Blowing up an affine scheme at a regular point

Question

I am reading Liu's Algebraic Geometry and Arithmetic Curves and get stuck at Lemma 8.1.2:

Let $A$ be a Noetherian ring an define for an ideal $I \subset A$ the $A$-algebra $$\tilde{A}:=\bigoplus_{d\geqslant 0} I^d, \qquad I^0:=A.$$ Let $f_1, \ldots, f_r$ be a system of generators of $I$ and let $t_i \in I = \tilde{A}_1$ denote the element $f_i$ considered as a homogeneous element of degree 1. We have a surjective homomorphism of graded $A$-algebras $$\phi: A[T_1, \ldots, T_n] \longrightarrow \tilde{A}, \qquad T_i \mapsto t_i.$$ Let $\tilde{X}:= \operatorname{Proj} \tilde{A}$.

My questions are the following:

1. He says: If $P$ is a polynomial with coefficients in $A$, then $P(t_1, \ldots, t_n)=0$ if and only if $P(f_1, \ldots, f_n)=0$. Why is this worth a remark, if $f_i=t_i$?

2. If $I$ is generated by a regular element, then $\tilde{A} \cong A[T]$. In the proof he says the homomorphism $\phi: A[T] \longrightarrow \tilde{A}$ from above is an isomorphism. But why? Is see, that the elements in $A[T]$ and in $\tilde{A}$ look very similar, but I cannot see, why the fact "regular" gives the claim.

I hope anybody can help me!

Thanks for helping me!

Answer 1

1. The author says homogeneous polynomial. I think he wants to emphasize on the fact that $P(t_1,\dots,t_n)$ is a homogeneous element in $\bar A$.

2. Let $f\in A[T]$, $f=a_0+a_1T+\cdots+a_nT^n$, such that $\phi(f)=0$. Suppose $(a)=I$. Then $f(a)=0$, that is, $a_0+a_1a+\cdots+a_na^n$. In $\bar A$ this leads to $a_ia^i=0$ for all $i$, so $a_i=0$ for all $i$ and thus $f=0$.