The Stacks project

Definition 82.32.1. Let $k$ be a field. Let $p : X \to \mathop{\mathrm{Spec}}(k)$ be a proper morphism of algebraic spaces. The degree of a zero cycle on $X$ is given by proper pushforward

(Lemma 82.16.3) composed with the natural isomorphism $\mathop{\mathrm{CH}}\nolimits _0(\mathop{\mathrm{Spec}}(k)) \to \mathbf{Z}$ which maps $[\mathop{\mathrm{Spec}}(k)]$ to $1$. Notation: $\deg (\alpha )$.