Witt group of Hermitian forms over a noncommutative discrete valuation ring

Witt group of Hermitian forms over a noncommutative discrete valuation ring

We investigate Hermitian forms on finitely generated torsion modules over a noncommutative discrete valuation ring. We also give some results for lattices, which still are satisfied even if the base ring is not commutative. Moreover, for a noncommutative discrete-valued division algebra D with valuation ring R and residual division algebra D̄, we prove that W(D̄) ∼=WT(R), where WT(R) denotes the ...

Witt Groups of Hermitian Forms over a Brauer Severi Variety

Let X be a Brauer Severi variety over a field k of characteristic not 2 and let D be a division algebra over k with a k-linear involution. We investigate Witt groups of certain hermitian forms over D ⊗k OX .

Fano Fibrations over a Discrete Valuation Ring

Using the Log minimal model program we construct nice models of del Pezzo fibrations which are relevant to the inductive study of del Pezzo fibrations. To investigate birational maps of Fano fibrations, we study elements in anticanonical linear systems of Fano varieties, which have the “worst” singularities. We also investigate certain special points on smooth hypersurfaces of degree n ≥ 3 in P...

Quadratic Forms over Fields Ii: Structure of the Witt Ring

Quadratic Forms over Discrete Valuation Fields

As a general rule, the theory of quadratic forms over a ring R goes much more smoothly if R is non-dyadic. Of course, if R is a field then this simply says that we wish to avoid characteristic 2, but in general there is more to it than this. For instance, the ring Z is dyadic according our definition whereas for an odd prime power q Fq[t] is not, and indeed the theory of quadratic forms is some...