On Borel fixed ideals generated in one degree

We construct a (shellable) polyhedral cell complex that supports a minimal free resolution of a Borel fixed ideal, which is minimally generated (in the Borel sense) by just one monomial in S=k[x_1,x_2,...,x_n]; this includes the case of powers of the homogeneous maximal ideal (x_1,x_2,...,x_n) as a special case. In our most general result we prove that for any Borel fixed ideal I generated in one degree, there exists a polyhedral cell complex that supports a minimal free resolution of I.Comment: 18 pages, 6 figure

Tropical types and associated cellular resolutions

An arrangement of finitely many tropical hyperplanes in the tropical torus leads to a notion of `type' data for points, with the underlying unlabeled arrangement giving rise to `coarse type'. It is shown that the decomposition of the tropical torus induced by types gives rise to minimal cocellular resolutions of certain associated monomial ideals. Via the Cayley trick from geometric combinatorics this also yields cellular resolutions supported on mixed subdivisions of dilated simplices, extending previously known constructions. Moreover, the methods developed lead to an algebraic algorithm for computing the facial structure of arbitrary tropical complexes from point data.Comment: minor correction