Harrison’s Witt ring of a generalized valuation ring

The Chow-Witt ring

The Reduced Witt Ring of a Formally

The reduced Witt rings of certain formally real fields are computed here in terms of some basic arithmetic invariants of the fields. For some fields, including the rational function field in one variable over the rational numbers and the rational function field in two variables over the real numbers, this is done by computing the image of the total signature map on the Witt ring. For a wider cl...

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 ...

Construction of the Ring of Witt Vectors

I will describe a functor A 7→ W (A) from the category of commutative rings to itself. The ring W (A) of ‘Witt vectors’ over A has many applications (to algebraic geometry, local rings, etc.), but I won’t discuss those. Convention: rings have 1’s that are respected by ring homomorphisms. By A I will always denote a commutative ring. The literature on the functor W is in a somewhat unsatisfactor...

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...