The Stacks project

Definition 31.19.5. Let $i : Z \to X$ be an immersion of schemes. The normal cone $C_ ZX$ of $Z$ in $X$ is

see Constructions, Definitions 27.7.1 and 27.7.2. The normal bundle of $Z$ in $X$ is the vector bundle

see Constructions, Definitions 27.6.1 and 27.6.2.