Q:

Integral dependence and (faithfully) flat ring extensions

Let $R\to S$ be a flat ring extension. By theorem 9.5 of the book Commutative Ring Theory  written by Matsumura the going-down theorem holds between $R$ and $S$. 
Is it true (or not) about these theorems: "INCOMPARABILITY THEOREM", "LYING-OVER THEOREM" and "GOING-UP THEOREM"? How about faithfully flat ring extensions?

A:

The lying over theorem holds for faithfully flat ring extensions. In fact, a flat ring map $R\to S$ is faithfully flat, if and only if $\text{Spec}(S)\to\text{Spec}(R)$ is surjective. So, a flat but unfaithful ring extension will not satisfy lying over.
For the incomparability, this is false on both counts. Consider $k\to k[x]$.
The going up theorem doesn't hold for faithfully flat ring extensions. For example, consider $\mathbb{Z}\to\mathbb{Z}[T]$. This is faithfully flat, since $\mathbb{Z}[T]$ is a free $\mathbb{Z}$-module. Consider then, for a prime $p$, the ideal $(1+pT)\subseteq\mathbb{Z}[T]$ sitting above $(0)$ in $\mathbb{Z}$. We can extend $(0)$ to $(p)$, but we can't extend $(1+pT)$ by $Q$ lying above $p$, else $Q$ would contain $1+pT$, and $pT$ and so would be $\mathbb{Z}[T]$.

