David Cook Ii

Cohen-Macaulay Graphs and Face Vectors of Flag Complexes

We introduce a construction on a flag complex that by means of modifying the associated graph generates a new flag complex whose

The Lefschetz properties of monomial complete intersections in positive characteristic

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Hyperplane sections and the subtlety of the Lefschetz properties

-vector is the face vector of the original complex. This construction yields a vertex-decomposable, hence Cohen-Macaulay, complex. From this we get a (nonnumerical) characterization of the face vectors of flag complexes and deduce also that the face vector of a flag complex is the

The structure of the Boij-Söderberg posets

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On Decomposing Betti Tables and O-Sequences

-vector of some vertex-decomposable flag complex. We conjecture that the converse of the latter is true and prove this, by means of an explicit construction, for

Partially ordered sets in Macaulay2

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The strong Lefschetz property in codimension two

-vectors of Cohen-Macaulay flag complexes arising from bipartite graphs. We also give several new characterizations of bipartite graphs with Cohen-Macaulay or Buchsbaum independence complexes.

The weak Lefschetz property, monomial ideals, and lozenges

Abstract Stanley proved that, in characteristic zero, all Artinian monomial complete intersections have the strong Lefschetz property. We provide a positive characteristic complement to Stanleyʼs result in the case of Artinian monomial complete intersections generated by monomials all of the same degree, and also for arbitrary Artinian monomial complete intersections in characteristic two. To establish these results, we first prove an a priori lower bound on the characteristics that guarantee the Lefschetz properties. We then use a variety of techniques to complete the classifications.

The uniform face ideals of a simplicial complex

Abstract The weak and strong Lefschetz properties are two basic properties that Artinian algebras may have. Both Lefschetz properties may vary under small perturbations or changes of the characteristic. We study these subtleties by proposing a systematic way of deforming a monomial ideal failing the weak Lefschetz property to an ideal with the same Hilbert function and the weak Lefschetz property. In particular, we lift a family of Artinian monomial ideals to finite level sets of points in projective space with the property that a general hyperplane section has the weak Lefschetz property in almost all characteristics, whereas a special hyperplane section does not have this property in any characteristic.

The weak Lefschetz property for monomial ideals of small type

Boij and Soderberg made a pair of conjectures, which were subsequently proven by Eisenbud and Schreyer and then extended by Boij and Soderberg, concerning the structure of Betti diagrams of graded modules. In the theory, a particular family of posets and their associated order complexes play an integral role. We explore the structure of this family. In particular, we show that the posets are bounded complete lattices and the order complexes are vertex-decomposable, hence Cohen-Macaulay and squarefree glicci.