Zipping Tate resolutions and exterior coalgebras

We conjecture what the cone of hypercohomology tables of bounded complexes of coherent sheaves on projective spaces is, when we have specified regularity conditions on the cohomology sheaves of this complex and its dual.There is an injection from this cone into the cone of homological data sets of squarefree modules over a polynomial ring k[x1,…,xn]k[x1,…,xn], and we conjecture that this is an isomorphism: The Tate resolutions of a complex of coherent sheaves on projective space P(W)P(W), and the exterior coalgebra on 〈x1,…,xn〉〈x1,…,xn〉 may be amalgamated together to form a complex of free Sym(⊕ixi⊗W⁎)Sym(⊕ixi⊗W⁎)-modules, a procedure introduced by Cox and Materov. Via a reduction ⊕ixi⊗W⁎→⊕ixi⊗k⊕ixi⊗W⁎→⊕ixi⊗k we get a complex of free modules over k[x1,…,xn]k[x1,…,xn]The extremal rays in the cone of squarefree complexes are conjecturally given by triplets of pure free squarefree complexes introduced in [15]. We describe the corresponding classes of hypercohomology tables, a class which generalizes vector bundles with supernatural cohomology.We also show how various pure resolutions in the literature, like resolutions of modules supported on determinantal varieties, and tensor complexes, may be obtained by the first part of the procedure.