"Reduction modulo $p$" of a scheme

Question

Suppose $\chi$ is a scheme of finite type over $\mathbb{Z}$. I sometimes see the notion of "reduction modulo $p$ of $\chi$" (which I will denote by $\chi_p$). What is meant here ? Is it just $\chi \otimes_{\mathrm{Spec}(\mathbb{Z})} \mathrm{Spec}(\mathbb{F}_p)$ ?

Answer 1

Yes, you are correct. As in the definition in II.3, p. 88, from Hartshorne's Algebraic Geometry, if $X$ is a scheme (over $\mathrm{Spec}(\Bbb Z)$) and $p \in \Bbb Z$ is a prime number, then the reduction mod $p$ of $X$ is $$X_p := X \times_{\mathrm{Spec}(\Bbb Z)} \mathrm{Spec}(\Bbb F_p)$$

This is just the fiber of the structural morphism $X \to \mathrm{Spec}(\Bbb Z)$ above $p\Bbb Z \in \mathrm{Spec}(\Bbb Z)$.

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As an example, if $X = \mathrm{Spec}(\Bbb Z[X_1, ..., X_n] / (f_1, ..., f_m))$ is of finite type over $\Bbb Z$, then $$X_p \cong \mathrm{Spec}\left( \Bbb F_p[X_1, ..., X_n] / (\overline{f_1}, ..., \overline{f_m}) \right),$$ as a scheme over $\Bbb F_p$, where $\overline{f_j}$ denotes the reduction mod $p$ of $f_j \in \Bbb Z[X_1, ..., X_n]$.

Said differently, the reduction of an affine variety (over $\Bbb Z$) defined by the equation, say, $y^2-x^3+1=0$ (over $\Bbb Q$, it is an elliptic curve) gives you the variety $y^2 - x^3 + [1]_p=0$ over $\Bbb F_p$ (which is not an elliptic curve if $p<5$).