Operations on Monomial Ideals

Abstract

In this chapter, we apply some of the operations of Appendix A to monomial ideals. We have already seen this theme for sums and products in Exercises 1.3.12 and 1.3.13. In Section 2.1 we show, for instance, that intersections of monomial ideals are monomial ideals. Since we are interested in decomposing monomial ideals into intersections, it is important to know that the set of monomial ideals in a fixed polynomial ring is closed under intersections. We show that generating sequences for intersections of monomial ideals are described using least common multiples (LCMs). This motivates the optional Section 2.2 on unique factorization domains (UFDs), which are rings where least common multiples are guaranteed to exist in general. Similarly, in Section 2.5 we show that colons of monomial ideals are monomial ideals, and we show how to compute monomial generating sequences for them, which is helpful for our work on parametric decompositions in Chapter 6.