29 research outputs found
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 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