Monomial Ideals and Duality

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 Alexander duality, corners, Möbius inversion and Poincaré series, minimal free resolutions, and the Cohen-Macaulay condition. The sections planned but not yet written will cover the Stanley-Reisner monomial ideals associated to simplicial complexes, Reisner’s criteria for such an ideal to be Cohen-Macaulay, injective resolutions, local cohomology, and Serre duality.