Abstract: 

First, I will present a simplified proof of descent for stably dominated types in ACVF. I will also state a more general version of descent for stably dominated types in any theory, dropping the hypothesis of the existence of invariant extensions. This first part is joint work with Pierre Simon. In the second part, motivated by the study of the space of definable types orthogonal to the value group in a henselian valued field and their cohomology; I will present a theorem that states that over an algebraically closed base of imaginary elements, a global invariant type is residually dominated (essentially controlled by the residue field) if and only if it is orthogonal to the value group , if and only if its reduct in ACVF is stably dominated. This is joint work with Pablo Cubides and Silvain Rideau-Kikuchi. The result extend to some valued fields with operators.

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