What property can be checked on local schemes $Y\times_X\operatorname{Spec}(\mathcal O_{X,x})$?

Question

Let $f:Y\rightarrow X$ be a $X$-scheme. For $x\in X$, we define the local scheme as the base change $Y\times_X\operatorname{Spec}(\mathcal O_{X,x})$.

I have some questions about this local scheme:

1. What property can be checked on local schemes? (I mean property $P$ holds for every local scheme implies it holds for $f$)

2. Is there general relation between stalks of $Y\times_X\operatorname{Spec}(\mathcal O_{X,x})$ and stalks of $X,Y$?

3. What's the intuitive difference between local scheme and fiber?

Thank you in advance!

Answer 1

Question: "What property can be checked on local schemes? (I mean property P holds for every local scheme implies it holds for f) Is there general relation between stalks of $Y×_X Spec(O_{X,x})$ and stalks of $X,Y$? What's the intuitive difference between local scheme and fiber?"

Answer: Let $f: A \rightarrow B$ be a map of commutative unital rings with $\mathfrak{q} \subseteq A, \mathfrak{p}\subseteq B$ prime ideals where $\mathfrak{p}\cap A=\mathfrak{q}$. There are canonical maps

$$B \rightarrow B_{\mathfrak{q}} \rightarrow B_{\mathfrak{p}}$$

and

$$B \rightarrow C:=\kappa(\mathfrak{q})\otimes_A B \rightarrow \kappa(\mathfrak{q})\otimes_{A_{\mathfrak{q}}}B_{\mathfrak{p}}.$$

Question: "What's the intuitive difference between local scheme and fiber?"

The local scheme is the spectrum $Spec(C_{\tilde{\mathfrak{p}}})$ of the local ring of the fiber ring $\kappa(\mathfrak{q})\otimes_A B$ at $\tilde{\mathfrak{p}}$: There is a prime ideal $\tilde{\mathfrak{p}} \subseteq C$ corresponding to $\mathfrak{p}$ and an isomorphism

$$C_{\tilde{\mathfrak{p}}}\cong \kappa(\mathfrak{q})\otimes_{A_{\mathfrak{q}}}B_{\mathfrak{p}} \cong B_{\mathfrak{p}}/\mathfrak{m}_{\mathfrak{q}}B_{\mathfrak{p}}.$$

Hence the local ring of the fiber ring $C$ at $\tilde{\mathfrak{p}}$ is a quotient of the local ring $B_{\mathfrak{p}}$. They both have the same residue field at $\mathfrak{p}$.

Answer: Hence the local scheme $Spec(\kappa(\mathfrak{q})\otimes_{A_{\mathfrak{q}}}B_{\mathfrak{p}})$ is the spectrum of the local ring of the fiber ring $C$ at $\tilde{\mathfrak{p}}$. You get a sequence of maps

$$\kappa(\mathfrak{q}) \rightarrow B_{\mathfrak{p}}/\mathfrak{m}_{\mathfrak{q}}B_{\mathfrak{p}} \rightarrow \kappa(\mathfrak{p}).$$

Lemma: If $A \rightarrow B$ with $B:=A[t_1,..,t_n]/(f_1,..,f_l)$ it follows $\Omega^1_{B/A}\otimes_{B_{\mathfrak{p}}} \kappa(\mathfrak{p})=0$ iff

the canonical map $\kappa(\mathfrak{q}) \subseteq B_{\mathfrak{p}}/\mathfrak{m}_{\mathfrak{q}}B_{\mathfrak{p}}$ is a finite separable extension of fields. There is an isomorphism

$$ B_{\mathfrak{p}}/\mathfrak{m}_{\mathfrak{q}}B_{\mathfrak{p}} \cong \kappa(\mathfrak{p}).$$

Proof: Theorem 3.1, Demazure/Gabriel - "Groupes algebriques".

Question: "What property can be checked on local schemes?"

Answer: You use the local scheme to define the notions "etale morphism " and "smooth morphisms".