nLab
Orlov spectrum

Idea

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.

Definitions

An object $E$ in a triangulated category $T$ defines the smallest triangulated subcategory $I_E\subset T$ which is closed under direct sum. Given two full triangulated subcategories $I_1$ and $I_2$ one defines the full triangulated category $I_1*I_2$ consisting of all $M$ such that there exist $M_1$ in $I_1$ and $M_2$ in $I_2$ such that $M_1\to M\to M_2$ is a distinguished triangle, and by $⟨I_1*I_2⟩$ the smallest full subcategory of $T$ containing $I_1*I_2$ and closed under finite coproducts, summands and shifts. Define by induction $⟨E{⟩}_1=I_E$ and $⟨E{⟩}_{k+1}=⟨⟨E{⟩}_k*I_E⟩$, $k>1$.

The dimension of a triangulated category $T$ is the minimal integer $d>0$ such that there is $E\in \mathrm{Ob}T$ such that $⟨E{⟩}_d=T$ or infinity otherwise. The generation time $d_E$ of an object $E$ in $T$ such that $⟨E{⟩}_{d+1}=T$ and $⟨E{⟩}_d\ne T$. $E$ is a strong generator if the generation time $d_E$ is finite.

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

Basic results

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

Literature

• Dmitri Orlov, Remarks on generators and dimensions of triangulated categories, arxiv/0804.1163
• R. Rouquier, Dimension of triangulated categories, J. K-Theory 1 (2008), no.2, 193-256, arXiv:math.CT/0310134
• Matthew Ballard, David Favero, Ludmil Katzarkov, Orlov spectra: bounds and gaps, arxiv/1012.0864
• A. Bondal, M. van den Bergh, Generators and representability of functors in commutative and non-commutative geometry, Mosc. Math. J. 3 (2003), no.1, 1-36, math.AG/0204218
• David Favero, Dimensions of triangulated categories, joint work with M. Ballard and L. Katzarkov, slides, Jan 2010, pdf