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...

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 ...

We discuss simple partitioning phenomena for intersection rings of certain toric varieties, and apply this to prove the following results. ̋ We generalize a result of Gräbe, proving that Stanley–Reisner rings of homology spheres are Gorenstein. ̋ We give a new and direct proof of a theorem of the authors due to which the cohomology rings of matroids are Poincare duality algebras. ̋ We prove a conj...

A certain squarefree monomial ideal HP arising from a finite partially ordered set P will be studied from viewpoints of both commutative algbera and combinatorics. First, it is proved that the defining ideal of the Rees algebra of HP possesses a quadratic Gröbner basis. Thus in particular all powers of HP have linear resolutions. Second, the minimal free graded resolution of HP will be construc...

A certain squarefree monomial ideal HP arising from a finite partially ordered set P will be studied from viewpoints of both commutative algebra and combinatorics. First, it is proved that the defining ideal of the Rees algebra of HP possesses a quadratic Gröbner basis. Thus in particular all powers of HP have linear resolutions. Second, the minimal free graded resolution of HP will be construc...

We study the intersections of general Schubert varieties Xw with permuted big cells, and give an inductive degeneration of each such “Schubert patch” to a StanleyReisner scheme. Similar results had been known for Schubert patches in various types of Grassmannians. Wemaintain reducedness using the results of [Knutson 2007] on automatically reduced degenerations, or throughmore standard cohomolog...