I'm reading the paper "How to use finite fields for problems concerning infinite fields" of Jean-Pierre Serre.
In pp. 2, Serre uses the fact that, if $\Lambda\subset\mathbb C$ is a ring finitely generated over $\mathbb Z$ and $\mathcal M$ is a maximal ideal of $\Lambda$, then $\Lambda/\mathcal M$ is a finite field.
Serre refers to p.68 of "Bourbaki, N., Algebre Commutative. Chapitre V. Entries, Hermann, Paris, 1964"
But in this book p. 68 I do not see the proof of this fact.
Can someone tell me where I can find a proof of this fact?
It suffices to prove that a field $R$, which is of finite type over $\mathbb{Z}$, is already a finite field.
Let $P$ be the prime field of $R$. Then $R$ is of finite type over $P$, hence finite since both are fields; this is a well-known application of Noether normalization. Now applying Proposition 7.8 in Atiyah-Macdonald to $\mathbb{Z} \to P \to R$ shows that $\mathbb{Z} \to P$ is of finite type. Hence $P \not\cong \mathbb{Q}$ and therefore $P$ is a finite prime field. Hence also $R$ is finite.
Actually more is true: A commutative ring is finite iff it is artinian and of finite type over $\mathbb{Z}$. See here.
If you want to know a reference for the above proof, you should find one in the context of the Weil conjectures since there it is crucial that a finite type scheme over $\mathbb{Z}$ has finite residue fields.