A property deducible from the generic initial ideal

Let $S_d$ be the vector space of monomials of degree $d$ in the variables $x_1, ..., x_s$. For a subspace V \sus S_d which is in general coordinates, consider the subspace \gin V \sus S_d generated by initial monomials of polynomials in $V$ for the revlex order. We address the question of what properties of $V$ may be deduced from \gin V. % This is an approach for understanding what algebraic or geometric properties of a homogeneous ideal I \sus k[x_1, ..., x_s] that may be deduced from its generic initial ideal \gin I.Comment: Completely revised compared to earlier hardcopy versions. AMS-Latex v1.2, 13 page

The cone of Betti diagrams of bigraded artinian modules of codimension two

We describe the positive cone generated by bigraded Betti diagrams of artinian modules of codimension two, whose resolutions become pure of a given type when taking total degrees. If the differences of these total degrees, p and q, are relatively prime, the extremal rays are parametrised by order ideals in N^2 contained in the region px + qy < (p-1)(q-1). We also consider some examples concerning artinian modules of codimension three.Comment: 15 page

Resolutions of letterplace ideals of posets

We investigate resolutions of letterplace ideals of posets. We develop topological results to compute their multigraded Betti numbers, and to give structural results on these Betti numbers. If the poset is a union of no more than $c$ chains, we show that the Betti numbers may be computed from simplicial complexes of no more than $c$ vertices. We also give a recursive procedure to compute the Betti diagrams when the Hasse diagram of $P$ has tree structure.Comment: 21 page

Surprising occurrences of order structures in mathematics

Order and symmetry are main structural principles in mathematics. We give five examples where on the face of it order is not apparent, but deeper investigations reveal that they are governed by order structures. These examples are finite topologies, associative algebras, subgroups of matrix groups, ideals in polynomial rings, and classes of bipartite graphs.Comment: 23 page

Filtering free resolutions

A recent result of Eisenbud-Schreyer and Boij-S\"oderberg proves that the Betti diagram of any graded module decomposes as a positive rational linear combination of pure diagrams. When does this numerical decomposition correspond to an actual filtration of the minimal free resolution? Our main result gives a sufficient condition for this to happen. We apply it to show the non-existence of free resolutions with some plausible-looking Betti diagrams and to study the semigroup of quiver representations of the simplest "wild" quiver.Comment: We correct a mistake in the proof of Corollary 4.2 in the published version of this paper. The mistake involves an incorrect definition for when two degree sequences are "sufficiently separated". The new definition weakens Theorem 1.3 somewhat, but the examples survive. We thank Amin Nematbakhsh and to Gunnar Floystad for bringing this mistake to our attention. We also correct some minor typo