Subcategories of weakly approximable triangulated categories and uniqueness of enhancements

A classical theorem by Rickard roughly states that the notion of derived equivalence for (noncommutative) rings is independent of the specific type of derived category: one can consider the unbounded derived category of all (left) modules, or any of its relevant triangulated subcategories (like the category of perfect complexes). It is natural to ask if Rickard's theorem can be generalized so as to include, for instance, also the geometric case where rings are replaced by (quasi-compact and quasi-separated) schemes. It will be shown that a positive answer can be given using the theory of weakly approximable triangulated categories, in combination with some results about uniqueness of dg enhancements. This is joint work, partly in progress, with A. Neeman and P. Stellari.

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