Tangent Spaces in Algebraic Geometry

This lecture introduces the concept of tangent spaces in algebraic geometry, focusing on defining derivations and the Zariski tangent space. The instructor explains how derivations are determined by their values on a generating set of regular functions, leading to a finite-dimensional subspace. The lecture also covers examples, such as the tangent space of CN and the isomorphism between derivations and directional derivatives. Additionally, an important lemma from algebraic geometry is presented, establishing an isomorphism between the tangent space and the dual space of the maximal ideal quotiented by its square. The lecture concludes by highlighting the significance of Zariski tangent spaces and hints at upcoming topics on computing them.