Expressing fiber product of affines via an ideal - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T13:52:19Z http://mathoverflow.net/feeds/question/18939 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18939/expressing-fiber-product-of-affines-via-an-ideal Expressing fiber product of affines via an ideal Qfwfq 2010-03-21T17:15:52Z 2010-03-22T05:58:51Z <p>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. $Y\rightarrow Z$).</p> <p>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).</p> <p>Question: how to express $W\subseteq k[x,y,z]$ explicitely in terms of $I$, $J$, $L$, $f^*$ and $g^*$?</p> <p>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. $g$).</p> http://mathoverflow.net/questions/18939/expressing-fiber-product-of-affines-via-an-ideal/18947#18947 Answer by Georges Elencwajg for Expressing fiber product of affines via an ideal Georges Elencwajg 2010-03-21T19:11:04Z 2010-03-21T19:11:04Z <p>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.</p> <p>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 </p> <p>$I^e$ is the extension of $I\subset k[x]$ to $I^e\subset k[x,y]$,</p> <p>$J^e$ is the extension of $J\subset k[y]$ to $J^e\subset k[x,y]$,</p> <p>$D$ is the ideal generated by the $s$ differences $f_i(x)-g_i(y),\quad (i=1,\ldots,s)$ </p> <p>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 $\mathbb A_k^s$ !</p>