Today we will talk about exceptional objects, or lack thereof, in the derived category of a smooth and proper curve, say \(C\). Recall, an exceptional object \(E\) in a k-linear triangulated category \(\mathcal{T}\) is an object such that \[ \mathcal{T}(E,E[i])\cong k[0]. \]

During my oral examinations, I was asked about the existence or non-existence of nontrivial semi-orthogonal decompositions of the derived category of a smooth and proper curve. Of course \(\mathbb{P}^1\) has Beilinson’s decomposition and Okawa proves that curves of genus larger than zero have no such decompositions in Semi-orthogonal decomposability of the Derived Category of a Curve. They were unsatsified with this answer and asked me to prove that exceptional objects do not exist if the genus is positive. We will reprove this fact in this post.

Let \(D(C)\) denote the bounded derived category of coherent sheaves over \(C\), where \(C\) is a smooth and proper curve of positive genus. This category is formal since \(C\) is dimension 1. Recall, the derived category is formal if all objects are formal, i.e. quasi-isomorphic to the cohomology objects. Instead of looking at complexes of sheaves, we can reduce the question to honest sheaves. That is, there is no sheaf \(\mathcal{F}\) on \(C\) such that \(\mathrm{Ext}^\ast_C(\mathcal{F,F})\cong k[0]\).

There are two types of sheaves on a curve \(C\): torsion sheaves and locally free sheaves. Any sheaf \(\mathcal{F}\) on \(C\) fits into an exact sequence \[ 0\to \mathcal{F}_{tor}\to \mathcal{F}\to \mathcal{E}\to 0 \] where \(\mathcal{E}\) is locally free and \(\mathcal{F}_{tor}\) is torsion. Moreover, the relevant extension group vanishes: \[ \mathrm{Ext}^1_C(\mathcal{E},\mathcal{F}_{tor}) \cong H^1(\mathcal{F}_{tor}\otimes \mathcal{E}^\vee)\cong H^1(\mathcal{F}_{tor}^{\oplus n}) = 0 \] where n is the rank of \(\mathcal{E}\). Thus any sheaf \(\mathcal{F}\) on \(C\) is the sum of a torsion sheaf and a locally free sheaf. It suffices to show that torsion sheaves are not exceptional and locally free sheaves are not exceptional.

Let’s start with locally free sheaves. Suppose \(\mathcal{E}\) is a vector bundle over \(C\), then there is an exact sequence of sheaves \[ 0\to \mathcal{K}\to \mathcal{End}(\mathcal{E})\xrightarrow{\mathrm{Trace}} \mathcal{O}_C\to 0. \]

Since \(C\) has positive genus, the mapping \[ H^1(\mathcal{End}(\mathcal{E}))\to H^1(\mathcal{O}_C)\to 0 \] is nontrivial and using the isomorphism \[ H^1(\mathcal{End}(\mathcal{E}))\cong \mathrm{Ext}^1(\mathcal{E},\mathcal{E}) \] we see that vector bundles cannot be exceptional.

For a torsion sheaf \(\tau\) to be exceptional, it must be supported at a single closed point, say p. Otherwise, \(\mathrm{Hom}_C(\tau,\tau)\) is more than one dimensional. As \(\tau\) is torsion, there is an isomorphism \(\tau\cong \mathcal{O}_{np}\). By \(\mathcal{O}_{np}\) I mean the cokernel in the exact sequence \[ 0\to \mathcal{O}(-np)\to \mathcal{O}\to \mathcal{O}_{np}\to 0. \] By applying \(\mathbf{R}\mathrm{Hom}_C(-,\mathcal{O}_{np})\), there is a nontrivial mapping: \[ 0\to \mathrm{Hom}_C(\mathcal{O}(-np),\mathcal{O}_{np})\to \mathrm{Ext}_C^1(\mathcal{O}_{np},\mathcal{O}_{np}). \] It is nontivial because \(\mathrm{Hom}_C(\mathcal{O}(-np),\mathcal{O}_{np})\neq 0\) and the isomorphism \[ \mathrm{Hom}_C(\mathcal{O}_{np},\mathcal{O}_{np})\cong \mathrm{Hom}_C(\mathcal{O},\mathcal{O}_{np}) \] induced by pullback.

Thus there are no exceptional torsion sheaves on a curve and this completes the argument.

This result is a shadow of a more general theme regarding semi-orthogonal decompositions. In the Fano case, one typically has an exceptional object or an exceptional collection, then a large piece characterized by the left or right semi-orthogonal complement. Smooth Fano curves are all isomorphic to \(\mathbb{P}^1\) and in this case the large piece is not present.

In the smooth and connected Calabi-Yau case, Bridgeland has proven there are no nontrivial semi-orthogonal decompositions. Roughly, any semi-orthogonal decomposition must be fully orthogonal by Serre duality. Then this contradicts connectedness.

In the general type case, they are more rare but interesting examples do exist. For example, the Beauville surface admits interesting semi-orthogonal decompositions.