We investigate the affine Grassmannian for the special linear group, and its Schubert varieties. In (positive) equal characteristic, we give an alternative way to construct Demazure resolutions of these Schubert varieties as subvarieties of a multigraded Hilbert scheme. In mixed characteristics, we investigate the approach by Haboush to construct in a similar way Schubert varieties for the p-adic affine Grassmannian. Moreover, we investigate which results on the affine Grassmannian can be carried over from the function field case to the p-adic case.