Commutative Algebra and Algebraic Geometry: The Fellowship of the Ring: Codepth, complete intersections, and quasi-cyclic modules

Seminar | February 26 | 3:45-4:45 p.m. | 939 Evans Hall

 Robin Hartshorne, UC Berkeley

 Department of Mathematics

Codepth is the dual notion to depth, being the greatest length of a coregular sequence for a module, meaning the first element maps the module surjectively, the second is subjective on the kernel of the first, and so on. For a curve in P3, let M be the local cohomology module of the graded coordinate ring with supports in the ideal of the curve. Then the theorem of Hellus says that C is a set theoretic complete intersection if and only if M has codepth 2. This criterion is not directly applicable, so we define the notion of a quasi-cyclic module, which is an increasing limit of cyclic modules. In this talk I will recall the still open problem of whether every irreducible nonsingular curve in P3 is a set theoretic complete intersection, and derive a number of consequences using the concepts introduced above.