Witt Ring Research Articles

Q-Deformations of statistical mechanical systems and motives over finite fields

We consider q-deformations of Witt rings, based on geometric operations on zeta functions of motives over finite fields, and we use these deformations to construct q-analogs of the Bost-Connes quantum statistical mechanical system. We show that the q-deformations obtained in this way can be related to Habiro ring constructions of analytic functions over F_1 and to categorifications of Bost-Connes systems.

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.

Real cohomology and the powers of the fundamental ideal in the Witt ring

Let A be a local ring in which 2 is invertible. It is known that the localization of the cohomology ring H et∗(A, ℤ∕2) with respect to the class (−1) ∈ H et1(A, ℤ∕2) is isomorphic to the ring C(sperA, ℤ∕2) of continuous ℤ∕2-valued functions on the real spectrum of A. Let In(A) denote the powers of the fundamental ideal in the Witt ring of symmetric bilinear forms over A. The starting point of this article is the “integral” version: the localization of the graded ring ⊕ n≥0In(A) with respect to the class 〈〈−1〉〉 := 〈1,1〉∈ I(A) is isomorphic to the ring C(sperA, ℤ) of continuous ℤ-valued functions on the real spectrum of A. This has interesting applications to schemes. For instance, for any algebraic variety X over the field of real numbers ℝ and any integer n strictly greater than the Krull dimension of X, we obtain a bijection between the Zariski cohomology groups HZar∗(X,ℐn) with coefficients in the sheaf ℐn associated to the n-th power of the fundamental ideal in the Witt ring W(X) and the singular cohomology groups Hsing∗(X(ℝ), ℤ).

The universal deformation of the Witt ring scheme

We determine the universal deformation over reduced base rings of the Witt ring scheme enhanced by a Frobenius lift and Verschiebung. It agrees with a -deformation introduced earlier by the second author; we also give a simpler description for this. In the appendix we discuss a Witt vector theory for ind-rings which may be of independent interest. Bibliography: 7 titles.

Witt equivalence of function fields of curves over local fields

ABSTRACTTwo fields are Witt equivalent if their Witt rings of symmetric bilinear forms are isomorphic. Witt equivalent fields can be understood to be fields having the same quadratic form theory. The behavior of finite fields, local fields, global fields, as well as function fields of curves defined over Archimedean local fields under Witt equivalence is well understood. Numbers of classes of Witt equivalent fields with finite numbers of square classes are also known in some cases. Witt equivalence of general function fields over global fields was studied in the earlier work [13] by the authors and applied to study Witt equivalence of function fields of curves over global fields. In this paper, we extend these results to local case, i.e. we discuss Witt equivalence of function fields of curves over local fields. As an application, we show that, modulo some additional assumptions, Witt equivalence of two such function fields implies Witt equivalence of underlying local fields.

𝔸1-homotopy invariance of algebraic K-theory with coefficients and du Val singularities

C. Weibel, and Thomason and Trobaugh, proved (under some assumptions) that algebraic K-theory with coefficients is A1-homotopy invariant. We generalize this result from schemes to the broad setting of dg categories. Along the way, we extend the Bass–Quillen fundamental theorem as well as Stienstra’s foundational work on module structures over the big Witt ring to the setting of dg categories. Among other cases, the above A1-homotopy invariance result can now be applied to sheaves of (not necessarily commutative) dg algebras over stacks. As an application, we compute the algebraic K-theory with coefficients of dg cluster categories using solely the kernel and cokernel of the Coxeter matrix. This leads to a complete computation of the algebraic K-theory with coefficients of the du Val singularities parametrized by the simply laced Dynkin diagrams. As a byproduct, we obtain vanishing and divisibility properties of algebraic K-theory (without coefficients).

On relative and overconvergent de Rham–Witt cohomology for log schemes

We construct the relative log de Rham-Witt complex. This is a generalization of the relative de Rham-Witt complex of Langer-Zink to log schemes. We prove the comparison theorem between the hypercohomology of the log de Rham-Witt complex and the relative log crystalline cohomology in certain cases. We construct the $p$-adic weight spectral sequence for relative proper strict semistable log schemes. When the base log scheme is a log point, We show it degenerates at $E_2$ after tensoring with the fraction field of the Witt ring. We also extend the definition of the overconvergent de Rham-Witt complex of Davis-Langer-Zink to log schemes $(X,D)$ associated with smooth schemes with simple normal crossing divisor over a perfect field. Finally, we compare its hypercohomology with the rigid cohomology of $X \setminus D$.

The homogeneous spectrum of Milnor–Witt K-theory

For any field F (of characteristic not equal to 2), we determine the Zariski spectrum of homogeneous prime ideals in K⁎MW(F), the Milnor–Witt K-theory ring of F. As a corollary, we recover Lorenz and Leicht's classical result on prime ideals in the Witt ring of F. Our computation can be seen as a first step in Balmer's program for studying the tensor triangular geometry of the stable motivic homotopy category.

Slices of hermitian K–theory and Milnor’s conjecture on quadratic forms

We advance the understanding of K-theory of quadratic forms by computing the slices of the motivic spectra representing hermitian K-groups and Witt-groups. By an explicit computation of the slice spectral sequence for higher Witt-theory, we prove Milnor's conjecture relating Galois cohomology to quadratic forms via the filtration of the Witt ring by its fundamental ideal. In a related computation we express hermitian K-groups in terms of motivic cohomology.

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.

Milnor–Witt K-groups of local rings

We introduce Milnor–Witt K-groups of local rings and show that the nth Milnor–Witt K-group of a local ring R which contains an infinite field of characteristic not 2 is the pull-back of the nth power of the fundamental ideal in the Witt ring of R and the nth Milnor K-group of R over the nth Milnor K-group of R modulo 2. This generalizes the work of Morel–Hopkins on Milnor–Witt K-groups of a field.

Automorphisms of Witt rings of local type

Automorphisms of Witt rings of local type

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.

Differential forms and bilinear forms under field extensions

Let F be a field of characteristic p>0. Let Ωn(F) be the F-vector space of n-differentials of F over Fp. Let K=F(g) be the function field of an irreducible polynomial g in m⩾1 variables over F. We derive an explicit description of the kernel of the restriction map Ωn(F)→Ωn(K). As an application in the case p=2, we determine the kernel of the restriction map when passing from the Witt ring (resp. graded Witt ring) of symmetric bilinear forms over F to that over such a function field extension K.

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.

On $$W_2$$ W 2 -lifting of Frobenius of algebraic surfaces

The classification of minimal algebraic surfaces in positive characteristics was accomplished by Bombieri and Mumford in the 1970s. In this work we decide completely which minimal algebraic surfaces in positive characteristics allow a lifting of their Frobenius over the truncated Witt rings of length 2. Besides, we show that the Frobenius morphism of many projective rational surfaces cannot be lifted to \(W_2(k)\).

A link invariant with values in the Witt ring

Using Maslov indices, we show the existence of oriented link in-variants with values in the Witt rings of certain fields. Various classical in-variants are closely related to this construction. We also explore a surprising connection with the Weil representation.

Zeta functions, Grothendieck groups, and the Witt ring

We prove some results connecting the zeta functions of varieties over finite fields with the big Witt ring over Z. We explore relations with motivic measures and a classical formula of Macdonald on invariants of symmetric products of a variety.

The Witt ring of a curve with good reduction over a non-dyadic local field

In this work, we present a generalization to varieties and sheaves of the fundamental ideal of the Witt ring of a field by defining a sheaf of fundamental ideals I˜ and a sheaf of Witt rings W˜ in the obvious way. The Milnor conjecture then relates the associated graded of W˜ to Milnor K-theory and so allows the classical invariants of a bilinear space over a field to be extended to our setting using étale cohomology. As an application of these results, we calculate the Witt ring of a smooth curve with good reduction over a non-dyadic local field.

On the arithmetic of the BC-system

For each prime p and each embedding \sigma of the multiplicative group of an algebraic closure of \mathbb{F}_p as complex roots of unity, we construct a p -adic indecomposable representation \pi_\sigma of the integral BC-system as additive endomorphisms of the big Witt ring of \bar{\mathbb{F}}_p . The obtained representations are the p -adic analogues of the complex, extremal KMS _\infty states of the BC-system. The role of the Riemann zeta function, as partition function of the BC-system over \mathbb{C} is replaced, in the p -adic case, by the p -adic L -functions and the polylogarithms whose values at roots of unity encode the KMS states. We use Iwasawa theory to extend the KMS theory to a covering of the completion \mathbb{C}_p of an algebraic closure of \mathbb{Q}_p . We show that our previous work on the hyperring structure of the adèle class space, combines with p -adic analysis to refine the space of valuations on the cyclotomic extension of \mathbb{Q} as a noncommutative space intimately related to the integral BC-system and whose arithmetic geometry comes close to fulfill the expectations of the “arithmetic site”. Finally, we explain how the integral BC-system appears naturally also in de Smit and Lenstra construction of the standard model of \bar{\mathbb{F}}_p which singles out the subsystem associated to the \hat{\mathbb{Z}} -extension of \mathbb{Q} .