The concept of smoothness is a major cornerstone in both mathematics and physics. In this talk, we will explore its definition in the context of algebraic geometry—that is, the study of shapes defined by polynomial equations. Our goal is to associate a number to a homomorphism of algebras, called the relative global dimension, in such a way that its finiteness implies the smoothness of the homomorphism. To achieve this, we will introduce some concepts from algebraic topology and develop a relative version of homological algebra, as introduced by Hochschild. This is based on a joint work with Iusenko–Marcos (IME–USP).