Quadratic and hermitian forms over rings
... Using descent theory, one can show that char A (x, a) is independent of S and the isomorphism above and lies in R [x]. Furthermore, the element a is invertible in A if and only if Nrd A (a), the reduced norm of a, is invertible in R (see [10], III.1.2, and [14], Theorem 4.3). ...
... where the automorphism ψ is the compositions of isomorphisms in the diagram. By a theorem of Artin (see, e.g., [10], §III, Lemma 1.2.1), one can find anètale faithfully flat S algebra T such that ψ ⊗ 1 : M n 2 (T ) → M n 2 (T ) is an inner automorphism. Now the determinant of the element a ⊗ 1 ⊗ 1 in the first row is N A/R (a) and in the second row is Nrd A (a) n and since ψ ⊗ 1 is inner, thus they coincide. ...
... and the R-algebra homomorphism i : A → M n 2 (A) where a maps to aI n 2 , where I n 2 is the identity matrix of M n 2 (A). Since R is a semilocal ring, the Skolem-Noether theorem is present in this setting (see [10], Prop. 5.2.3) and thus there is g ∈ GL n 2 (A) such that f (a) = gi(a)g −1 . ...
Let A be an Azumaya algebra of constant rank n^2 over a Hensel pair (R,I) where R is a semilocal ring with n invertible in R. Then the reduced Whitehead group SK(A) coincides with its reduction SK(A/IA).
... Then we immediately get N S R (c) ∈ D(q). 5 Grothendieck-Serre's conjecture for the case of spinor group 5.1. Let R be a local domain with the residue field k of characteristic different form 2 and q be a regular quadratic form over R. Following [3] we define the spinor group (scheme) Spin q to be Spin q (R) = {x ∈ SΓ q (R)|xσ(x) = 1}, where σ is the canonical involution. ...
... Let R be a local regular ring and K be it's quotient field. We have the following commutative digram (see [3]): ...
... Observe that by definition of the spinor norm [3] it is the same as to show that the norm map commutes with the functor D : S → q(S) (that sends any R-algebra S to the group of values of the quadratic form q on S), i.e., N S R (D(S)) ⊂ D(R). Hence, we have to prove the analog of Knebusch's Norm Principle for quadratic forms in the case of finite etale extensions of semi-local rings. ...
We prove the version of Knebusch's Norm principle for simple extensions of (semi-)local rings. As an application we prove the Grothedieck-Serre's conjecture on principal homogeneous spaces for the split case of the spinor group.
... In this section, we will recall some needed definitions and theorems from [11]. ...
... Proof. This is [11,Chapter I,2.3]. □ Proof. ...
... □ x → m x is an isomorphism. By [11,Chapter I,9.6], M has a ϵ-hermitian form h for ϵ = ±1. Then L can be given the structure of involution by the map ...
Let X be a smooth geometrically connected projective curve of genus at least 2 over a field of characteristic zero. We compute the essential dimension of the moduli stack of symplectic bundles over X. Unlike the case of vector bundles, we are able to precisely compute the essential dimension as the generic gerbe of the moduli stack has period 2 over it's moduli space.
... Nesta seção introduzimos aspectos gerais da TAFH. Nossos esforços são somados com propostas e resultados trabalhados nas referências [10] e [11] para as definições básicas, conceitos geométricos e introdução do grupo de Witt. ...
... Mais alguns resultados sobre este produto são enunciados na proposição abaixo (veja [10], Lema 8. 1.1 na página 48). Proposição 4.28. ...
... Se (M, h)é um espaço λ−hermitiano, dizemos queM ′ , h M ′ é um subespaço hermitiano. A forma h M ′é a restrição de h ao submódulo M ′ .O teorema a seguir (veja[10] lemas 3.6.1 e 3.6.2) tambémé válido no contexto de formas quadráticas (pode ser generalizado neste contexto).Teorema 4.22. Seja (M, h) um espaço λ−hermitiano. ...
O objetivo deste trabalho é duplo: fornecer uma introdução para os leitores que não estão familiarizados com a teoria algébrica das formas quadráticas sobre corpos (TAFQ) e, na sequência, com o referente construído na TAFQ introduzir a teoria algébrica de formas hermitinas com coeficientes em algébras associativas munidas de uma involução (TAFH).
... Building on this foundation, future work will focus on developing a real spectrum for non-commutative rings with involution, as a preparation for establishing an abstract theory of Hermitian forms ( [16]). ...
... On the other hand, Item 2 specifies a reciprocal condition; that is, each element, x ∈ A 0 , has a norm lying in the center. In the classical theory of rings with involution (see, for instance, [16]), involution with traces x + σ(x) and norms xσ(x) lying in the center are called standard. This justifies the notation above. ...
... This is an area we intend to explore in future work. Moreover, the continued development of the theory on non-commutative multialgebras with involution should lay a robust foundation for establishing an abstract theory of Hermitian forms ( [16]), similar to how the theory of special groups ( [22]) serves as an abstract theory of quadratic forms. ...
The primary motivation for this work is to develop the concept of Marshall’s quotient applicable to non-commutative multi-rings endowed with involution, expanding upon the main ideas of the classical case—commutative and without involution—presented in Marshall’s seminal paper. We define two multiplicative properties to address the involutive case and characterize their Marshall quotient. Moreover, this article presents various cases demonstrating that the “multi” version of rings with involution offers many examples, applications, and relatives in (multi)algebraic structures. Therefore, we established the first steps toward the development of an expansion of real algebra and real algebraic geometry to a non-commutative and involutive setting.
... Le fait de se placer dans le cadre des algèbres à involution permet de symétriser complètement la situation : sur une algèbre à involution (A, σ), on peut définir des formes ε-hermitiennes (V, h), et alors (End A (V ), σ h ) est encore une algèbre à involution. On renvoie à [16] ainsi qu'à la partie 2.1 pour plus de détails sur ces considérations, mais on peut déjà constater que les algèbres à involution forment un cadre stable pour étudier les formes hermitiennes et leurs algèbres d'endomorphismes. ...
... Le fait que la composition des morphismes dans Br h (K) soit bien définie et associative est une simple reformulation de la théorie de Morita hermitienne telle qu'elle est présentée dans [16]. ...
... condition, et V g = L g u g ⊂ V . Comme dim k (L g ) = 4, on a g∈H V g de dimension 16 sur k, donc V = g∈H V g . De plus, cette décomposition est orthogonale pour T τ puisque les Lu g sont orthogonaux pour la forme trace (en effet, Trd B (λu g ) = 0 si g = 1). ...
In order to study certain algebraic objects, and notably algebraic groups, Serre introduced the notion on invariants, in particular cohomological invariants. The construction of non-trivial cohomological invariants of algebraic groups is an active area of modern research, and very few invariants are known in degree greater than 3. In the first chapter, we give a complete description of Witt and cohomological invariants of the functors In as combinations of fundamental invariants behaving like divided powers, whose construction relies crucially on lambda operations in the Grothendieck-Witt ring. We also study the behaviour of these invariants with respect to various operations such as products or similitudes. The second chapter is dedicated to the construction of a "mixed" Witt ring associated to an algebra with involution : the fundamental idea is to define the product of two -hermitian forms using a Morita equivalence given by the involution trace form of the algebra. We also define lambda operations on the mixed Grothendieck-Witt ring, as well as a fundamental filtration imitating the split case. A particular attention is given to explicit calculations in the case of algebras of index 2. We use those tools in the third chapter to mimic the constructions of chapter 1 in the framework of hermitian forms, and thus construct cohomological invariants of algebras with involution, with more detailed results in index 2. The main interest is to be in principle able to define non-trivial invariants of arbitrarily high degree, although the index constitues a form of obstruction.
... Let (E , q, L ) be a quadratic form of rank n on a scheme S, and C 0 = C 0 (E , q, L ) be its even Clifford algebra (see [15] or [3, §1.8]), and Z = Z (E , q, L ) be its center. Then C 0 is a locally free O S -algebra of rank 2 n−1 , cf. [40,IV.1.6]. The associated finite morphism f : T → S is called the discriminant cover. ...
... The associated finite morphism f : T → S is called the discriminant cover. We remark that if S is locally factorial and q is generically regular of even rank then Z is a locally free O S -algebra of rank two, by (the remarks preceding) [40,IV Prop. 4.8.3], ...
... is finite flat of degree two over S, the first claim is verified. For the second claim, by [40,IV Prop. 7.3.1], ...
... There is an excellent book, [14], that deal with quadratic forms in an style near to that was presented in Lam's classical books [15] and [16], in the most general possible setting. And of course, some abstract theories appears trying to deal with this question. ...
... The generalization of the theory quadratic forms to general coefficients in rings is a hard step. The book [14] cover some basic aspects in the most general setting possible, and we have Marshall's theory of abstract real spectra ( [19]) and its algebraic counterpart, the real semigroups of Dickmann and Petrovich ( [4]) given a nice approach for the reduced theory of quadratic forms on rings, but, most of the relevant aspects of quadratic forms, like Witt rings, Pfister forms and etc, are uncovered. In the forthcoming [22], we propose the fundamentals for a non reduced and first-order abstract quadratic forms theory in general coefficients on rings, with the intuition and machinery of multirings and multifields, inspired by the functorial picture described here. ...
We provide, explicitly, equivalences and dual equivalences between categories of abstract quadratic forms theories and subcategories of multifields and multirings, that will bring new perspectives and methods to the abstract theories of quadratic forms in forthcoming papers.
... Clifford algebra and Azumaya algebra. A good introduction to Clifford algebras is provided by [16], where Clifford algebras are defined in the context of modules over commutative rings. In the same spirit, one can associate a sheaf of Z-graded algebras to each line bundle valued quadratic form by following a construction proposed in [10]. ...
... and is called an extension if X 1 is isomorphic to X 2 , and f 0 is the identity. Since the pullback of an Azumaya algebra is still Azumaya, see [16] III.5.1, one can define also Azumaya varieties as noncommutative varieties (X, B) in which the algebra B is Azumaya. Given a noncommutative variety (X, B), the so-called twisted derived category D b (X, B) can be defined in the usual way starting with the category of coherent sheaves of right B-modules on X as objects. ...
We describe an Azumaya algebra on the resolution of singularities of the double cover of a plane ramified along a nodal sextic associated to a non generic cubic fourfold containing a plane. We show that the derived category of such a resolution, twisted by the Azumaya algebra, is equivalent to the Kuznetsov component in the semiorthogonal decomposition of the derived category of the cubic fourfold.
... The second way is to define it with different signs [32,39,53]: ...
... Theorem 4.7. We have the following equivalent definitions of the group P Λ p,q,r : 39) and the group P ±Λ p,q,r : ...
In this paper, we introduce and study several Lie groups in degenerate (Clifford) geometric algebras. These Lie groups preserve the even and odd subspaces under the adjoint representation and the twisted adjoint representation. The considered Lie groups are interesting for the study of spin groups and their generalizations in degenerate case.
... Proof. In the case of an infinite field k this is a consequence of Ojanguren's Lemma [Knu,Ch.VIII,Corollary 3.5.2]. In the case of a finite field k this is a consequence of Lindel's Lemma [L,Proposition 2]. ...
Let D be a DVR of mixed characteristic.Let G be a reductive D-group scheme.Then the Grothendieck--Serre conjecture is true for the D-group scheme G and any geometrically regular local D-algebra R. Also we prove a version of Lindel--Ojanguren--Gabber's geometric presentation lemma in the DVR context.
... By definition the unitary group relative Λ is the group U ǫ 2n (R, Λ) := {A ∈ GL 2n (R) : h(Ax, Ay) = h(x, y), q(Ax) = q(x), x, y ∈ R}. For more general definitions and the properties of these spaces and groups see [2]. ...
In this paper homology stability for unitary groups over a ring with finite unitary stable rank is established. Homology stability of symplectic groups and orthogonal groups appears as a special case of our results.
... Let W (A, σ) denote the Witt group of (A, σ), i.e. the W (F )-module of Witt equivalence classes of nondegenerate hermitian forms h : M × M → A, where M is a finitely generated right A-module (cf. [18,Chap. I] or [32,Chap. ...
We provide a coherent picture of our efforts thus far in extending real algebra and its links to the theory of quadratic forms over ordered fields in the noncommutative direction, using hermitian forms and "ordered" algebras with involution.
... e.g. [Knu91]. ...
The paper discusses stably trivial torsors for spin and orthogonal groups over smooth affine schemes over infinite perfect fields of characteristic unequal to 2. We give a complete description of all the invariants relevant for the classification of such objects over schemes of dimension at most 3, along with many examples. The results are based on the -representability theorem for torsors and transfer of known computations of -homotopy sheaves along the sporadic isomorphisms to spin groups.
... There are examples of non constant G-torsors P over affine spaces; see Ojanguren-Sridharan [OS] (cf. also [K,VII.10]). Proposition 8.1 tells us that in these examples the twisted groups P G do not carry maximal tori. ...
Let G be a reductive affine group scheme defined over a semilocal ring k. Assume that either G is semisimple or k is normal and noetherian. We show that G has a finite k-subgroup S such that the natural map H^1(R, S) --> H^1(R, G) is surjective for every semilocal ring R containing k. In other words, G-torsors over Spec(R) admit reduction of structure to S. We also show that the natural map H^1(X, S) --> H^1(X, G) is surjective in several other contexts, under suitable assumptions on the base ring k, the scheme X/k and the group scheme G/k. These results have already been used to study loop algebras as well as essential dimension of connected algebraic groups in prime characteristic. Additional applications are presented at the end of this paper.
... ( [11] I Theorem 6.1, [3], [15]). §3. ...
We study the conjugate gradient method for solving a system of linear equations with coefficients which are measurable functions and establish the rate of convergence of this method. Consider a computer program used to give numerical solutions of stochastic differential equations which are the backbones of financial mathematics today. The input data are supposed to be the values at certain points of the input random variable and the output which will also believe to be the values of the output random variable, in fact the output data points are correlated to give an random variable which is then used to calculate the values at other intermediate points. If the program used for this computation contains subroutines based on certain numerical methods which may corrupt random variables, then when we input a random variable we are no longer certain the output is still a random variable. Though we all know a trader at the desk only cares about the numbers but a quantitative analyst would certainly feel better if he knows at least theoretically that, when the input is a random variable, the computer programs do output a random variable and not just a function pretending to be an instance of a random variable. We show in this paper that a numerical computation of random variables using the conjugate gradient method is robust-input a random variable it will output a random
... The main purpose of the present work is to provide the fundamental steps to expand Marshall's seminal paper [2] to the context of non-commutative multirings with involutions: this concerns mainly in provide and study the expansion of the notion of "Mashall's quotient" (see [1]), since it is a fundamental construction in abstract notions of real algebra and real algebraic geometry (space of signs [6]; abstract real spectra [7]; real semigroups [8]; real reduced multirings [2]). Building over this basis, a future work will be devoted to developing a "real spectra" for non-commutative rings with involutions, a preparation to establish an abstract theory of hermitian forms ( [9]). ...
The primary motivation for this work is to develop the concept of Marshall's quotient (\cite{ribeiro2016functorial}) applicable to non-commutative multirings endowed with involution, expanding the main ideas of the classical (= commutative, without involution) case presented in Marshall's seminal paper \cite{marshall2006real}. We define two multiplicative properties to deal with the involutive case and characterize their Marshall quotient. Besides, this article presents various cases showing that the ''multi'' version of rings with involution offers many examples, applications, and relatives in (multi) algebraic structures.
... 31.1], [24,Ch. IV], [27,Ch. ...
We provide a generalized definition for the quantized Clifford algebra introduced by Hayashi using another parameter k that we call the twist. For a field of characteristic not equal to 2, we provide a basis for our quantized Clifford algebra, show that it can be decomposed into rank 1 components, and compute its center to show it is a classical Clifford algebra over the group algebra of a product of cyclic groups of order 2k. In addition, we characterize the semisimplicity of our quantum Clifford algebra in terms of the semisimplicity of a cyclic group of order 2k and give a complete set of irreducible representations. We construct morphisms from quantum groups and explain various relationships between the classical and quantum Clifford algebras. By changing our generators, we provide a further generalization to allow k to be a half integer, where we recover certain quantum Clifford algebras introduced by Fadeev, Reshetikhin, and Takhtajan as a special case.
... v -Formas quadráticas não se restringem a coeficientes em corpos. Para uma abordagem generalizada de formas bilineares em anéis, pode-se consultar [16]. Para uma teoria geral de formas quadráticas sobre anéis que ainda tem a vantagem de ser descrita em linguagem da lógica de primeira ordem, consulte [17]. ...
Esta é uma pequena introdução lúdica à Teoria Algébrica de Formas Quadráticas, como apresentada em [1], e intermediada por fábulas ou anedotas do Candomblé brasileiro contadas em [2], no intuito de expor e fortalecer a Teoria de Formas Quadráticas perante a comunidade brasileira, dado que esta é uma teoria abrangente dentro da matemática (por exemplo, com conexões em teoria dos números e geometria algébrica real), e com importantes contribuições dadas por matemáticos latino-americanos, como por exemplo, as contribuições dos professores F. Miraglia e M. Dickmann nos artigos [3] e [4], e a contribuição do professor M. Spira no artigo [5]. O texto se concentra em apresentar os conceitos iniciais da teoria, como forma quadrática, espaços quadráticos, elementos representados por uma forma, discriminante, hiperbolicidade, anisotropia e diagonalização de formas. Após isso é apresentada uma fábula (inspirada pelo estilo de R. Smullyan em [6], e pelos jogos topológicos) envolvendo uma disputa entre Orixás solucionada através de um jogo que utiliza elementos da aritmética de formas quadráticas, como forma lúdica de envolver/interessar o leitor na bela teoria de formas quadráticas através de elementos da cultura afro-brasileira.
... There exist several constructions [1,4,6,7,9,12,17] generalizing Chevalley groups of classical types A ℓ , B ℓ , C ℓ , and D ℓ . Unitary groups of A. Bak [1] constructed by modules over form rings with hermitian and quadratic forms generalize Chevalley groups of types A ℓ , C ℓ , D ℓ , and some of their twisted forms. ...
Isotropic odd unitary groups generalize the Chevalley groups of classical types over commutative rings and their twisted forms. Such groups have root subgroups parameterized by a root system and may be constructed by so-called odd form rings with Peirce decompositions. We show the converse: if a group G has root subgroups indexed by roots of and satisfying natural conditions, then there is a homomorphism inducing isomorphisms on the root subgroups, where is the odd unitary Steinberg group constructed by an odd form ring with a Peirce decomposition. For groups with root subgroups indexed by (the already known case) the resulting odd form ring is essentially a generalized matrix ring.
... The second way is to define it with different signs [32,39,53]: ...
In this paper, we introduce and study five families of Lie groups in degenerate Clifford geometric algebras. These Lie groups preserve the even and odd subspaces and some other subspaces under the adjoint representation and the twisted adjoint representation. The considered Lie groups contain degenerate spin groups, Lipschitz groups, and Clifford groups as subgroups in the case of arbitrary dimension and signature. The considered Lie groups can be of interest for various applications in physics, engineering, and computer science.
... where g A ⊗q is the "twisted goldman element" (Id ⊗σ ⊗q )(g A ⊗q ) ∈ A ⊗2q . is given by the bilinear form (x, y) → Trd A ⊗q (σ ⊗q (x)y) on A ⊗q , which is given the twisted A ⊗2q -action on the left. From the explicit description of the inverse of a hermitian Morita equivalence (see from instance [9]), we need to check that if x, y, z ∈ A ⊗q , then ...
The Grothendieck-Witt ring of a field is known to be a -ring, where the -operations are induced by the exterior powers of bilinear spaces. We give a similar construction on the mixed Grothendieck-Witt ring of a central simple algebra with involution of the first kind over a field. In doing so we also develop a general framework for pre--ring structures on semi-rings graded over a monoid. Some explicit computations of even -powers are given in terms of restrictions of trace forms, and we explain how determinants of hermitian forms (which for us are maximal -powers) induce a duality similar to the one well-known for bilinear forms.
... The second way is to define it with different signs [32,39,53]: ...
In this paper, we introduce and study five families of Lie groups in degenerate Clifford geometric algebras. These Lie groups preserve the even and odd subspaces and some other subspaces under the adjoint representation and the twisted adjoint representation. The considered Lie groups contain degenerate spin groups, Lipschitz groups, and Clifford groups as subgroups in the case of arbitrary dimension and signature. The considered Lie groups can be of interest for various applications in physics, engineering, and computer science.
We prove the Bloch-Ogus Theorem for regular local rings geometrically regular over a discrete valuation ring. In particular, we prove the Bloch-Ogus Theorem for regular local rings of mixed characteristic that are essentially smooth over a discrete valuation ring.
We provide a new proof of the analogue of the Artin-Springer theorem for groups of type that can be represented by similitudes over an algebra of Schur index 2: an anisotropic generalized quadratic form over a quaternion algebra Q remains anisotropic after generic splitting of Q, hence also under odd degree field extensions of the base field. Our proof is characteristic free and does not use the excellence property.
We interpret in the setting of modules over schemes classical results pertaining to involutions of the first kind on algebras End R ( V ) , where V is a faithfully projective R -module.
Structural properties of unitary groups over local, not necessarily commutative, rings are developed, with applications to the computation of the orders of these groups (when finite) and to the degrees of the irreducible constituents of the Weil representation of a unitary group associated to a ramified extension of finite local rings.
For a simple linear algebraic group G acting faithfully on a vector space V and under mild assumptions, we show: if V is large enough, then the Lie algebra of G acts generically freely on V. That is, the stabilizer in the Lie algebra of G of a generic vector in V is zero. The bound on grows like and holds with only mild hypotheses on the characteristic of the underlying field. The proof relies on results on generation of Lie algebras by conjugates of an element that may be of independent interest. We use the bound in subsequent works to determine which irreducible faithful representations are generically free, with no hypothesis on the characteristic of the field. This in turn has applications to the question of which representations have a stabilizer in general position as well as the determination of the invariants of the representation.
We define an analogue of Shapovalov forms for Q-type Lie superalgebras and factorize the corresponding Shapovalov determinants which are responsible for simplicity of highest weight modules. We apply the factorization to obtain a description of the centres of Q-type Lie superalgebras.
An involution # on an associative ring R is \textit{formally real} if a sum of nonzero elements of the form r^# r where is nonzero. Suppose that R is a central simple algebra (i.e. for some integer n and central division algebra D) and # is an involution on R of the form r^# = a^{-1} r^\ast a, where is some transpose involution on R and a is an invertible matrix such that . In section 1 we characterize formal reality of # in terms of a and . In later sections we apply this result to the study of formal reality of involutions on crossed product division algebras. We can characterize involutions on that extend to a formally real involution on the split algebra . Every such involution is formally real but we show that there exist formally real involutions on D which are not of this form. In particular, there exists a formally real involution # for which the hermitian trace form x \mapsto \tr(x^#x) is not positive semidefinite.
Let G be an adjoint simple algebraic group of inner type. We express the Chow motive (with integral coefficients) of some anisotropic projective G-homogeneous varieties in terms of motives of simpler G-homogeneous varieties, namely, those that correspond to maximal parabolic subgroups of G. We decompose the motive of a generalized Severi-Brauer variety SB_2(A), where A is a division algebra of degree 5, into a direct sum of two indecomposable motives. As an application we provide another counter-example to the uniqueness of a direct sum decomposition in the category of motives with integral coefficients.
To an orthogonal or unitary involution on a central simple algebra of degree 4, or to a symplectic involution on a central simple algebra of degree 8, we associate a Pfister form that characterises the decomposability of the algebra with involution. In this way we obtain a unified approach to known decomposability criteria for several cases, and a new result for symplectic involutions on degree-8 algebras in characteristic 2.
Quadratic descent of generalized quadratic forms over a division algebra with involution of the first kind in characteristic two is investigated. Using the notion of transfer, it is shown that a system of quadratic forms, associated to such a generalized quadratic form, can be used to characterize its descent properties.
Let K be a field of characteristic other than 2, and let be the algebra deduced from by n successive Cayley–Dickson processes. Thus is provided with a natural basis indexed by the subsets E of . Two questions have motivated this paper. If a subalgebra of dimension 4 in is spanned by 4 elements of this basis, is it a quaternion algebra? The answer is always “yes”. If a subalgebra of dimension 8 in is spanned by 8 elements of this basis, is it an octonion algebra? The answer is more often “no” than “yes”. The present article establishes the properties and the formulas that justify these two answers, and describes the fake octonion algebras.
We prove that the hermitian Gersten-Witt complex is exact for Azumaya algebras with involution of the first- or second kind over a regular local ring, which is essentially smooth over a field, or over a discrete valuation ring.
The Witt group of skew hermitian forms over a division algebra D with symplectic involution is shown to be canonically isomorphic to the Witt group of symmetric bilinear forms over the Severi-Brauer variety of D with values in a suitable line bundle. In the special case where D is a quaternion algebra we extend previous work by Pfister and by Parimala on the Witt group of conics to set up two five-terms exact sequences relating the Witt groups of hermitian or skew-hermitian forms over D with the Witt groups of the center, of the function field of the Severi-Brauer conic of D, and of the residue fields at each closed point of the conic.
Trialitarian triples are triples of central simple algebras of degree 8 with orthogonal involution that provide a convenient structure for the representation of trialitarian algebraic groups as automorphism groups. This paper explicitly describes the canonical "trialitarian'' isomorphisms between the spin groups of the algebras with involution involved in a trialitarian triple, using a rationally defined shift operator that cyclically permutes the algebras. The construction relies on compositions of quadratic spaces of dimension 8, which yield all the trialitarian triples of split algebras. No restriction on the characteristic of the base field is needed.
We introduce the concept of an embedding of a quadratic space in an associative algebra. Using Clifford Algebras we derive some fundamental properties that any embedding should satisfy. Conversely, there is a simple description of the Clifford Algebra and the corresponding Spin groups in terms of the algebra in which the quadratic space is embedded. Though Clifford Algebras have been studied in detail, they may not always be easy to work with. Sometimes it may be useful to switch to a more concrete embedding to study low dimensional Spin and Epin (or Elementary Spin) groups.
We provide a geometric model for the classifying space of automorphism groups of Hermitian vector bundles over a ring with involution R such that \frac{1}{2} \in R ; this generalizes a result of M. Schlichting and G. S. Tripathi [Math. Ann. 362, No. 3–4, 1143–1167 (2015; Zbl 1331.14028)]. We then prove a periodicity theorem for Hermitian K -theory and use it to construct an E_\infty motivic ring spectrum \mathbf{KR}^{\text{alg}} representing homotopy Hermitian K -theory. From these results, we show that \mathbf{KR}^{\text{alg}} is stable under base change, and cdh descent for homotopy Hermitian K -theory of rings with involution is a formal consequence.
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