Indecomposable and noncrossed product division algebras over function fields of smooth p-adic curves

Indecomposable and Noncrossed Product Division Algebras over Curves over Complete Discrete Valuation Rings

Let T be a complete discrete valuation ring and $\hat{X}$ a smooth projective curve over $S=\spec(T)$ with closed fibre $X$. Denote by $F$ the function field of $\hat{X}$ and by $\hat{F}$ the completion of $F$ with respect to the discrete valuation defined by $X$, the closed fibre. In this paper, we construct indecomposable and noncrossed product division algebras over $F$. This is done by defining an index preserving group homomorphism $s:\br(\hat{F})'\to\br(F)'$, and using it to lift indecomposable and noncrossed product division algebras over $\hat{F}$.

Tame Covers and Cohomology of Relative Curves over Complete Discrete Valuation Rings, with Applications to the Brauer Group

We prove the existence of noncrossed product and indecomposable division algebras over the function field of a smooth p-adic curve, especially when the curve does not admit a smooth model over Z_p. Thus we generalize arXiv 0907.0670. To make our constructions, we investigate the lifting of cohomology classes from the total fraction ring of the closed fiber to the function field of the curve, over an arbitrary discrete valuation ring of mixed characteristic.

Cyclicity and indecomposability in the Brauer group of a $p$-adic curve

For a $p$-adic curve $X$, we study conditions under which all classes in the $n$-torsion of $Br(X)$ are $\mathbb{Z}/n$-cyclic. We show that in general not all classes are $\mathbb{Z}/n$-cyclic classes. On the other hand, if $X$ has good reduction and $n$ is prime to $p$, of if $X$ is an elliptic curve over $\mathbb{Q}_p$ with split multiplicative reduction and $n$ is a power of $p$, then we prove that all order $n$ elements of $Br(X)$ are $\mathbb{Z}/n$-cyclic. Finally, if $X$ has good reduction and its function field $K(X)$ contains all $p^2$-th roots of $1$, we show the existence of indecomp…

分次环上分次模的不可分解性

Indecomposability of graded modules over a graded ring

本文证明了在 Henselian 局部环上的非负分次环及其有限型分次代数中，模的不可分解性在分次、局部化和完备化意义下等价，并给出了分次模分解的 Krull-Schmidt 唯一性定理及其在特征 p 环 FFRT 性质上的应用。