabolic Poincaré objects, and establish a version of Ranicki’s algebraic Thom construction. For derived infty -categories of rings, we classify all Poincaré structures and study in detail the process of deriving them from classical input, thereby locating the usual setting of forms over rings within our framework. We also develop the example of visible Poincaré structures on infty -categories of parametrised spectra, recovering the visible signature of a Poincaré duality space. We conduct a thorough investigation of the global structural properties of Poincaré infty -categories, showing in particular that they form a bicomplete, closed symmetric monoidal infty -category. We also study the process of tensoring and cotensoring a Poincaré infty -category over a finite simplicial complex, a construction featuring prominently in the definition of the {text {L}}- and Grothendieck-Witt spectra that we consider in the next instalment. Finally, we define already here the 0th Grothendieck-Witt group of a Poincaré infty -category using generators and relations. We extract its basic properties, relating it in particular to the 0th {text {L}}- and algebraic {text {K}}-groups, a relation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applications.

The projective bundle formula for Grothendieck-Witt spectra

Grothendieck-Witt spectra represent higher Grothendieck-Witt groups and higher Hermitian K-theory in particular. A description of the Grothendieck-Witt spectrum of a finite dimensional projective bundle $\mathbb{P}(\mathcal{E})$ over a base scheme $X$ is given in terms of the Grothendieck-Witt spectra of the base, using the dg category of strictly perfect complexes, provided that $X$ is a scheme over $\text{Spec }\mathbb{Z}[1/2]$ and satisfies the resolution property, e.g. if $X$ has an ample family of line bundles.

THE BOREL CHARACTER

AbstractThe 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.

Grothendieck–Witt groups of some singular schemes

We establish some structural results for the Witt and Grothendieck-Witt groups of schemes over $\mathbb{Z}[1/2]$, including homotopy invariance for Witt groups and a formula for the Witt and Grothendieck-Witt groups of punctured affine spaces over a scheme. All these results hold for singular schemes and at the level of spectra.

On the motivic commutative ring spectrum $\mathbf {BO}$

We construct an algebraic commutative ring T -spectrum BO which is stably fibrant and (8, 4)-periodic and such that on SmOp/S the cohomology theory (X, U) 7→ BO(X+/U+) and Schlichting’s hermitian K-theory functor (X, U) 7→ KO [q] 2q−p(X, U) are canonically isomorphic. We use the motivic weak equivalence Z×HGr ∼ −→ KSp relating the infinite quaternionic Grassmannian to symplectic K-theory to equip BO with the structure of a commutative monoid in the motivic stable homotopy category. When the base scheme is SpecZ[ 1 2 ], this monoid structure and the induced ring structure on the cohomology theory BO are the unique structures compatible with the products KO [2m] 0 (X)× KO [2n] 0 (Y ) → KO [2m+2n] 0 (X × Y ). on Grothendieck-Witt groups induced by the tensor product of symmetric chain complexes. The cohomology theory is bigraded commutative with the switch map acting on BO(T∧T ) in the same way as multiplication by the Grothendieck-Witt class of the symmetric bilinear space 〈−1〉.

Hermitian K-theory, derived equivalences and Karoubi's fundamental theorem

Within the framework of dg categories with weak equivalences and duality that have uniquely 2-divisible mapping complexes, we show that higher Grothendieck–Witt groups (aka. hermitian K-groups) are invariant under derived equivalences and that Morita exact sequences induce long exact sequences of Grothendieck–Witt groups. This implies an algebraic Bott sequence and a new proof and generalisation of Karoubi's Fundamental Theorem. For the higher Grothendieck–Witt groups of vector bundles of (possibly singular) schemes X with an ample family of line-bundles such that 12∈Γ(X,OX), we obtain Mayer–Vietoris long exact sequences for Nisnevich coverings and blow-ups along regularly embedded centres, projective bundle formulas, and a Bass fundamental theorem. For coherent Grothendieck–Witt groups, we obtain a localization theorem analogous to Quillen's K′-localization theorem.

Twisted K-theory, Real A-bundles and Grothendieck–Witt groups

We introduce a general framework to unify several variants of twisted topological K-theory. We focus on the role of finite dimensional real simple algebras with a product-preserving involution, showing that Grothendieck–Witt groups provide interesting examples of twisted K-theory. These groups are linked with the classification of algebraic vector bundles on real algebraic varieties.

The Homotopy Fixed Point Theorem and the Quillen–Lichtenbaum conjecture in Hermitian K-theory

We settle two conjectures for computing higher Grothendieck–Witt groups (also known as Hermitian K-groups) of noetherian schemes X, under some mild conditions. It is shown that the comparison map from the Hermitian K-theory of X to the homotopy fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence. We also prove that the mod 2ν comparison map between the Hermitian K-theory of X and its étale version is an isomorphism on homotopy groups in the same range as for the Quillen–Lichtenbaum conjecture in K-theory. Applications compute higher Grothendieck–Witt groups of complex algebraic varieties and rings of 2-integers in number fields, and hence values of Dedekind zeta-functions.

Geometric models for higher Grothendieck–Witt groups in $$\mathbb {A}^1$$ A 1 -homotopy theory

We show that the higher Grothendieck–Witt groups, a.k.a. algebraic hermitian $$K$$ -groups, are represented by an infinite orthogonal Grassmannian in the $$\mathbb {A}^1$$ -homotopy category of smooth schemes over a regular base for which $$2$$ is a unit in the ring of regular functions. We also give geometric models for various $$\mathbb {P}^1$$ - and $$S^1$$ -loop spaces of hermitian $$K$$ -theory.

Witt groups of curves and surfaces

We study Witt groups of smooth curves and surfaces over algebraically closed fields of characteristic not two. In both dimensions, we determine both the classical Witt group and Balmer's shifted Witt groups. In the case of curves, the results are supplemented with a complete description of the (shifted) Grothendieck-Witt groups. In a second step, we analyse the relationship of Witt groups of smooth complex curves and surfaces with their real topological K-groups. They turn out to be surprisingly close: for all curves and for all projective surfaces of geometric genus zero, the Witt groups may be identified with the quotients of their even KO-groups by the images of their complex topological K-groups under realification.

Finiteness theorems for the shifted Witt and higher Grothendieck–Witt groups of arithmetic schemes

For smooth varieties over finite fields, we prove that the shifted ( aka derived) Witt groups of surfaces are finite and the higher Grothendieck–Witt groups ( aka Hermitian K-theory) of curves are finitely generated. For more general arithmetic schemes, we give conditional results, for example, finite generation of the motivic cohomology groups implies finite generation of the Grothendieck–Witt groups.

Hermitian K-theory of exact categories

AbstractWe study the theory of higher Grothendieck-Witt groups, alias algebraic hermitian K-theory, of symmetric bilinear forms in exact categories, and prove additivity, cofinality, dévissage and localization theorems – preparing the ground for the theory of higher Grothendieck-Witt groups of schemes as developed in [Sch08a] and [Sch08b]. No assumption on the characteristic is being made.

The excess intersection formula for Grothendieck–Witt groups

We prove the excess intersection and self intersection formulae for Grothendieck–Witt groups.

Chow–Witt groups and Grothendieck–Witt groups of regular schemes

Let A be a noetherian commutative Z [ 1 / 2 ] -algebra of Krull dimension d and let P be a projective A-module of rank d. We use derived Grothendieck–Witt groups and Euler classes to detect some obstructions for P to split off a free factor of rank one. If d ⩽ 3 , we show that the vanishing of its Euler class in the corresponding Grothendieck–Witt group is a necessary and sufficient condition for P to have a free factor of rank one. If d is odd, we also get some results in that direction. If A is regular, we show that the Chow–Witt groups defined by Morel and Barge appear naturally as some homology groups of a Gersten-type complex in Grothendieck–Witt theory. From this, we deduce that if d = 3 then the vanishing of the Euler class of P in the corresponding Chow–Witt group is a necessary and sufficient condition for P to have a free factor of rank one.

A vanishing theorem for oriented intersection multiplicities

Let A be a regular local ring containing 1/2, which is either equicharacteristic, or is smooth over a d.v.r. of mixed characteristic. We prove that the product maps on derived Grothendieck-Witt groups of A satisfy the following property: given two elements with supports which do not intersect properly, their product vanishes. This gives an analogue for oriented intersection multiplicities of Serre's vanishing result for intersection multiplicities. It also suggests a Vanishing Conjecture for arbitrary regular local rings containing 1/2, which is analogous to Serre's (which was proved independently by Roberts, and Gillet and Soule).