Brave new local moduli for ordinary K3 surfaces

It is shown that the K3 spectra which refine the local rings of the moduli stack of ordinary p-primitively polarized K3 surfaces in characteristic p allow for an Eoo structure which is unique up to equivalence. This uses the Eoo obstruction theory of Goerss and Hopkins and the description of the deformation theory of such K3 surfaces in terms of their Hodge F-crystals due to Deligne and Illusie. Furthermore, all automorphism of such K3 surfaces can be realized by Eoo maps which are unique up to homotopy, and this can by rigidified to an action if the automorphism group is tame.