Quotients are ubiquitous in Mathematics, and a general question is whether a certain category of sets allows quotients. For the category of definable sets in a given structure, the model theoretic approach is called elimination of imaginaries. For algebraically closed fields, Chevalley’s theorem and the existence of a field of definition of a variety imply that a quotient of a Zariski constructible set by a Zariski constructible equivalence relation is again a constructible set. Similar results hold for other classes of fields, such as differentially closed fields.
In this talk, we will focus on separably closed fields of positive characteristic p, and particularly those of infinite imperfection degree. Such a field K has infinite linear dimension over K p . It is unknown whether the theory SCF p,∞ of separably closed fields of positive characteristic p and infinite imperfection degree has elimination of imaginaries. In joint work with Martin Ziegler, we will provide a natural expansion of the language to achieve this, by showing that the theory SCF p,∞ is equational. Equationality, introduced by Srour, and later considered by Srour and Pillay, is a
generalisation of local noetherianity. We will present the main ideas of the proof, without assuming a deep knowledge of model theory.