Hermitian Forms over Algebras with Involution and Hermitian Categories

This thesis is concerned with the algebraic theory of hermitian forms. It is organized in two parts. The first, consisting of the first two chapters, deals with some descent properties of unimodular hermitian forms over central simple algebras with involution. The second, which consists of the last two chapters, generalizes several classical properties of unimodular hermitian forms over rings with involution to the setting of sesquilinear forms in hermitian categories. The original results established in this thesis are joint work with Professor Eva Bayer-Fluckiger. The first chapter contains an introduction to the algebraic theory of unimodular ε-hermitian forms over fields with involution. One knows that if L/K is an extension of odd degree (where char(K) ≠ 2) then the restriction map rL/K : W(K) →W(L) is injective. In addition, if the extension is purely inseparable then the map rL/K is bijective. In the second chapter we first introduce the basic notions and techniques of the theory of unimodular ε-hermitian forms over algebras with involution, in particular the technique of Morita equivalence. Let L/K be a finite field extension, τ an involution on L and A a finite-dimensional K-algebra endowed with an involution α such that αœK = τœK. E. Bayer-Fluckiger and H.W. Lenstra proved that if L/K is of odd degree and αœK = idK then the restriction map rL/Kε : Wε(A, α) → Wε(A ⊗K L, α ⊗ τ) is injective for any ε = ±1. This holds also if αœK ≠ idK. We prove that if, in addition, L/K is purely inseparable and A is a central simple K-algebra, then the above map is actually bijective. The proof proceeds via induction on the degree of the algebra and uses in an essential way an exact sequence of Witt groups due to R. Parimala, R. Sridharan and V. Suresh, later extended by N. Gernier-Boley and M.G. Mahmoudi. The third chapter contains a survey of the theory of hermitian and quadratic forms in hermitian categories. In particular, we cover the transfer between two hermitian categories, the reduction by an ideal, the transfer into the endomorphism ring of an object, as well as the Krull-Schmidt-Azumaya theorem and some of its applications. In the fourth chapter we prove, adapting the ideas developed by E. Bayer-Fluckiger and L. Fainsilber, that the category of sesquilinear forms in a hermitian category ℳ is equivalent to the category of unimodular hermitian forms in the category of double arrows of ℳ. In order to obtain this equivalence of categories we associate to a sesquilinear form the double arrow consisting of its two adjoints, equipped with a canonical unimodular hermitian form. This equivalence of categories allows us to define a notion of Witt group for sesquilinear forms in hermitian categories. This generalizes the classical notion of a Witt group of unimodular hermitian forms over rings with involution. Using the above equivalence of categories we deduce analogues of the Witt cancellation theorem and Springer's theorem for sesquilinear forms over certain algebras with involution. We also extend some finiteness results due to E. Bayer-Fluckiger, C. Kearton and S.M. J. Wilson. In addition, we study the weak Hasse-Minkowski principle for sesquilinear forms over skew fields with involution over global fields. We prove that this principle holds for systems of sesquilinear forms over a skew field over a global field and endowed with a unitary involution. Two systems of sesquilinear forms are hence isometric if and only if they are isometric over all the completions of the base field. This result has already been known for unimodular hermitian and skew-hermitian forms over rings with involution, under the same hypothesis. Finally, we study the behaviour of the Witt group of a linear hermitian category under extension of scalars. Let K be a field of characteristic different from 2, L a finite extension of K and ℳ a K-linear hermitian category. We define the extension of ℳ to L as being the category with the same objects as ℳ and with morphisms given by the morphisms of ℳ extended to L. We obtain an L-linear hermitian category, denoted by ℳL. The canonical functor of scalar extension ℛL/K : ℳ → ℳL induces for any ε = ±1 a group homomorphism Wε(ℳ) →Wε(ℳL). We prove that if all the idempotents of the category ℳ split and the extension L/K is of odd degree then this homomorphism is injective. This result has already been known in the case when ℳ is the category of finite-dimensional K-vector spaces.

Annihilating Polynomials for Quadratic Forms

Klaas-Tido Rühl

Let K be a field with char(K) ≠ 2. The Witt-Grothendieck ring (K) and the Witt ring W (K) of K are both quotients of the group ring ℤ[𝓖(K)], where 𝓖(K) := K*/(K*)2 is the square class group of K. Since ℤ[𝓖(K)] is integral, the same holds for (K) and W(K). The subject of this thesis is the study of annihilating polynomials for quadratic forms. More specifically, for a given quadratic form φ over K, we study polynomials P ∈ ℤ[X] such that P([φ]) = 0 or P({φ}) = 0. Here [φ] ∈ (K) denotes the isometry class and {φ} ∈ W(K) denotes the equivalence class of φ. The subset of ℤ[X] consisting of all annihilating polynomials for [φ], respectively {φ}, is an ideal, which we call the annihilating ideal of [φ], respectively {φ}. Chapter 1 is dedicated to the algebraic foundations for the study of annihilating polynomials for quadratic forms. First we study the general structure of ideals in ℤ[X], which later on allows us to efficiently determine complete sets of generators for annihilating ideals. Then we introduce a more natural setting for the study of annihilating polynomials for quadratic forms, i.e. we define Witt rings for groups of exponent 2. Both (K) and W(K) are Witt rings for the square class group 𝓖(K). Studying annihilating polynomials in this more general setting relieves us to a certain extent from having to distinguish between isometry and equivalence classes of quadratic forms. In Section 1.1 we study the structure of ideals in R[X], where R is a principal ideal domain. For an ideal I ⊂ R[X] there exist sets of generators, which can be obtained in a natural way by considering the leading coefficients of elements in I. These sets of generators are called convenient. By discarding super uous elements we obtain modest sets of generators, which under certain assumptions are minimal sets of generators for I. Let G be a group of exponent 2. In Section 1.2 we study annihilating polynomials for elements of ℤ[G]. With the help of the ring homomorphisms Hom(ℤ[G],ℤ) it is possible to completely classify annihilating polynomials for elements of ℤ[G]. Note that an annihilating polynomial for an element f ∈ ℤ[G] also annihilates the image of f in any quotient of ℤ[G]. In particular, Witt rings for G are quotients of ℤ[G]. In Section 1.3 we use the ring homomorphisms Hom(ℤ[G],ℤ) to describe the prime spectrum of ℤ[G]. The obtained results can then be employed for the characterisation of the prime spectrum of a Witt ring R for G. Section 1.4 is dedicated to proving the structure theorems for Witt rings. More precisely, we generalise the structure theorems for Witt rings of fields to the general setting of Witt rings for groups of exponent 2. Section 1.5 serves to summarise Chapter 1. If R is a Witt ring for G, then we use the structure theorems to determine, for an element x ∈ R, the specific shape of convenient and modest sets of generators for the annihilating ideal of x. In Chapter 2 we study annihilating polynomials for quadratic forms over fields. More specifically, we first consider fields K, over which quadratic forms can be classified with the help of the classical invariants. Calculations involving these invariants allow us to classify annihilating ideals for isometry and equivalence classes of quadratic forms over K. Then we apply methods from the theory of generic splitting to study annihilating polynomials for excellent quadratic forms. Throughout Chapter 2 we make heavy usage of the results obtained in Chapter 1. Let K be a field with char(K) ≠ 2. Section 2.1 constitutes an introduction to the algebraic theory of quadratic forms over fields. We introduce the Witt-Grothendieck ring (K) and the Witt ring W(K), and we show that these are indeed Witt rings for 𝓖(K). In addition we adapt the structure theorems to the specific setting of quadratic forms. In Section 2.2 we introduce Brauer groups and quaternion algebras, and in Section 2.3 we define the first three cohomological invariants of quadratic forms. In particular we use quaternion algebras to define the Clifford invariant. In Section 2.4 we begin our actual study of annihilating polynomials for quadratic forms. Henceforth it becomes necessary to distinguish between isometry and equivalence classes of quadratic forms. We start by classifying annihilating ideals for quadratic forms over fields K, for which (K) and W(K) have a particularly simple structure. Subsequently we use calculations involving the first three cohomological invariants to determine annihilating ideals for quadratic forms over a field K such that I3(K) = {0}, where I(K) ⊂ W(K) is the fundamental ideal. Local fields, which are a special class of such fields, are studied in Section 2.5. By applying the Hasse-Minkowski Theorem we can then determine annihilating ideals of quadratic forms over global fields. Section 2.6 serves as an introduction to the elementary theory of generic splitting. In particular we introduce Pfister neighbours and excellent quadratic forms, which are the subjects of study in Section 2.7. We use methods from generic splitting to study annihilating polynomials for Pfister neighbours. The obtained result can be applied inductively to obtain annihilating polynomials for excellent quadratic forms. We conclude the section by giving an alternative, elementary approach to the study of annihilating polynomials for excellent forms, which makes use of the fact that (K) and W(K) are quotients of ℤ[𝓖(K)].

EPFL2010

Hermitian Categories, Extension Of Scalars And Systems Of Sesquilinear Forms

Eva Bayer Fluckiger, Daniel Arnold Moldovan

We prove that the category of systems of sesquilinear forms over a given hermitian category is equivalent to the category of unimodular 1-hermitian forms over another hermitian category. The sesquilinear forms are not required to be unimodular or defined on a reflexive object (i.e., the standard map from the object to its double dual is not assumed to be bijective), and the forms in the system can be defined with respect to different hermitian structures on the given category. This extends an earlier result of the first and third authors. We use the equivalence to define a Witt group of sesquilinear forms over a hermitian category and to generalize results such as Witt's cancellation theorem, Springer's theorem, the weak Hasse principle, and finiteness of genus to systems of sesquilinear forms over hermitian categories.

Pacific Journal Mathematics2014

Categorical Foundations for K-theory

Nicolas Michel

K-Theory was originally defined by Grothendieck as a contravariant functor from a subcategory of schemes to abelian groups, known today as K0. The same kind of construction was then applied to other fields of mathematics, like spaces and (not necessarily commutative) rings. In all these cases, it consists of some process applied, not directly to the object one wants to study, but to some category related to it: the category of vector bundles over a space, of finitely generated projective modules over a ring, of locally free modules over a scheme, for instance. Later, Quillen extracted axioms that all these categories satisfy and that allow the Grothendieck construction of K0. The categorical structure he discovered is called today a Quillen-exact category. It led him not only to broaden the domain of application of K-theory, but also to define a whole K-theory spectrum associated to such a category. Waldhausen next generalized Quillen's notion of an exact category by introducing categories with weak equivalences and cofibrations, which one nowadays calls Waldhausen categories. K-theory has since been studied as a functor from the category of suitably structured (Quillen-exact, Waldhausen, symmetric monoidal) small categories to some category of spectra1. This has given rise to a huge field of research, so much so that there is a whole journal devoted to the subject. In this thesis, we want to take advantage of these tools to begin studying K-theory from another perspective. Indeed, we have the impression that, in the generalization of topological and algebraic K-theory that has been started by Quillen, something important has been left aside. K-theory was initiated as a (contravariant) functor from the various categories of spaces, rings, schemes, …, not from the category of Waldhausen small categories. Of course, one obtains information about a ring by studying its Quillen-exact category of (finitely generated projective) modules, but still, the final goal is the study of the ring, and, more globally, of the category of rings. Thus, in a general theory, one should describe a way to associate not only a spectrum to a structured category, but also a structured category to an object. Moreover, this process should take the morphisms of these objects into account. This gives rise to two fundamental questions. What kind of mathematical objects should K-theory be applied to? Given such an object, what category "over it" should one consider and how does it vary over morphisms? Considering examples, we have made the following observations. Suppose C is the category that is to be investigated by means of K-theory, like the category of topological spaces or of schemes, for instance. The category associated to an object of C is a sub-category of the category of modules over some monoid in a monoidal category with additional structure (topological, symmetric, abelian, model). The situation is highly "fibred": not only morphisms of C induce (structured) functors between these sub-categories of modules, but the monoidal category in which theses modules take place might vary from one object of C to another. In important cases, the sub-categories of modules considered are full sub-categories of "locally trivial" modules with respect to some (possibly weakened notion of) Grothendieck topology on C . That is, there are some specific modules that are considered sufficiently simple to be called trivial and locally trivial modules are those that are, locally over a covering of the Grothendieck topology, isomorphic to these. In this thesis, we explore, with K-theory in view, a categorical framework that encodes these kind of data. We also study these structures for their own sake, and give examples in other fields. We do not mention in this abstract set-theoretical issues, but they are handled with care in the discussion. Moreover, an appendix is devoted to the subject. After recalling classical facts of Grothendieck fibrations (and their associated indexed categories), we provide new insights into the concept of a bifibration. We prove that there is a 2-equivalence between the 2-category of bifibrations over a category ℬ and a 2-category of pseudo double functors from ℬ into the double category of adjunctions in CAT. We next turn our attention to composable pairs of fibrations , as they happen to be fundamental objects of the theory. We give a characterization of these objects in terms of pseudo-functors ℬop → FIBc into the 2-category of fibrations and Cartesian functors. We next turn to a short survey about Grothendieck (pre-)topologies. We start with the basic notion of covering function, that associate to each object of a category a family of coverings of the object. We study separately the saturation of a covering function with respect to sieves and to refinements. The Grothendieck topology generated by a pretopology is shown to be the result of these two steps. We define then, inspired by Street [89], the notion of (locally) trivial objects in a fibred category P : ℰ → ℬ equipped with some notion of covering of objects of the base ℬ. The trivial objects are objects chosen in some fibres. An object E in the fibre over B ∈ ℬ is locally trivial if there exists a covering {fi : Bi → B}i ∈ I such the inverse image of E along fi is isomorphic to a trivial object. Among examples are torsors, principal bundles, vector bundles, schemes, locally constant sheaves, quasi-coherent and locally free sheaves of modules, finitely generated projective modules over commutative rings, topological manifolds, … We give conditions under which locally trivial objects form a subfibration of P and describe the relationship between locally trivial objects with respect to subordinated covering functions. We then go into the algebraic part of the theory. We give a definition of monoidal fibred categories and show a 2-equivalence with monoidal indexed categories. We develop algebra (monoids and modules) in these two settings. Modules and monoids in a monoidal fibred category ℰ → ℬ happen to form a pair of fibrations . We end this thesis by explaining how to apply this categorical framework to K-theory and by proposing some prospects of research. ______________________________ 1 Works of Lurie, Toën and Vezzosi have shown that K-theory really depends on the (∞, 1)-category associated to a Waldhausen category [94]. Moreover, topological K-theory of spaces and Banach algebras takes the fact that the Waldhausen category is topological in account [62, 70].

EPFL2010

Annihilating Polynomials for Quadratic Forms

Klaas-Tido Rühl

EPFL2010

Categorical Foundations for K-theory

Nicolas Michel

EPFL2010