Exceptional sequences over graded Cohen-Macaulay rings (Q2711340)

Let \(R\) be a local ring of a simple plane curve singularity. It is well-known that up to shift, the category \(\text{CM}(R)\) of graded maximal Cohen-Macaulay modules over \(R\) has only finitely many indecomposables. A complex \(E^\bullet\in\text{D}^b(R\text{-mod})\) is said to be exceptional if \(R\text{Hom}(E^\bullet,E^\bullet)=k:=\) the 0-part of \(R\) which is assumed to be an algebraically closed field. A sequence \((E^\bullet_i)\) of exceptional complexes is said to be exceptional if \(R\text{Hom}(E^\bullet_i,E^\bullet_j)=0\) for \(i>j\). As usual, there is a natural braid group action on exceptional sequences, generated by mutations. In contrast to the case of quiver representations, exceptional sequences for \(\text{CM}(R)\) may be infinite. In fact, the author shows that there is always a maximal (infinite) exceptional sequence for any such \(R\). Let \(B\) be the braid group with generators \(\sigma_i\), \(i\in\mathbb{Z}\). Apart from the action of \(B\), the two shift operations on exceptional sequences give rise to an action of \(\mathbb{Z}\) and \(\mathbb{Z}^{(\mathbb{Z})}\) on them. Therefore, the group \(G:=\mathbb{Z}\ltimes(B\ltimes\mathbb{Z}^{(\mathbb{Z})})\) acts on the exceptional sequences \((E^\bullet_i)_{i\in\mathbb{Z}}\). By an inspection of the Auslander-Reiten quivers for each \(R\), the author shows first that the indecomposables in \(\text{CM}(R)\) are exceptional. In each case he then finds a (necessarily maximal) exceptional sequence \(\mathcal E\) in \(\text{CM}(R)\) such that every indecomposable module in \(\text{CM}(R)\) occurs in \(\sigma{\mathcal E}\) for some \(\sigma\in G\).