Related papers: On the residue fields of Henselian valued stable f…

We classify all possible extensions of a valuation from a ground field $K$ to a rational function field in one or several variables over $K$. We determine which value groups and residue fields can appear, and we show how to construct…

A basic finite dimensional algebra over an algebraically closed field $k$ is isomorphic to a quotient of a tensor algebra by an admissible ideal. The category of left modules over the algebra is isomorphic to the category of representations…

In this project, we will study the Brauer group that was first defined by R. Brauer. The elements of the Brauer group are the equivalence classes of finite dimensional central simple algebra. Therefore understanding the structure of the…

The present paper shows that if $q \in \mathbb P$ or $q = 0$, where $\mathbb P$ is the set of prime numbers, then there exist characteristic $q$ fields $E _{q,k}\colon \ k \in \mathbb N$, of Brauer dimension Brd$(E _{q,k}) = k$ and infinite…

Let $A$ be an abelian variety over the function field $K$ of a curve over a finite field. We describe several mild geometric conditions ensuring that the group $A(K^{\rm perf})$ is finitely generated and that the $p$-primary torsion…

For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the $n$-th valued hyperfields of $K_1$ and $K_2$ are isomorphic over $p$ for each $n\ge1$, then $K_1$ and $K_2$…

For a smooth and proper variety $X$ over an algebraically closed field $k$ of characteristic $p>0$, the group $Br(X)[p^\infty]$ is a direct sum of finitely many copies of $\mathbb{Q}_p/\mathbb{Z}_p$ and an abelian group of finite exponent.…

In this paper we show that if two central simple $k$-algebras generate the same cyclic subgroup in $\mathrm{Br}(k)$, then there are rational maps between varieties associated to these algebras, such as Brauer--Severi varieties, norm…

We give a process to construct non-split, three-dimensional simple Lie algebras from involutions of sl(2,k), where k is a field of characteristic not two. Up to equivalence, non-split three-dimensional simple Lie algebras obtained in this…

Let $K$ be the function field of a $p$-adic curve, $G$ a semisimple simply connected group over $K$ and $X$ a $G$-torsor over $K$. A conjecture of Colliot-Th\'el\`ene, Parimala and Suresh predicts that if for every discrete valuation $v$ of…

Let $(R,\mathfrak{m}_R,k)$ be a one-dimensional complete local reduced $k$-algebra over a field of characteristic zero. R. Berger conjectured that $R$ is regular if and only if the universally finite module of differentials $\Omega_R$ is…

We classify group schemes in terms of their Cartier modules. We also prove the equivalence of different definitions of the tangent space and the dimension for these group schemes; in particular, the minimal dimension of a formal group law…

We show that every central simple algebra A over a field k is Brauer equivalent to a quotient of a finite dimensional Hopf algebra over the same field (that is- A is Hopf Schur). If the characteristic of the field is zero, or if the algebra…

Let $E$ be a field, $R$ a finite separable extension of $E$, and $R_{\rm ab}$ the maximal abelian subextension of $E$ in $R$. The main result of this paper shows that the norm groups $N(R/E)$ and $N(R_{\rm ab}/E)$ are equal in each of the…

We prove in arbitrary characteristic that an immediate valued algebraic function field $F$ of transcendence degree 1 over a tame field $K$ is contained in the henselization of $K(x)$ for a suitably chosen $x\in F$. This eliminates…

In this paper we classify triangular semisimple and cosemisimple Hopf algebras over any algebraically closed field k. Namely, we construct, for each positive integer N, relatively prime to the characteristic of k if it is positive, a…

Let $X$ be a smooth projective integral variety over a finitely generated field $k$ of characteristic $p>0$. We show that the finiteness of the exponent of the $p$-primary part of $\mathrm{Br}(X_{k^s})^{G_k}$ is equivalent to the Tate…

We look at connection Laplacians L,g defined by a field h:G to K, where G is a finite set of sets and K is a normed division ring which does not need to be commutative, nor associative but has a conjugation leading to the norm as the square…

We give an interpretation of the Brauer group of a purely inseparable extension of exponent 1, in terms of restricted Lie-Rinehart cohomology. In particular, we define and study the category $p$-$\rm{LR}(A)$ of restricted Lie-Rinehart…

Given a global field K and a positive integer n, there exists an abelian extension L/K (of exponent n) such that the local degree of L/K is equal to n at every finite prime of K, and is equal to two at the real primes if n=2. As a…