Some Homological Properties of Lattice Ideals

Abstract

The interplay of algebra and combinatorics is fruitful in both fields: combinatorics provides algebraic structures with tractable realizations, while algebra underpins combinatorial objects with a rigorous framework. Pioneered by Hochster and Stanley, interest in combinatorial commutative algebra has grown rapidly, often including techniques from simplicial topology and convex geometry.

This thesis presents two main results that combine commutative algebra and combinatorics. The first result considers the Cohen–Macaulayness of a lattice ideal and its associated toric ideal. Despite the deep algebraic connection between these two ideals, we produce infinitely many examples, in every codimension, of pairs where one of these ideals is Cohen–Macaulay but the other is not.

The second result describes the free resolution of the ground field over the quotient ring by a specific type of lattice ideal, that defining a rational normal 2-scroll. This chapter also includes a computation of the Betti numbers of the ground field when resolved over the ring coming from an arbitrary rational normal k-scroll.