The study of the topological connectivity of algebraic sets is a fundamental subject in algebraic geometry. Local cohomology is a powerful tool in this field. In this chapter we shall use this tool to prove some results on connectivity which are of basic significance. Our main result will be the Connectedness Bound for Complete Local Rings, a refinement of Grothendieck's Connectedness Theorem. We shall apply this result to projective varieties in order to obtain a refined version of the Bertini-Gothendieck Connectivity Theorem. Another central result of this chapter will be the Intersection Inequality for Connectedness Dimensions of Affine Algebraic Cones. As an application it will furnish a refined version of the Connectedness Theorem for Projective Varieties due to W. Barth, to W. Fulton and J. Hansen, and to G. Faltings. The final goal of the chapter will be a ring-theoretic version of Zariski's Main Theorem.

The crucial appearances of local cohomology in this chapter are just in two proofs, but the resulting far-reaching consequences in algebraic geometry illustrate again the power of local cohomology as a tool in the subject. We shall use little more from local cohomology than the Mayer-Vietoris Sequence 3.2.3 and its graded version 13.1.4, the Lichtenbaum-Hartshorne Vanishing Theorem 8.2.1 and the graded version 13.1.16, and the vanishing result of 3.3.3. The use of these techniques in this context originally goes back to Hartshorne [21] and has been pushed further by J. Rung (see [5]).

The connectedness dimension

To begin, we have to introduce a measure for the connectivity of an algebraic set, or, more generally, of a Noetherian topological space.