We develop the notion of deformations using a valuation ring as ring of coefficients. This permits to consider in particular the classical Gerstenhaber deformations of associative or Lie algebras as infinitesimal deformations and to solve the equation of deformations in a polynomial frame. We consider also the deformations of the enveloping algebra of a rigid Lie algebra and we define valued de...

We prove that NIP valued fields of positive characteristic are henselian, and we begin to generalize the known results on dp-minimal dp-finite fields. On any unstable field K, define a type-definable group “infinitesimals,” corresponding canonical topology (K,+). reduce classification construction non-trivial Aut(K/A)-invariant valuation rings.

In this paper we compute the topological Hochschild homology of quotients discrete valuation rings (DVRs). Along way give a short argument for Bökstedt periodicity and generalizations over various other bases. Our strategy also gives very efficient to redo computations $\operatorname {THH}$ (respectively, logarithmic ) complete DVRs originally due Lindenstrauss Madsen Hesselholt Madsen).

In this paper, we undertake a systematic model and valuation theoretic study of the class ordered fields which are dense in their real closure. We apply to determine definable henselian valuations on fields, language rings. light our results, re-examine Shelah-Hasson Conjecture (specialised fields) provide an example limiting its conclusions.

Adjoining to the language of rings function symbols for splitting coefficients, relative $p$-coordinate functions, and division predicate a valuation, some theories pseudo-algebraically closed non-trivially valued fields admit quantifier elimination. It is also shown that in same theory given exponent imperfection does not elimination, due Galois theoretic obstructions.