Is projective morphism over a discrete valuation ring always flat?

Question

Let $A$ be a discrete valuation ring with a uniformizing parameter $t$. Let $f:X\to \DeclareMathOperator{\Spec}{Spec}\Spec A$ be a projective morphism (i.e. $X$ is a closed subscheme of $\mathbb{P}^n_A$ for some $n\geq 0$). Do we have that $f$ is flat?

Answer 1

$\mathrm{Spec}\ \mathbb F_p\hookrightarrow \mathrm{Spec}\ \mathbb Z_p$ is not flat. Indeed, $p\colon\mathbb Z_p\to\mathbb Z_p$ is injective but the tensor with $\mathbb F_p$ is not.