Tuesday 13 October 2020
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The green groupoid and its action on derived categories
(joint work with Milen Yakimov)
To see the Abstract with mathematical symbols correctly displayed, please click here.
We introduce the green groupoid G of a 2-Calabi–Yau triangulated category C. It is an augmentation of the mutation graph of C, which is defined by means of silting theory.
The green groupoid G has certain key properties:
- If C is the stable category of a Frobenius category E, then G acts on the derived categories of the endomorphism rings E(m,m) where m is a maximal rigid object.
- G can be obtained geometrically from the g-vector fan of C.
- If the g-vector fan of C is a hyperplane arrangement H, then G specialises to the Deligne groupoid of H, and G acts faithfully on the derived categories of the endomorphism rings E(m,m).
The situation in (3) occurs if ΣC2, the square of the suspension functor, is the identity. It recovers results by Donovan, Hirano, and Wemyss where E is the category of maximal Cohen–Macaulay modules over a suitable singularity.