The notion of fibre in algebraic geometry

Question

Let me recall from here the following. Let $A,B,$ and $R$ be $k$-algebras. There is a functor from the category of $k$-algebras to the category of sets, called $\operatorname{Spec}A$, defined by $R\mapsto\operatorname{Hom}_{k\text{-alg}}(A;R)$. Elements of $\operatorname{Spec}A(R)$, i.e., $k$-algebra homomorphisms $A\rightarrow R$, are called $R$-points of $\operatorname{Spec}A$.

Now I'm trying to understand the notion of fibre. Below is an extract from lecture notes that were handed out.

Given $\operatorname{Spec}A$, $\operatorname{Spec}B$, and an $R$-point $p:\operatorname{Spec}R\rightarrow \operatorname{Spec}A$, the fibre is defined as $f^{-1}(p)=\operatorname{Spec}(R\otimes_A B)$.

There are a few questions, which are listed below. Please to use as simple mathematical language as you can, i.e., try not to invoke complicated mathematical concepts, if possible.

1. What is '$f$'?
2. The fibre of what? I mean I can understand what the fibre of a point under a map is, but I cannot understand what this plain term "fibre" means here.
3. As noted at the very beginning, an $R$-point is a homomorphism of $k$-algebras $A\rightarrow R$ (from the same notes). However, here $p$ is a map $\operatorname{Spec}R\rightarrow \operatorname{Spec}A$. Is it the same? Moreover, is it possible to explain briefly what is this map $p$ from the quote? (I am not an expert in category theory, nor am I supposed to be one -- only definitions of a category and a functor were touched upon in the course.) If not, just say so.
4. Is the above terminology common? I've been trying to find a book/other lecture notes where similar terminology is used, i.e., where $\operatorname{Spec}A$ is considered as a functor, where the same definition of an $R$-point is given, but I didn't succeed. Any ideas about where this terminology is used?

Answer 1

You seem to have forgotten that you're talking about a map of schemes $f : \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$.

$\operatorname{Spec}$ maps colimits of rings to limits of schemes; in particular, it sends the pushout diagram

$$ \begin{matrix} A &\to& R \\ \downarrow & & \downarrow \\ B &\to& R \otimes_A B \end{matrix}$$

to the pullback diagram

$$ \begin{matrix} \mathop{\mathrm{Spec}}(R \otimes_A B) &\to& \mathop{\mathrm{Spec}}(B) \\ \downarrow & & \ \ \downarrow f \\ \mathop{\mathrm{Spec}}(R) &\xrightarrow{p}& \mathop{\mathrm{Spec}}(A) \end{matrix}$$

That is, the map $\mathop{\mathrm{Spec}}(R \otimes_A B) \to \mathop{\mathrm{Spec}}(B)$ is the pullback by $f$ of the map $\mathop{\mathrm{Spec}}(R) \to \mathop{\mathrm{Spec}}(A)$.

In Top, the fiber of a bundle $\pi : E \to B$ is given the same way; letting $*$ denote the one-point space, a point $B$ is the same thing as a map $p:* \to B$, and the fiber $E_p$ over $p$ is the pullback

$$\begin{matrix} E_p &\to& E \\ \downarrow & & \ \ \ \downarrow \pi \\ * &\to& B \end{matrix} $$

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3. As noted at the very beginning, an $R$-point is a homomorphism of $k$-algebras $A\rightarrow R$ (from the same notes). However, here $p$ is a map $\operatorname{Spec}R\rightarrow \operatorname{Spec}A$. Is it the same?

Yes; ring homomorphisms and morphisms between affine schemes are the same thing. $\operatorname{Spec}$ and $\mathcal{O}$ is actually an equivalence between categories $\mathbf{CRing}^\text{op} \equiv \mathbf{AffSch}$.

In particular, $\operatorname{Spec}$ is a functor, so if you have a map $p : A \to R$, applying $\operatorname{Spec}$ gives a map $\operatorname{Spec}(p) : \operatorname{Spec}(R) \to \operatorname{Spec}(A)$. We often notate $\operatorname{Spec}(p)$ as just $p$, or maybe $p^*$

$k$-algebra homomorphisms correspond to morphisms of schemes over $\operatorname{Spec}(k)$.

Moreover, is it possible to explain briefly what is this map $p$ from the quote? (I am not an expert in category theory, nor am I supposed to be one -- only definitions of a category and a functor were touched upon in the course.) If not, just say so.

Anything. Our inspiration is drawn from a wealth of examples where the "elements" of an object are given by homomorphisms; e.g.

• In Set, there is a natural bijection $S \cong S^1$
• In Top, there is a natural bijection $|X| \cong C(*, X)$ (where $|X|$ is the set of points)
• In CRing, there is a natural bijection $|R| \cong \hom(\mathbf{Z}[x], R)$ (where $|R|$ is the set of elements)

and so forth. Generalizing these, category theory teaches us that it is useful to consider arbitrary homorphisms with codomain $A$ as generalized elements of $A$. This is especially fruitful in algebraic geometry. ("generalized element" is a standard terminology)

All of the points of the locally ringed space approach to schemes are of this form; e.g. every prime ideal $\mathfrak{p}$ of a ring $A$ corresponds to a map $\operatorname{Spec}(A_\mathfrak{p}/\mathfrak{p}_\mathfrak{p}) \to \operatorname{Spec}(A)$. (which is the map you get by applying $\operatorname{Spec}$ to the canonical map $A \to A_\mathfrak{p} / \mathfrak{p}_\mathfrak{p}$)

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4. Is the above terminology common? I've been trying to find a book/other lecture notes where similar terminology is used, i.e., where $\operatorname{Spec}A$ is considered as a functor, where the same definition of an $R$-point is given, but I didn't succeed. Any ideas about where this terminology is used?

Yes, the terminology is common. For considering schemes as functors, they keyphrase is "functor of points"