Applications of Local Cohomology

Local cohomology was discovered in the 1960s as a tool to study sheaves and their cohomology in algebraic geometry, but have since seen wide use in commutative algebra. An example of their use is to answer the question: how many elements are necessary to generate a given ideal, up to radical? For example, consider two planes in 4-space meeting at a point. The vanishing ideal I = (x, y) ∩ (u, v) ⊆ k[x, y, u, v] can be generated up to radical by xu, yv, xv+yu. Krull’s Hauptidealsatz implies that one element is not enough, but local cohomology is used to show two elements also do not work. The main sources for this talk are [Hun07] and [Eis05, App. 1]. For a more “homological” introduction, see [Wei94, §4.6]. The speaker would like to point out that the algebro-geometric literature on the topic takes a different approach via sheaf cohomology; see [Har67; Har77].