We prove that the bounded derived category of coherent sheaves on a quasicompact separated quasiexcellent scheme of finite dimension has a strong generator in the sense of Bondal-Van den Bergh. This extends a recent result of Neeman and is new even in the affine case. The main ingredient includes Gabber's weak local uniformization theorem and the notions of boundedness and descendability of a morphism of schemes.

Strong generators in $\mathbf {D}^{\mathrm {perf}}(X)$ and $\mathbf {D}^b_{\mathrm {coh}}(X)$

Quasiexcellence implies strong generation

Abstract We develop a purely categorical theory of action filtrations and their associated growth invariants. When specialized to categories of geometric interest, such as the wrapped Fukaya category of a Weinstein manifold, and the bounded derived category of coherent sheaves on a smooth algebraic variety, our categorical action filtrations essentially recover previously studied filtrations of geometric origin. Our approach is built around the notion of a smooth categorical compactification. We prove that a smooth categorical compactification induces well-defined growth invariants, which are moreover preserved under zig-zags of such compactifications. The technical heart of the paper is a method for computing these growth invariants in terms of the growth of certain colimits of (bi)modules. In practice, such colimits arise in both geometric settings of interest. The main applications are: (1) A “quantitative” refinement of homological mirror symmetry, which relates the growth of the Reeb-length filtration on the symplectic geometry side with the growth of filtrations on the algebraic geometry side defined by the order of pole at infinity (often these can be expressed in terms of the dimension of the support of sheaves). (2) A proof that the Reeb-length growth of symplectic cohomology and wrapped Floer cohomology on a Weinstein manifold are at most exponential. (3) Lower bounds for the entropy and polynomial entropy of certain natural endofunctors acting on Fukaya categories. 

Categorical action filtrations via localization and the growth as a symplectic invariant

Let $\mathcal T$ be a well generated triangulated category, and let $S\subset\mathcal T$ be a set of objects. We prove that there is a t-structure on $\mathcal T$ with ${\mathcal T}^{\leq0}=\overline{\langle S\rangle}^{(-\infty,0]}$. 
This article is an improvement on the main result of a 2003 article by Alonso, Jeremias and Souto---in that article the theorem was proved under the assumption that $\mathcal T$ has a nice enough model. It should be mentioned that the theorem of Alonso, Jeremias and Souto has been influential---it turns out to be interesting to study all of these t-structures.

The t-structures generated by objects

Abstract We prove that if X is a regular quasi-projective variety of dimension d, the set of line bundles { 𝒪 X ⁢ ( n ) } n ∈ ℤ {\{\mathcal{O}_{X}(n)\}_{n\in{\mathbb{Z}}}} generates the bounded derived category of X in d steps. This proves new cases of a conjecture of Orlov as well as a conjecture of Elagin and Lunts.

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Strong generators in $\mathbf {D}^{\mathrm {perf}}(X)$ and $\mathbf {D}^b_{\mathrm {coh}}(X)$

Citations

Quasiexcellence implies strong generation

Categorical action filtrations via localization and the growth as a symplectic invariant

The t-structures generated by objects

Ample line bundles and generation time