Tangent Spaces and Obstruction Theories

These are notes from my series of 8 lectures on tangent spaces and obstruction theories, which were part of a MSRI summer workshop on Deformation Theory and Moduli in Algebraic Geometry (July 23 – August 3, 2007). Nothing in these notes is original. For a list of references where this material (and much more!) can be found see the end of the notes. These are the (slightly cleaned up) notes prepared in advance of the lectures. The actual content of the lectures may differ slightly (and in particular several of the examples included here were left out of the lectures due to time constraints). Lecture 1. The ring of dual numbers. 1.1. Motivation. Let X be a scheme over a field k, and let x ∈ X(k) be a point. The tangent space T X (x) of X at x is the dual of the k-vector space m/m 2 , where m ⊂ O X,x is the maximal ideal. If X represents some functor F : (schemes) op → Set, then the point x ∈ X(k) corresponds to an element of F (Spec(k)), and elements of the tangent space should correspond to infinitesimal deformations of x. At first approximation, the purpose of my lecture series is to understand from a functorial point of view this tangent space as well as the obstruction spaces that arise if X is singular at x.