Definition 81.32.1. 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

$p_* : \mathop{\mathrm{CH}}\nolimits _0(X) \longrightarrow \mathop{\mathrm{CH}}\nolimits _0(\mathop{\mathrm{Spec}}(k)) \longrightarrow \mathbf{Z}$

(Lemma 81.16.3) 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: $\deg (\alpha )$.

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