Dimension of Algebraic Variety

This lecture covers the concept of dimension of an algebraic variety, which is defined as the largest integer n such that there exists a chain of distinct irreducible closed subsets. The instructor explains how the dimension of an algebraic set is related to the dimension theory for rings, specifically focusing on the (Krull)-dimension of a ring. Various propositions are presented, including the relationship between the dimension of an affine algebraic variety and its coordinate ring. The lecture also discusses computing dimensions using tools from commutative algebra, such as transcendence degree and prime ideals. Additionally, the precise dimension of varieties in different scenarios, like hypersurfaces, is explored.