Orlov spectrum


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 EE in a triangulated category TT defines the smallest triangulated subcategory I ETI_E\subset T which is closed under direct sum. Given two full triangulated subcategories I 1I_1 and I 2I_2 one defines the full triangulated category I 1*I 2I_1 \ast I_2 consisting of all MM such that there exist M 1M_1 in I 1I_1 and M 2M_2 in I 2I_2 such that M 1MM 2M_1\to M \to M_2 is a distinguished triangle, and by I 1*I 2\langle I_1 \ast I_2\rangle the smallest full subcategory of TT containing I 1*I 2I_1 \ast I_2 and closed under finite coproducts, summands and shifts. Define by induction E 1=I E\langle E\rangle_1 = I_E and E k+1=E k*I E\langle E\rangle_{k+1} = \langle \langle E\rangle_k \ast I_E \rangle, k>1k\gt 1.

The dimension of a triangulated category TT is the minimal integer d>0d\gt 0 such that there is EObTE\in Ob T such that E d=T\langle E\rangle_d = T or infinity otherwise. The generation time d Ed_E of an object EE in TT such that E d+1=T\langle E\rangle_{d+1} = T and E dT\langle E\rangle_d \neq T. EE is a strong generator if the generation time d Ed_E is finite.

The dimension spectrum of TT is the set σ(T)\sigma(T) of generation times of all strong generators of TT.

Basic results

By a result of Rouquier, the dimension of the triangulated category D b(CohX)D^b(Coh X) for a separated scheme XX of finite type over a perfect field is finite and, if XX is in addition reduced, it is equal or bigger than the Krull dimension of XX but smaller or equal the double dimension 2dim(X)2 dim(X). By a conjecture of Orlov, for any smooth variety it should be in fact equal to dim(X)dim(X).