Remarks on generators and dimensions of triangulated categories

In this paper we prove that the dimension of the bounded derived category of coherent sheaves on a smooth quasi-projective curve is equal to one. We also discuss dimension spectrums of these categories.

💡 Research Summary

The paper studies the notion of generators and the dimension of triangulated categories, focusing on the bounded derived category of coherent sheaves on smooth quasi‑projective curves. After recalling the definitions of a classical generator, a strong generator, and the dimension of a triangulated category (the minimal integer d such that some object E satisfies ⟨E⟩_{d+1}=𝒯), the author reviews known results: Bondal‑Van den Bergh proved that Perf (X) for a quasi‑compact, quasi‑separated scheme X always has a classical generator, and Rouquier showed that for a separated scheme of finite type the bounded derived category D^b(coh X) possesses a strong generator, hence finite dimension.

The main contribution is a precise computation for smooth projective curves C of genus g ≥ 1. Choosing a line bundle L with degree ≥ 8g, the author defines the object
E = L^{‑1} ⊕ 𝒪_C ⊕ L ⊕ L^{2}.
The goal is to prove that ⟨E⟩_{2}=D^b(coh C), i.e. the category is generated in two steps, which implies dim D^b(coh C)=1. The proof splits into two parts: torsion sheaves and vector bundles.

For torsion sheaves T, Lemma 7 constructs an exact sequence (L^{‑1})^{⊕r₁} → 𝒪_C^{⊕r₀} → T → 0 using global sections, showing T lies in ⟨E⟩_{2}. For a vector bundle F, the Harder–Narasimhan filtration is used. Selecting an index i where the slope jumps across 4g, Lemma 8 provides two exact sequences: one expressing the subbundle F_i via (L^{‑1}) and 𝒪_C, and another expressing the quotient F/F_i via L and L^{2}. Lemma 9 guarantees that for line bundles of sufficiently large degree, cohomology H¹ vanishes and the bundles are globally generated, which is essential for constructing the exact sequences.

With these sequences, the author builds a morphism φ: (L^{‑1})^{⊕r₁} ⊕ L^{⊕s₀}