the aim of this paper is to study orders over a valuation ring v with arbitrary rank in acentral simple f-algebra q. the relation between all of the orders is explained with a diagram. it is thenshown that inside bezout order, extremal v-orders are precisely semi-hereditary. in the last section, theeffect of henselization on maximal and semi-hereditary orders is examined.

In his 1974 text, Commutative Ring Theory, Kaplansky states that among the examples of non-Dedekind Prüfer domains, the main ones are valuation domains, the ring of entire functions and the integral closure of a Prüfer domain in an algebraic extension of its quotient field [Kap74, p.72]. A similar list today would likely include Kronecker function rings, the ring of integervalued polynomials an...

For the whole paper, K denotes an algebraically closed field endowed with a nontrivial non-archimedean complete absolute value | |. The corresponding valuation is v := − log | | with value group Γ := v(K). The valuation ring is denoted by K. Note that the residue field K̃ is algebraically closed. In Theorem 1.3, §8 and in the second part of §9, we start with a field K endowed with a discrete val...

We prove R = T theorems for certain reducible residual Galois representations. We answer in the positive a question of Gross and Lubin on whether certain Hecke algebras T are discrete valuation rings. In order to prove these results we determine (using the theory of Breuil modules) when two finite flat group schemes G and H of order p over an arbitrarily tamely ramified discrete valuation ring ...

We describe an alternate construction of some of the basic rings introduced by Fontaine in p-adic Hodge theory. In our construction, the central role is played by the ring of p-typical Witt vectors over a p-adic valuation ring, rather than theWitt vectors over a ring of positive characteristic. This suggests the possibility of forming a meaningful global analogue of p-adic Hodge theory.