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 $E$ 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 $I_1 \ast I_2$ 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 $\langle I_1 \ast I_2\rangle$ the smallest full subcategory of $T$ containing $I_1 \ast I_2$ and closed under finite coproducts, summands and shifts. Define by induction $\langle E\rangle_1 = I_E$ and $\langle E\rangle_{k+1} = \langle \langle E\rangle_k \ast I_E \rangle$, $k\gt 1$.

The **dimension of a triangulated category** $T$ is the minimal integer $d\gt 0$ such that there is $E\in Ob T$ such that $\langle E\rangle_d = T$ or infinity otherwise. The **generation time** $d_E$ of an object $E$ in $T$ such that $\langle E\rangle_{d+1} = T$ and $\langle E\rangle_d \neq T$. $E$ 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 $T$.

By a result of Rouquier, the dimension of the triangulated category $D^b(Coh X)$ 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 $dim(X)$.

- 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