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Full-Text Articles in Algebra
Lattice Simplices: Sufficiently Complicated, Brian Davis
Lattice Simplices: Sufficiently Complicated, Brian Davis
Theses and Dissertations--Mathematics
Simplices are the "simplest" examples of polytopes, and yet they exhibit much of the rich and subtle combinatorics and commutative algebra of their more general cousins. In this way they are sufficiently complicated --- insights gained from their study can inform broader research in Ehrhart theory and associated fields.
In this dissertation we consider two previously unstudied properties of lattice simplices; one algebraic and one combinatorial. The first is the Poincar\'e series of the associated semigroup algebra, which is substantially more complicated than the Hilbert series of that same algebra. The second is the partial ordering of the elements of …
Boij-Söderberg Decompositions, Cellular Resolutions, And Polytopes, Stephen Sturgeon
Boij-Söderberg Decompositions, Cellular Resolutions, And Polytopes, Stephen Sturgeon
Theses and Dissertations--Mathematics
Boij-Söderberg theory shows that the Betti table of a graded module can be written as a linear combination of pure diagrams with integer coefficients. In chapter 2 using Ferrers hypergraphs and simplicial polytopes, we provide interpretations of these coefficients for ideals with a d-linear resolution, their quotient rings, and for Gorenstein rings whose resolution has essentially at most two linear strands. We also establish a structural result on the decomposition in the case of quasi-Gorenstein modules. These results are published in the Journal of Algebra, see [25].
In chapter 3 we provide some further results about Boij-Söderberg decompositions. We …