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Remarks on generators and dimensions of triangulated categories. (English) Zbl 1197.18004

An object \(E\) in a triangulated category \(\mathcal{T}\) is called a classical generator if the smallest triangulated category of \(\mathcal{T}\) containing \(E\) and closed under direct summands is equal to \(\mathcal{T}\). A classical generator is called strong if it generates the category in a finite number of steps. This roughly means that any object of \(\mathcal{T}\) can be reached from \(E\) taking a finite number of cones (and shifts, finite direct sums and direct summands). An example of a strong generator is provided by the object \(\mathcal{O}\oplus \mathcal{O}(1)\) on \(\mathbb{P}^1\). The dimension of a triangulated category is roughly the smallest number of cones needed to build the whole category starting from some strong generator. The notion was introduced by R. Rouquier [J. K-Theory 1, No. 2, 193–256 (2008); Erratum ibid. 257–258 (2008; Zbl 1165.18008)], where in particular several results relating the dimension of a scheme \(X\) with the dimension of the bounded derived category \({\text D}^{\text b}(X)\) of \(X\) were proved. For example, if \(X\) is a reduced and separated scheme of finite type, then \(\dim({\text D}^{\text b}(X))\geq \dim(X)\). If \(X\) is a smooth affine scheme, then we have an equality.
One of the main results in the paper under review is that for any smooth projective curve \(C\) of genus \(g\geq 1\) the equality \(\dim({\text D}^{\text b}(C))=1\) holds (the case \(g=0\) is clear by the example above). The author furthermore conjectures that in fact for any smooth quasi-projective scheme \(X\) of dimension \(n\) the equality \(\dim({\text D}^{\text b}(X))=n\) holds.
The strategy of the proof in the case of a curve is roughly the following. Any object in \({\text D}^{\text b}(C)\) is the direct sum of its cohomologies and thus one can work with sheaves. Furthermore, any sheaf is the direct sum of a torsion sheaf and a vector bundle. The proof consists of the verification of the claim that the object \(E=\mathcal{L}^{-1}\oplus \mathcal{O}_C\oplus \mathcal{L}\oplus \mathcal{L}^2\), where \(\mathcal{L}\) is a line bundle such that \(\deg(\mathcal{L})\geq 8g\), is indeed a strong generator (in one step). For any torsion sheaf and any vector bundle the author produces exact sequences involving direct summands of \(E\) to prove this claim.
The paper also contains several results on dimension spectra of derived categories of curves. Here, the dimension spectrum is roughly the subset of \(\mathbb{Z}\) consisting of integers \(d\) with the property that there exists an object generating the category in precisely \(d\) steps. Furthermore, the author gives an explicit description of a classical generator of the category of perfect complexes on a quasi-projective scheme of dimension \(n\).
Reviewer: Pawel Sosna (Bonn)

MSC:

18E30 Derived categories, triangulated categories (MSC2010)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)

Citations:

Zbl 1165.18008
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