Witt Groups Research Articles

Witt kernels and Brauer kernels for quartic extensions in characteristic two

Let F be a field of characteristic 2 and let E/F be a field extension of degree 4. We determine the kernel Wq(E/F) of the restriction map WqF→WqE between the Witt groups of nondegenerate quadratic forms over F and over E, completing earlier partial results by Ahmad, Baeza, Mammone and Moresi. We also deduce the corresponding result for the Witt kernel W(E/F) of the restriction map WF→WE between the Witt rings of nondegenerate symmetric bilinear forms over F and over E from earlier results by the first author. As an application, we describe the 2-torsion part of the Brauer kernel for such extensions.

Witt Groups of P1

Witt Groups of P1

Topology and purity for torsors

We study the homotopy theory of the classifying space of the complex projective linear groups to prove that purity fails for \operatorname{PGL}_p -torsors on regular noetherian schemes when p is a prime. Extending our previous work when p=2 , we obtain a negative answer to a question of Colliot-Thélène and Sansuc, for all \operatorname{PGL}_p . We also give a new example of the failure of purity for the cohomological filtration on the Witt group, which is the first example of this kind of a variety over an algebraically closed field.

Witt groups of smooth toric surfaces

Witt groups of smooth toric surfaces

Derived Witt group formalism

In this article we establish some formalism of derived Witt theory for resolving subcategories of abelian categories. Results directly apply to noetherian schemes.

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.

Nonsplittability of the rational homology cobordism group of 3-manifolds

The smooth rational homology cobordism group of rational homology three spheres, T, contains subgroups T_p generated by 3-manifolds with first homology p-torsion, where p is a prime. Rochlin's theorem and gauge theoretic methods show that the inclusion of the direct sum of the T_p into T has infinitely generated kernel. We use Heegaard-Floer methods to show that the cokernel is infinitely generated. On the level of Witt groups of linking forms and conjecturally in the topological category, the inclusion is an isomorphism. One application is the demonstration of the failure in dimension three of an algebraic and higher dimensional theorem of Stoltzfus regarding primary splittings in the knot concordance group. We also build on work of the second author with Hedden and Ruberman to prove that the homology cobordism group of rational homology three spheres that bound topological rational homology 4-balls remains infinitely generated when modded out by the homology cobordism group of integral homology 3-spheres.

Which algebraic groups are Picard varieties?

We show that every connected commutative algebraic group over an algebraically closed field of characteristic 0 is the Picard variety of some projective variety having only finitely many non-normal points. In contrast, no Witt group of dimension at least 3 over a perfect field of prime characteristic is isogenous to a Picard variety obtained by this construction.

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.

Twisted Witt Groups of Flag Varieties

AbstractCalmès and Fasel have shown that the twisted Witt groups of split flag varieties vanish in a large number of cases. For flag varieties over algebraically closed fields, we sharpen their result to an if-and-only-if statement. In particular, we show that the twisted Witt groups vanish in many previously unknown cases. In the non-zero cases, we find that the twisted total Witt group forms a free module of rank one over the untwisted total Witt group, up to a difference in grading.Our proof relies on an identification of the Witt groups of flag varieties with the Tate cohomology groups of their K-groups, whereby the verification of all assertions is eventually reduced to the computation of the (twisted) Tate cohomology of the representation ring of a parabolic subgroup.

Quadratic Forms and Linear Algebraic Groups

Topics discussed at the workshop “Quadratic Forms and Linear Algebraic Groups” included besides the algebraic theory of quadratic and Hermitian forms and their Witt groups several aspects of the theory of linear algebraic groups and homogeneous varieties, as well as some arithmetic aspects pertaining to the theory of quadratic forms over function fields or number fields.

Witt Groups of Algebraic Groups

Let G_k be a split reductive group over a subfield k of \mathbb C corresponding to a Lie group G . Let T be a maximal torus of G . We show the isomorphism W^*(G_k)\cong W^*(G_k/T_k) of Balmer–Witt groups. When k is algebraically closed, we prove that W^*(G_k) is isomorphic to the topological K -theory KO^{2*-1}(G/T) of the flag manifold G/T . Then we compute it explicitly by using the fact that W^*(G_k) is a Hopf algebra.

On the non-injectivity of the Vaserstein symbol in dimension three

L.N. Vaserstein proved over a two dimensional ring that the orbit space of unimodular rows of length three modulo elementary action has a Witt group structure. R.A. Rao and W. van der Kallen showed that the Vaserstein symbol need not be injective over three dimensional affine algebras over the real field, but is injective over three dimensional affine algebras over a field of cohomological dimension one whose characteristic ≠2,3. R.G. Swan, R.A. Rao and J. Fasel gave another example of a real affine algebra of dimension three for which the Vaserstein symbol is not injective. We demonstrate an uncountably infinite family of non-isomorphic affine algebras of dimension three over the real field for which the Vaserstein symbol is not injective.

Weil representation and transfer factor

This paper concerns the Weil representation of the semidirect product of the metaplectic and Heisenberg groups. First we present a canonical construction of the metaplectic group as a central extension of the symplectic group by a subquotient of the Witt group. This leads to simple formulas for the character, for the inverse Weyl transform, and for the transfer factor appearing in J. Adams’s work on character lifting. Along the way, we give formulas for outer automorphisms of the metaplectic group induced by symplectic similitudes. The approach works uniformly for finite and local fields.

Bicategories for Boundary Conditions and for Surface Defects in 3-d TFT

We analyze topological boundary conditions and topological surface defects in three-dimensional topological field theories of Reshetikhin-Turaev type based on arbitrary modular tensor categories. Boundary conditions are described by central functors that lift to trivializations in the Witt group of modular tensor categories. The bicategory of boundary conditions can be described through the bicategory of module categories over any such trivialization. A similar description is obtained for topological surface defects. Using string diagrams for bicategories we also establish a precise relation between special symmetric Frobenius algebras and Wilson lines involving special defects. We compare our results with previous work of Kapustin-Saulina and of Kitaev-Kong on boundary conditions and surface defects in abelian Chern-Simons theories and in Turaev-Viro type TFTs, respectively.

Relationship between algebraic MSL-cobordisms and derived Witt groups

Relationship between algebraic MSL-cobordisms and derived Witt groups

Isomorphism criteria for Witt rings of real fields

We prove isomorphism criteria for Witt rings and reduced Witt rings of certain types of real fields. Refined criteria are obtained under the additional assumption that the field be SAP. This leads to a generalization of a result by Koprowski on Witt equivalence of function fields of transcendence degree 1 over a real closed field. Isomorphism criteria are also obtained for Witt rings of hermitian forms over a quadratic extension of a real base field and for Witt groups of hermitian forms over a quaternion algebra with a real field as center. All these criteria are expressed in terms of properties involving topological subspaces of the space of orderings of the base field.

The graded Witt group kernel of biquadratic extensions in characteristic two

Baeza showed that when char(F)=2 if E/F is the separable biquadratic extension E=F[℘−1(b1),℘−1(b2)], thenker[Wq(F)→Wq(E)]=WF⋅[1,b1]+WF⋅[1,b2]. Here we give the analogous result for the graded Witt group. Specifically we obtain an exact sequence νF(n,1)⊕νF(n,1)→H2n+1F→H2n+1E from which the result for GWqF follows by the isomorphisms of Kato. Applications to 2-algebras of exponent and index 4 are also given.

On coherent hermitian Witt groups

We prove homotopy invariance for coherent hermitian Witt groups of Azumaya algebras with involutions over separated Gorenstein schemes of finite Krull dimension. We also generalize the zero theorem for the transfer and show that the Gersten conjecture holds for the hermitian Gersten–Witt complex of flat coherent algebras over a regular local ring which contains a field.

BASES OF TOTAL WITT GROUPS AND LAX-SIMILITUDE

We explain how to work with total Witt groups, more specifically, how to circumvent the classical embarrassment of making choices for line bundles up to isomorphisms and up to squares.