Related papers: Purity for Similarity Factors
We consider the following problem: Under what assumptions do one or more of the following are equivalent for a ring $R$: (A) $R$ is Morita equivalent to a ring with involution, (B) $R$ is Morita equivalent to a ring with an…
Let $R$ be a valuation ring with fraction field $K$ and $2\in R^\times$. We give an elementary proof of the following known result: Two unimodular quadratic forms over $R$ are isometric over $K$ if and only if they are isometric over $R$.…
We consider a series of questions that grew out of determining when two quantum planes are isomorphic. In particular, we consider a similar question for quantum matrix algebras and certain ambiskew polynomial rings. Additionally, we modify…
The main theorem (Theorem 4.1) of this paper claims that any ring morphism from an Azumaya algebra of constant rank over a commutative ring to another one of the same constant rank and over a reduced commutative ring induces a ring morphism…
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.
In this paper we prove that two idempotent rings are Morita equivalent if every corner of one of them is isomorphic to a corner of a matrix ring of the other one. We establish the converse (which is not true in general) for $\sigma$-unital…
We give a geometrical criterion to determine when a quaternion algebra over the function field of a stable elliptic surface X is an Azumaya algebra over X.
Let $R$ be a semilocal principal ideal domain. Two algebraic objects over $R$ in which scalar extension makes sense (e.g. quadratic spaces) are said to be of the same genus if they become isomorphic after extending scalars to all…
Suppose $A$ is an Azumaya algebra over a ring $R$ and $\sigma$ is an involution of $A$ extending an order-$2$ automorphism $\lambda:R\to R$. We say $\sigma$ is extraordinary if there does not exist a Brauer-trivial Azumaya algebra…
In this paper we continue our investigation of signatures of hermitian forms over Azumaya algebras with involution over commutative rings. We show that the approach used in an earlier paper for central simple algebras can be extended to…
The entanglement characteristics of two qubits are encoded in the invariants of the adjoint action of SU(2) x SU(2) group on the space of density matrices defined as the space of positive semi-definite Hermitian matrices. The corresponding…
We equip the complex polynomial algebra C[t] with the involution which is the identity on C and sends t to -t. Answering a question raised by V.G. Kac, we show that every hermitian or skew-hermitian matrix over this algebra is congruent to…
Let $(A,\sigma)$ be an Azumaya algebra with orthogonal involution over a ring $R$ with $2\in R^\times$. We show that if $(A,\sigma)$ admits an improper isometry, i.e., an element $a\in A$ with $\sigma(a)a=1$ and $\mathrm{Nrd}_{A/R}(a)=-1$,…
Given an Azumaya algebra with involution $(A,\sigma)$ over a commutative ring $R$ and some auxiliary data, we construct an $8$-periodic chain complex involving the Witt groups of $(A,\sigma)$ and other algebras with involution, and prove it…
It is proved that whenever two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, their associated complexity functions are asymptotically equivalent in a certain sense (which implies, if the…
There has been a long-standing question about whether being perfectoid for an algebra is local in the analytic topology. We provide affirmative answers for the algebras (e.g., over $\overline{\mathbb{Z}_p}$) whose spectra are inverse limits…
We prove a number of structure and isomorphism results concerning the non-commutative Natsume-Olsen spheres $\mathbb{S}^{2n-1}_{\theta}$ deformed along a skew-symmetric matrix $\theta\in \mathbb{R}$. These include (a) the fact that two…
Let $R$ be a semilocal Dedekind domain. Under certain assumptions, we show that two (not necessarily unimodular) hermitian forms over an $R$-algebra with involution, which are rationally ismorphic and have isomorphic semisimple coradicals,…
We consider the general circumstance of an Azumaya algebra $A$ of degree $n$ over a locally ringed topos $(\mathbf{X}, {\mathcal{O}}_{\mathbf{ X}})$ where the latter carries a (possibly trivial) involution, denoted $\lambda$. This…
Conjugation spaces are equipped with an involution such that the fixed points have the same mod 2 cohomology (as a graded vector space, a ring, and even an unstable algebra) but with all degrees divided by 2, generalizing the classical…