In recent work, Clark and Tchernev introduced the notion of monomial resolution supported on a poset. In this talk, I will review this notion, and the related notion of monomial resolution supported on a CW complex. I will discuss a result of mine that shows that resolutions supported on a CW complex are also supported on the face poset of a CW complex. This relates to the work of Clark and Tch...

We study blowups of affine n-space with center an arbitrary monomial ideal and call monomial ideals that render smooth blowups tame ideals. We give a combinatorial criterion to decide whether the blowup is smooth and apply this criterion to discuss a smoothing procedure proposed by Rosenberg, monomial building sets and permutohedra.

Alexander duality has, in the past, made its way into commutative algebra through Stanley-Reisner rings of simplicial complexes. This has the disadvantage that one is limited to squarefree monomial ideals. The notion of Alexander duality is generalized here to arbitrary monomial ideals. It is shown how this duality is naturally expressed by Bass numbers, in their relations to the Betti numbers ...

Monomial ideals form an important link between commutative algebra and combinatorics. In this chapter, we demonstrate how to implement algorithms in Macaulay 2 for studying and using monomial ideals. We illustrate these methods with examples from combinatorics, integer programming, and algebraic geometry.

These are lecture notes, in progress, on monomial ideals. The point of view is that monomial ideals are best understood by drawing them and looking at their corners, and that a combinatorial duality satisfied by these corners, Alexander duality, is key to understanding the more algebraic duality theories at play in algebraic geometry and commutative algebra. Sections written so far cover Alexan...

Classically, Gröbner bases are computed by first prescribing a fixed monomial order. Moss Sweedler suggested an alternative in the mid 1980s and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on K(x, y) that are suitable for this framework. We then perform such computations for i...

In this paper we primarily study monomial ideals and their minimal free resolutions by studying their associated lcm-lattices. In particular, we formally define the notion of coordinatizing a finite atomic lattice P to produce a monomial ideal whose lcm-lattice is P , and we give a characterization of all such coordinatizations. We prove that all relations in the lattice L(n) of all finite atom...