Consider two morphisms $T\to Z$ and $Y\to Z$ of varieties over an algebraically closed field $k$ where $Z$ is an affine space. If $Y\to Z$ is flat, is it always true that the fiber product of $T\times\_Z Y$ is a complete intersection in $T\times\_k Y$?


The motivation comes from an argument of Knop in his paper [On the Set of Orbits for a Borel Subgroup](https://pdfs.semanticscholar.org/cb26/a628ee401c9d413a3b4c7b6d914097f15f44.pdf). Let $G$ be a reductive group with Lie algebra $\mathfrak{g}$, and $X$ be a spherical variety on which $G$ acts. In the proof of Lemma 6.5, where $Y=\mathfrak{t}$ is a Cartan subalgebra of $\mathfrak{g}$, $Z=\mathfrak{t}/W$ is the quotient by the Weyl group, and $T=T^\ast X$ is the cotangent bundle over $X$, the above statement is claimed for $$T^\ast X\times\_{\mathfrak{t}/W}\mathfrak{t}\subset T^\ast X\times\_k \mathfrak{t}.$$ EDIT: I have edited to include a flatness assumption to avoid simple counterexamples, as pointed out by @Alexander Braverman.


I would like to understand this point a bit better. Is this a standard argument, and if so is there a good reference?



Complete intersection argument
Let $Z=\mathbb{A}^n$. The map $T\times \_ZY\rightarrow T$ is the pull back over $T$ of $Y\rightarrow Z$, hence it is flat, of relative dimension $\dim(Y)-n$. Therefore $$\dim(T\times \_ZY)=\dim(T)+(\dim(Y)-n)=\dim(T\times Y)-n\, .$$ On the other hand, 
$T\times \_ZY$ is defined in $T\times Y$ by the $n$ equations $f\_i-g\_i=0$, where 
$(f\_1,\ldots ,f\_n)$ and $(g\_1,\ldots ,g\_n)$ are the two maps from $T$ and $Y$ to $Z=\mathbb{A}^n$. Thus $T\times \_ZY$ is a (global) complete intersection in $T\times Y$.


