# Expressing fiber product of affines via an ideal

Let $X$ (resp. $Y$) be the affine $k$-scheme defined by the ideal $I$ (resp. $J$) in the polynomial ring $k[x_1,...x_n]$ (resp. $k[y_1,...,y_m]$). Let $Z$ be the affine scheme defined by the ideal $L$ in $k[z_1,...z_s]$, and let $f^\*:k[z]/L\rightarrow k[x]/I$ (resp. $g^\*:k[z]/L\rightarrow k[y]/J$) be $k$-homomorphisms, where $x=(x_1,...,x_n)$ and so forth, corresponding to scheme morphisms $f:X\rightarrow Z$ (resp. $Y\rightarrow Z$).

Then it should be possible to express the fiber product $X\times_{f,Z,g}Y$ via an ideal $W$ in the polinomial ring $k[x,y,z]$ [edit: actually, $W$ should be an ideal in $k[x,y]$] (where $x$ stands for the string of variables $x_1,...,x_n$, and so on).

Question: how to express $W\subseteq k[x,y,z]$ explicitely in terms of $I$, $J$, $L$, $f^\*$ and $g^\*$?

Edit: You can express things explicitely in terms of some polynomials $F_i$, $G_i$ and $H_i$ such that $I=(F_1,...,F_N)$, $J=(G_1,...,G_M)$ and $L=(H_1,...,H_S)$, and in terms of the components $(f_1,...,f_s)$ (resp. $(g_1,...,g_s)$) of $f$ (resp. $g$).

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I'm assuming that you're familiar with the fact that the fiber product of the affine schemes Spec(A) and Spec(B) over Spec(C) is given by $Spec(A \underset{C}{\otimes} B)$? So your question is about a nice description of this tensor product for various finitely generated k-algebras? Also, I'm not sure why you would expect your fiber product to be a closed subscheme of k[x,y,z]. I don't see why it's anything more than an algebra over k[z]/L. – Mike Skirvin Mar 21 '10 at 17:45
@Mike: Yes, you can assume familiarity with the tensor product thing. The answer is "yes" also to your second question, in some sense. As far as I understand, the euristic interpretation of fiber product is: "construct a space $P$ over $Z$ such that its fiber $P_z$ over the point $z\inZ$ is just the absolute (over $Spec(k)=$ the residue field at $z$) product of the fibers: $X_z\timesY_z$". Does it sound correct? – Qfwfq Mar 21 '10 at 20:17
Ah, can assume $k$ is a field if you want: just to simplify matters, but perhaps it's not relevant. – Qfwfq Mar 21 '10 at 20:18
@unknown: not sure why you seem to think a space isn't needed between a LaTeX control sequence and a following letter! – Reid Barton Mar 22 '10 at 6:13

Dear unknown, let me first congratulate you on the clearness of your question and the quality of your notation, which I'm now going to use.

The fibre product $X\times_Z Y$ is the subscheme of $\mathbb A_k^n \times \mathbb A_k^m$ described by an ideal $\mathfrak A \subset k[x,y]$. That ideal is $\mathfrak A=I^e + J^e + D$, where

$I^e$ is the extension of $I\subset k[x]$ to $I^e\subset k[x,y]$,

$J^e$ is the extension of $J\subset k[y]$ to $J^e\subset k[x,y]$,

$D$ is the ideal generated by the $s$ differences $f_i(x)-g_i(y),\quad (i=1,\ldots,s)$

I find it clearer not to use generators for $I$ and $J$ and, strangely, $L$ is not used at all: this is because the fibre product is the same whether considered over $Z$ or over $\mathbb A_k^s$ !

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Interesting answer: I will read more carefully later. Also, I suspect I made a mistake: the fiber product, as a scheme over $k$, should really sit inside the product (X\timesY), NOT inside $X\timesY\timesZ$ as it's written in my question. Should I edit? – Qfwfq Mar 21 '10 at 20:30
Dear unknown, LaTeX doesn't render your comment very well for me: I get a message "Unknown control sequence" after your two words "NOT inside". Anyway, you may edit if you like, of course, but I think your question is fairly clear. Anyone interested in technical details about the exact scheme over which to take the fibre product might look them up in: EGA I,chapter 1, 3.3 Propriétés formelles du produit; changement de préschéma de base . – Georges Elencwajg Mar 21 '10 at 21:57
Ok, I just wrote "NOT inside XxYxZ". Because, as you pointed out, it's a subscheme of XxY. – Qfwfq Mar 22 '10 at 6:00