The cohomology table of a coherent sheaf on a projective variety is numerical data of the dimension of each cohomology group of each twist of the sheaf. Eisenbud--Schreyer give a description of the cone of cohomology tables of vector bundles and coherent sheaves on projective spaces. This leads to their proof of the Boij--Soderberg theory which describes the cone spanned by the Betti tables of finite length graded modules over polynomial rings. In this talk, we present some extensions of these results of Eisenbud--Schreyer to arbitrary projective varieties and arbitrary standard graded rings. Our method is to use a sequence of coherent sheaves that behave like an Ulrich sheaf asymptotically, which we call a lim Ulrich sequence of sheaves. This talk is based on joint work with Srikanth Iyengar and Mark Walker.