Orlov’s dimension spectrum (or simply Orlov spectrum) is an invariant of a triangulated category, introduced by Dmitri Orlov. When the triangulated category is of geometric origin (e.g. the bounded derived category of coherent sheaves on a projective variety, or the Fukaya category associated to a symplectic manifold) then the Orlov spectrum reflects some geometric information. The spectrum is defined in terms of counting the extensions needed to generate all the objects from a fixed object, with some other operations, like competing under direct coproducts and summands not counted.
An object in a triangulated category defines the smallest triangulated subcategory which is closed under direct sum. Given two full triangulated subcategories and one defines the full triangulated category consisting of all such that there exist in and in such that is a distinguished triangle, and by the smallest full subcategory of containing and closed under finite coproducts, summands and shifts. Define by induction and , .
The dimension of a triangulated category is the minimal integer such that there is such that or infinity otherwise. The generation time of an object in such that and . is a strong generator if the generation time is finite.
The dimension spectrum of is the set of generation times of all strong generators of .
By a result of Rouquier, the dimension of the triangulated category for a separated scheme of finite type over a perfect field is finite and, if is in addition reduced, it is equal or bigger than the Krull dimension of but smaller or equal the double dimension . By a conjecture of Orlov, for any smooth variety it should be in fact equal to .