Tangent Spaces in Algebraic Geometry

This lecture introduces the concept of tangent spaces in algebraic geometry, focusing on derivations of the ring of regular functions at a point on a variety. The instructor explains how to define a tangent space using directional derivatives and the Leibniz rule, leading to a finite-dimensional subspace of derivations. The lecture covers examples in CN space and discusses an isomorphism between tangent spaces and dual spaces of maximal ideals. Additionally, an intrinsic definition of tangent spaces using the dual space of the maximal ideal quotiented by its square is presented, highlighting the local nature of tangent spaces in algebraic geometry.