Related papers: Azumaya Algebras With Orthogonal Involution Admitt…

We classify all two-dimensional simple algebras (which may be non-associative) over an algebraically closed field. For each two-dimensional algebra $\mathcal{A}$, we describe a minimal (with respect to inclusion) generating set for the…

This paper is dedicated to a further study of derived Azumaya algebras. The first result we obtain is a Beauville-Laszlo-style property for such objects (considered up to Morita equivalence), which is consequence of a more general…

Let $\Gamma$ be a $T$-ideal of identities of an affine PI-algebra over an algebraically closed field $F$ of characteristic zero. Consider the family $\mathcal{M}_{\Gamma}$ of finite dimensional algebras $\Sigma$ with $Id(\Sigma) = \Gamma$.…

Let $V$ be a $6$-dimensional complex vector space with an involution $\sigma$ of trace $0$, and let $W \subset \Sym^2 V^\vee$ be a generic $3$-dimensional subspace of $\sigma$-invariant quadratic forms. To these data we can associate an…

We show that the group of proper projective similitudes of a totally decomposable algebra with involution of the first kind over a field of characteristic different from 2 is R-trivial.

Given systems of two (inhomogeneous) quadratic equations in four variables, it is known that the Hasse principle for integral points may fail. Sometimes this failure can be explained by some integral Brauer-Manin obstruction. We study the…

Let $X$ be an Enriques surface. Using Beauville's result about the triviality of the Brauer map of $X$, we define a new involution on the category of coherent sheaves on the canonically covering K3 surface $\overline{X}$. We relate the…

We consider moduli spaces of Azumaya algebras on K3 surfaces and construct an example. In some cases we show a derived equivalence which corresponds to a derived equivalence between twisted sheaves. We prove if $A$ and $A'$ are Morita…

We describe all Witt invariants of anti-hermitian forms over a quaternion algebra with its canonical involution, and in particular all Witt invariants of orthogonal groups $O(A,\sigma)$ where $(A,\sigma)$ is an central simple algebra with…

The Umehara algebra is studied with motivation on the problem of the non-existence of common complex submanifolds. In this paper, we prove some new results in Umehara algebra and obtain some applications. In particular, if a complex…

Let A be an introverted algebra with mean value. We prove that its spectrum \Delta (A) is a compact topological semigroup, and that the kernel K(\Delta (A)) of \Delta (A) is a compact topological group over which the mean value on A can be…

Using the technique of inductive resolution introduced in arXiv:2303.07979, we prove that the homology of Rook-Brauer Algebra, interpreted as appropriate Tor-group, is isomorphic to that of symmetric group for all degrees under the…

We show that for a given Nakayama algebra $\Theta$, there exist countably many cyclic Nakayama algebras $\Lambda_i$, where $i \in \mathbb{N}$, such that the syzygy filtered algebra of $\Lambda_i$ is isomorphic to $\Theta$ and we describe…

Let T -> S be a finite flat morphism of degree two between regular integral schemes of dimension at most two (and with 2 invertible), having regular branch divisor D. We establish a bijection between Azumaya quaternion algebras on T and…

Let A be a central simple algebra with involution sigma of first or second kind. Let v be a valuation on the sigma-fixed part F of Z(A). A sigma-special v-gauge g on A is a kind of value function on A extending v on F, such that g(sigma(x)…

The so-called Tits class, associated to an adjoint absolutely almost simple algebraic group, provides a cohomological obstruction for this group to admit an outer automorphism. If the group has inner type, this obstruction is the only one.…

Let $X$ be a connected CW complex. Let $\mathcal{V}$ be a symplectic vector bundle of rank $2mn$ over $X$, and let $\mathcal{A}$ be a topological Azumaya algebra of degree $2mn$ with a symplectic involution over a $X$. We give conditions…

Let $\sigma$ be an endomorphism and $\delta$ an $\sigma$-derivation of a ring $R$. In this paper, we show that if $R$ is $(\sigma,\delta)$-skew Armendariz and $a\sigma(b)=0$ implies $ab=0$ for $a,b\in R$. Then $R$ is symmetric…

We classify conjugacy classes of involutions in the isometry groups of nondegenerate, symmetric bilinear forms over the field of two elements. The new component of this work focuses on the case of an orthogonal form on an even dimensional…

We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an…