---- > [!definition] Definition. ([[scheme-theoretic fiber]]) > In $\mathsf{Set}$, the fiber of $y \in Y$ under a function $f:X \to Y$ is characterized per the following [[categorical pullback|fiber product]]: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRADMA9YAWgEYAvgAoAngEoAvAB0pwETIEACGXgC28APoBNRQA0QA0uky58hFH3JVajFm32Hj2PASJk+1+s1aIQMuQoGRiAYzmZElh7UXna+WgbWMFAA5vBEoOwAThCqSJYgOBBIAMzRtj4cQRnZuYgATNSFJY4cNUhkBUX1LVk57Y1dghQCQA > \begin{tikzcd} > f^{-1}(y)=\{y\} \times_Y X \arrow[d] \arrow[r] & X \arrow[d, "f"] \\ > \{y\} \arrow[r] & Y > \end{tikzcd} > \end{document} > ``` > We can use this philosophy to *define* fibers for [[scheme]] [[morphism of locally ringed spaces|morphisms]]. > > So let $f:X \to Y$ be a [[morphism of locally ringed spaces|morphism of schemes]]. Given $y \in Y$, consider the [[residue field]] $k(y)=\frac{\mathcal{O}_{Y,y}}{\mathfrak{m}_{y}}$; [[scheme|recall]] that $k(y)$ encodes a canonical [[morphism of locally ringed spaces|morphism]] $\text{Spec }k(y) \to Y$ with image $y$. The **(scheme-theoretic) fiber at $y$** is $X_{y}:=\text{Spec }k(y) \times_{Y} X.$ > > This is the same idea as in $\mathsf{Set}$, except that defining a 'point' takes more work in $\mathsf{Sch}$. > Let $f:X=\text{Spec }k[x] \to Y=\text{Spec }k[t]$ be induced by$\begin{align} k[t] &\to k[x] \\ t & \mapsto x^{2} \end{align}.$ For $y=\langle t-a \rangle \subset k[t]$ and $a\in k$, the residue field is $k(y)=$ ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```