Witt Groups Research Articles

Level bounds for exceptional quantum subgroups in rank two

There is a long-standing belief that the modular tensor categories [Formula: see text], for [Formula: see text] and finite-dimensional simple complex Lie algebras [Formula: see text], contain exceptional connected étale algebras (sometimes called quantum subgroups) at only finitely many levels [Formula: see text]. This premise has known implications for the study of relations in the Witt group of nondegenerate braided fusion categories, modular invariants of conformal field theories, and the classification of subfactors in the theory of von Neumann algebras. Here, we confirm this conjecture when [Formula: see text] has rank 2, contributing proofs and explicit bounds when [Formula: see text] is of type [Formula: see text] or [Formula: see text], adding to the previously known positive results for types [Formula: see text] and [Formula: see text].

Some comments on motivic nilpotence

We discuss some results and conjectures related to the existence of the non-nilpotent motivic maps η \eta and μ 9 \mu _9 . To this purpose, we establish a theory of power operations for motivic H ∞ H_{\infty } -spectra. Using this, we show that the naive motivic analogue of the unstable Kahn-Priddy theorem fails. Over the complex numbers, we show that the motivic T T -spectrum S [ η − 1 , μ 9 − 1 ] S[\eta ^{-1},\mu _9^{-1}] is closely related to higher Witt groups, where S S is the motivic sphere spectrum and η \eta and μ 9 \mu _9 are explicit elements in π ∗ ∗ ( S ) \pi _{**}(S) .

Cohomological invariants for quadratic forms over local rings

Let A be local ring in which 2 is invertible and let n be a non-negative integer. We show that the nth cohomological invariant of quadratic forms is a well-defined homomorphism from the nth power of the fundamental ideal in the Witt ring of A to the degree n etale cohomology of A with mod 2 coefficients, which is surjective and has kernel the ( $$\hbox {n}+1$$ )th power of the fundamental ideal. This is obtained by proving the Gersten conjecture for Witt groups in an important mixed-characteristic case.

Classification of $${{\mathfrak{sl}}_3}$$ sl 3 Relations in the Witt Group of Nondegenerate Braided Fusion Categories

The Witt group of nondegenerate braided fusion categories $\mathcal{W}$ contains a subgroup $\mathcal{W}_\text{un}$ consisting of Witt equivalence classes of pseudo-unitary nondegenerate braided fusion categories. For each finite-dimensional simple Lie algebra $\mathfrak{g}$ and positive integer $k$ there exists a pseudo-unitary category $\mathcal{C}(\mathfrak{g},k)$ consisting of highest weight integerable $\hat{g}$-modules of level $k$ where $\hat{\mathfrak{g}}$ is the corresponding affine Lie algebra. Relations between the classes $[\mathcal{C}(\mathfrak{sl}_2,k)]$, $k\geq1$ have been completely described in the work of Davydov, Nikshych, and Ostrik. Here we give a complete classification of relations between the classes $[\mathcal{C}(\mathfrak{sl}_3,k)]$, $k\geq1$ with a view toward extending these methods to arbitrary simple finite dimensional Lie algebras $\mathfrak{g}$ and positive integer levels $k$.

ESSENTIAL DIMENSION OF GENERIC SYMBOLS IN CHARACTERISTIC

In this article the$p$-essential dimension of generic symbols over fields of characteristic$p$is studied. In particular, the$p$-essential dimension of the length$\ell$generic$p$-symbol of degree$n+1$is bounded below by$n+\ell$when the base field is algebraically closed of characteristic$p$. The proof uses new techniques for working with residues in Milne–Kato$p$-cohomology and builds on work of Babic and Chernousov in the Witt group in characteristic 2. Two corollaries on$p$-symbol algebras (i.e, degree 2 symbols) result from this work. The generic$p$-symbol algebra of length$\ell$is shown to have$p$-essential dimension equal to$\ell +1$as a$p$-torsion Brauer class. The second is a lower bound of$\ell +1$on the$p$-essential dimension of the functor$\operatorname{Alg}_{p^{\ell },p}$. Roughly speaking this says that you will need at least$\ell +1$independent parameters to be able to specify any given algebra of degree$p^{\ell }$and exponent$p$over a field of characteristic$p$and improves on the previously established lower bound of 3.

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.

Finite homological dimension and a derived equivalence

For a Cohen-Macaulay ring R R , we exhibit the equivalence of the bounded derived categories of certain resolving subcategories, which, amongst other results, yields an equivalence of the bounded derived category of finite length and finite projective dimension modules with the bounded derived category of projective modules with finite length homologies. This yields isomorphisms of K K -theory and Witt groups (amongst other invariants) and improves on terms of associated spectral sequences and Gersten complexes.

The Witt group of real algebraic varieties

Let $V$ be an algebraic variety over $\mathbb R$. The purpose of this paper is to compare its algebraic Witt group $W(V)$ with a new topological invariant $WR(V_{\mathbb C})$, based on symmetric forms on Real vector bundles (in the sense of Atiyah) on the space of complex points of $V$, This invariant lies between $W(V)$ and the group $KO(V_{\mathbb R})$ of $\mathbb R$-linear topological vector bundles on $V_{\mathbb R}$, the set of real points of $V$. We show that the comparison maps $W(V)\to WR(V_{\mathbb C})$ and $WR(V_{\mathbb C})\to KO(V_{\mathbb R})$ that we define are isomorphisms modulo bounded 2-primary torsion. We give precise bounds for the exponent of the kernel and cokernel of these maps, depending upon the dimension of $V.$ These results improve theorems of Knebusch, Brumfiel and Mah\'e. Along the way, we prove a comparison theorem between algebraic and topological Hermitian $K$-theory, and homotopy fixed point theorems for the latter. We also give a new proof (and a generalization) of a theorem of Brumfiel.

Comparing Euler Classes

We establish the equality of two definitions of a Euler class in algebraic geometry: the first definition is as a ‘characteristic class’ with values in Chow–Witt theory, while the second definition is as an ‘obstruction class.’ Along the way, we refine Morel's relative Hurewicz theorem in A1-homotopy theory, and show how to define (twisted) Chow–Witt groups for geometric classifying spaces.

Modular Extensions of Unitary Braided Fusion Categories and 2+1D Topological/SPT Orders with Symmetries

A finite bosonic or fermionic symmetry can be described uniquely by a symmetric fusion category $\mathcal{E}$. In this work, we propose that 2+1D topological/SPT orders with a fixed finite symmetry $\mathcal{E}$ are classified, up to $E_8$ quantum Hall states, by the unitary modular tensor categories $\mathcal{C}$ over $\mathcal{E}$ and the modular extensions of each $\mathcal{C}$. In the case $\mathcal{C}=\mathcal{E}$, we prove that the set $\mathcal{M}_{ext}(\mathcal{E})$ of all modular extensions of $\mathcal{E}$ has a natural structure of a finite abelian group. We also prove that the set $\mathcal{M}_{ext}(\mathcal{C})$ of all modular extensions of $\mathcal{C}$, if not empty, is equipped with a natural $\mathcal{M}_{ext}(\mathcal{E})$-action that is free and transitive. Namely, the set $\mathcal{M}_{ext}(\mathcal{C})$ is an $\mathcal{M}_{ext}(\mathcal{E})$-torsor. As special cases, we explain in details how the group $\mathcal{M}_{ext}(\mathcal{E})$ recovers the well-known group-cohomology classification of the 2+1D bosonic SPT orders and Kitaev's 16 fold ways. We also discuss briefly the behavior of the group $\mathcal{M}_{ext}(\mathcal{E})$ under the symmetry-breaking processes and its relation to Witt groups.

Quadratic forms and Clifford algebras on derived stacks

In this paper we present an approach to quadratic structures in derived algebraic geometry. We define derived n-shifted quadratic complexes, over derived affine stacks and over general derived stacks, and give several examples of those. We define the associated notion of derived Clifford algebra, in all these contexts, and compare it with its classical version, when they both apply. Finally, we prove three main existence results for derived shifted quadratic forms over derived stacks, define a derived version of the Grothendieck–Witt group of a derived stack, and compare it to the classical one.

Cohomology and graded Witt group kernels for extensions of degree four in characteristic two

In earlier work the authors determined the graded Witt kernel GWq(E/F)=ker(GWqF→GWqE) when E/F is a biquadratic extension in characteristic 2 by calculating the cohomological kernel H2⁎(E/F)=ker(H2⁎F→H2⁎E). In this paper this result is extended to the cases where [E:F]=4 and E is either cyclic or has dihedral Galois closure. In addition, the use of Izhboldin's Q-groups is generalized to obtain six-term exact sequences that describe the behavior of these graded rings whenever four-term exact sequences and homotopies describe the arithmetic of the extension. These tools are valid in characteristic p, although the applications here are in characteristic 2.

On the relation of special linear algebraic cobordism to Witt groups

We reconstruct derived Witt groups via special linear algebraic cobordism. There is a morphism of ring cohomology theories which sends the canonical Thom class in special linear cobordism to the Thom class in the derived Witt groups. We show that for every smooth variety X this morphism induces an isomorphism between MSL^{*,*}(X)[h^{-1}] with the extended coefficient ring MSL^{4*,2*}(pt) -> W^{2*}(pt) and Laurent polynomial ring over the derived Witt groups W^*(X), where h is the stable Hopf map. This result is an analogue of the result by Panin and Walter reconstructing hermitian K-theory using symplectic algebraic cobordism.

Graded Witt kernels of the compositum of multiquadratic extensions with the function fields of Pfister forms

Let F be a field of characteristic 2 and Wq(F) be the Witt group of nonsingular quadratic forms over F. Let φ be a bilinear Pfister form over F and L be a multiquadratic extension of F of separability degree less than of equal to 2. In this paper we compute the kernel of the natural homomorphism H2m+1(F)⟶H2m+1(L(φ)), where H2m+1(F) is the cokernel of the Artin–Schreier operator ℘:ΩFm⟶ΩFm/dΩFm−1 given by xdx1x1∧⋯∧dxmxm↦(x2−x)dx1x1∧⋯∧dxmxm, where ΩFm is the space of m-differential forms over F, and F(φ) is the function field of the affine quadric given by the diagonal quadratic form associated to the bilinear form φ. As a consequence, we deduce the kernel of the natural homomorphisms Iqm+1‾(F)⟶Iqm+1‾(L(φ)) and Iqm+1(F)⟶Iqm+1(L(φ)), where Iqm+1‾(F) denotes the quotient Iqm+1(F)/Iqm+2(F) such that Iqm+1(F)=ImF⊗Wq(F) and ImF is the m-th power of the fundamental ideal IF of the Witt ring of F-bilinear forms. We also include some results concerning the case where φ is replaced by a bilinear Pfister neighbor or a quadratic Pfister form.

On the push-forwards for motivic cohomology theories with invertible stable Hopf element

We present a geometric construction of push-forward maps along projective morphisms for cohomology theories representable in the stable motivic homotopy category assuming that the element corresponding to the stable Hopf map is inverted in the coefficient ring of the theory. The construction is parallel to the one given by Nenashev for derived Witt groups. Along the way we introduce cohomology groups twisted by a formal difference of vector bundles as cohomology groups of a certain Thom space and compute twisted cohomology groups of projective spaces.

Signatures of hermitian forms and “prime ideals” of Witt groups

In this paper a further study is made of H-signatures of hermitian forms, introduced previously by the authors. It is shown that a tuple of reference forms H may be replaced by a single form and that the H-signature is invariant under Morita equivalence of algebras with involution. The “prime ideals” of the Witt group are studied, obtaining results that are analogues of the classification of prime ideals of the Witt ring by Harrison and Lorenz–Leicht. It follows that H-signatures canonically correspond to morphisms into the integers.

Metabolic modules of finite group algebras over finite fields of characteristic two

Let G be a finite group and let F be a finite field of characteristic 2. We introduce F-special subgroups and F-special elements of G. In the case where F contains a pth primitive root of unity for each odd prime p dividing the order of G (e.g. it is the case once F is a splitting field for all subgroups of G), the F-special elements of G coincide with real elements of odd order. We prove that a symmetric FG-module V is metabolic if and only if the restriction VD of V to every F-special subgroup D of G is metabolic, and also, if and only if the characteristic polynomial on V defined by every F-special element of G is a square of a polynomial over F. Some immediate applications to characters, self-dual codes and Witt groups are given.

Witt kernels of quadratic forms for multiquadratic extensions in characteristic 2

Let $F$ be a field of characteristic $2$ and let $K/F$ be a purely inseparable extension of exponent $1$. We show that the extension is excellent for quadratic forms. Using the excellence we recover and extend results by Aravire and Laghribi who computed generators for the kernel $W_q(K/F)$ of the natural restriction map $W_q(F)\to W_q(K)$ between the Witt groups of quadratic forms of $F$ and $K$, respectively, where $K/F$ is a finite multiquadratic extension of separability degree at most $2$.

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.