On the proof of elimination of imaginaries in algebraically closed valued fields

ACVF is the theory of non-trivially valued algebraically closed valued f ields. This theory is the model companion of the theory of valued fields. ACVF does not have elimination of imaginaries in the home sort (the valued field sort). Nevertheless, Haskell, Hrushovski, and Macpherson in [2] were able to find a collection of “geometric sorts” in which elimination of imaginaries holds. Let K be a model of ACVF, with valuation ring O and residue field k. A lattice in K is an O-submodule Λ ⊂ K isomorphic to O. Let Sn denote the set of lattices in K. This is an interpretable set; it can be identified with GLn(K)/GLn(O). For each lattice Λ ⊂ K, let res Λ denote Λ⊗O k, a k-vector space of dimension n. Let Tn be