The following two on commutative algebra are true?
Let $S$ be a f.g. algebra over a field $k$. Let $e$ be an integer. Then (1) There is an ideal $I\subset S$ such that if $Q$ is a maximal ideal of $S$ then
$dim S_Q\ge e$ iff $Q\supset I.$
(2) if $S=S_0\oplus S_1\oplus \cdots$ is a graded algebra, f.g. over $S_0=k$ then
$dim S\ge e.$
EDIT. (1) is done by Matt. I rewite (2).
Is it trivially true when $R$ is a field? If not, how should we modify it?
Theorem 14.8b(Eisenbud CA p316)
Let $S$ be a f.g. algebra over a Noetheian ring $R$. Let $e$ be an integer. Then
(2) if $S=S_0\oplus S_1\oplus \cdots$ is a graded algebra, f.g. over $S_0=R$ then there exists an ideal $J$ of $R$ such that for any prime ideal $P\subset R$,
$dim K(R/P)\otimes S\ge e$ iff $P\supset J.$
Here $K(\cdot)$ means a quotient field.