Zipping Tate resolutions and exterior coalgebras

We conjecture what the cone of hypercohomology tables of bounded complexes of coherent sheaves on projective spaces are, when we have specified regularity conditions on the cohomology sheaves of this complex and its dual. There is an injection from the this cone into the cone of homological data sets of squarefree modules over a polynomial ring \kk[x_1, \ldots, x_n], and we conjecture that this is an isomorphism: The Tate resolutions of a complex of coherent sheaves and the exterior coalgebra on $\langle x_1, \ldots, x_n \rangle$ may be amalgamated together to form a complex of free \Sym(\oplus_i x_i \te W^*)-modules, a procedure introduced by Cox and Materov. Via a reduction \oplus_i x_i \te W^* \pil \oplus_i x_i \te \kk we get a complex of free modules over \kk[x_1, \ldots, x_n] The extremal rays in the cone of squarefree complexes are conjecturally given by triplets of pure free squarefree complexes introduced in \cite{FlTr}. 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.Comment: 46 pages. Minor change

The linear space of Betti diagrams of multigraded artinian modules

We study the linear space generated by the multigraded Betti diagrams of Z^n-graded artinian modules of codimension n whose resolutions become pure of a given type when taking total degrees. We show that the multigraded Betti diagram of the equivariant resolution constructed by D.Eisenbud, J.Weyman, and the author, and all its twists, form a basis for this linear space.Comment: 15 pages, some modifications and added materia

Geometric properties derived from generic initial spaces

For a vector space V of homogeneous forms of the same degree in a polynomial ring, we investigate what can be said about the generic initial ideal of the ideal generated by V, from the form of the generic initial space gin(V) for the revlex order. Our main result is a considerable generalisation of a previous result by the first author.Comment: Improved presentation, 8 pages, to appear in Proceedings of AM

Triplets of pure free squarefree complexes

On the category of bounded complexes of finitely generated free squarefree modules over the polynomial ring S, there is the standard duality functor D = Hom_S(-, omega_S) and the Alexander duality functor A. The composition AD is an endofunctor on this category, of order three up to translation. We consider complexes F of free squarefree modules such that both F, AD(F) and (AD)^2(F) are pure, when considered as singly graded complexes. We conjecture i) the existence of such triplets of complexes for given triplets of degree sequences, and ii) the uniqueness of their Betti numbers, up to scalar multiple. We show that this uniqueness follows from the existence, and we construct such triplets if two of them are linear.Comment: 25 pages, minor improvement