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Fantechi gives a few good examples in her notes and differentiates between abelian and regular cones. One can generalize her computations though: given a complete intersection ideal of we can consider the cone of the ideal . If we write the quotient , then the associated cone is given by the relative spec of the sheaf of algebras
This example should motivate the examples for sheaves on DM-stacks. The main geometric examples I know of are the weighted projective stacks with for and complete intersection substacks. If we take the ring then the weights determine a grading of the ring. If we take the stacky proj, then we get a DM-stack. It should be obvious from here that there are associated sheaves from the action, and they are what you would expect. For example, on we have