# Complete intersection argument

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. 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?

• The answer is obviously "no" in general. For example, consider the case when $Y$ is one point in $Z$. Then $T\times _Z Y$ is the preimage of this point in $T$ which is not necessarily a complete intersection in $T$ in general. However, I think that the statemet is true if one of the maps is flat (and the map ${\mathfrak t}\to {\mathfrak t}/W$ is flat. – Alexander Braverman Oct 4 '18 at 21:23
• Yes of course, thanks for the simple counterexample. I have edited the question to include a flatness assumption. – WSL Oct 4 '18 at 21:33
• If $Y=Z$ then $T\times_ZY=T$ so you need $T$ to be a CI in T\times_kY$. That is, you need a point to be CI in$Y$, so it is necessary that$Y\$ be smooth. – inkspot Oct 4 '18 at 21:53

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 $$T\times Y$$.