Qregularity and an extension of the Evans-Griffiths criterion to vector bundles on quadrics

Here we define the concept of Qregularity for coherent sheaves on a smooth quadric hypersurface Qn⊂Pn+1. In this setting we prove analogs of some classical properties. We compare the Qregularity of coherent sheaves on Qn with the Castelnuovo-Mumford regularity of their extension by zero in Pn+1. We also classify the coherent sheaves with Qregularity −∞. We use our notion of Qregularity in order to prove an extension of the Evans-Griffiths criterion to vector bundles on quadrics. In particular, we get a new and simple proof of Knörrer's characterization of ACM bundles.