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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 $\chi \otimes_{\mathrm{Spec}(\mathbb{Z})} \mathrm{Spec}(\mathbb{F}_p)$ ?

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    $\begingroup$ Yes${}{}{}{}{}$. $\endgroup$ Dec 21, 2017 at 10:30
  • $\begingroup$ What means $\chi \otimes_{\mathrm{Spec}(\mathbb{Z})} \mathrm{Spec}(\mathbb{F}_p)$ ? (when say $\chi = Spec(\mathbb{Z}[x,y]/(y^2-x^3-x)$) @LordSharktheUnknown $\endgroup$
    – reuns
    Dec 21, 2017 at 10:40
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    $\begingroup$ @reuns It can be seen in the category of commutative rings, in where fibered product of affine schemes is translated into tensor product of algebras. $\endgroup$
    – Phil. Z
    Dec 21, 2017 at 11:57
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    $\begingroup$ @Phil.Z So you are saying I should look at $\mathbb{Z}[x,y]/(y^2-x^3-x) \otimes_{\mathbb{Z}} \mathbb{Z}/p\mathbb{Z}$ ? Then $p u\otimes_{\mathbb{Z}} a=u\otimes_{\mathbb{Z}} a p =u\otimes_{\mathbb{Z}} 0 = 0 \otimes_{\mathbb{Z}}0$ so that $\mathbb{Z}[x,y]/(y^2-x^3-x) \otimes_{\mathbb{Z}} \mathbb{Z}/p\mathbb{Z} \cong \mathbb{Z}/p\mathbb{Z}[x,y]/(y^2-x^3-x)$, and something similar should happen for the local rings $\endgroup$
    – reuns
    Dec 21, 2017 at 12:31
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    $\begingroup$ @Boccherini : you might accept the answer, if it is fine to you. Otherwise, you could tell me how I can improve it. Many thanks! $\endgroup$
    – Watson
    Jan 8, 2018 at 12:35

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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 $p\Bbb Z \in \mathrm{Spec}(\Bbb Z)$.


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 $p<5$).

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  • $\begingroup$ When I say "it is an elliptic curve", I mean that the projectivization $y^2z - x^3 + z^3 = 0$ is an elliptic curve. $\endgroup$
    – Watson
    Jan 25, 2019 at 15:22

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