Identifying morphisms of affine $k$-scheme with solutions of polynomials

Question

Could you please explain the last equality? Could you briefly recall this identification/isomorphism with the zeros of polynomials?

Answer 1

In general, let $R$ be a ring, $C$ an $R$-algebra, and $I$ an ideal of $R[t_1, ... t_n]$ generated by polynomials $f_1, ... , f_r$. The natural map

$$\operatorname{Hom}_{\textrm{$R$-alg}}(R[t_1,... , t_n], C) \rightarrow C^n \tag{1}$$

$$\phi \mapsto (\phi(t_1), ... , \phi(t_n))$$

is a bijection.

We can identify

$$\operatorname{Hom}_{\textrm{$R$-alg}}(R[t_1, ... , t_n]/I,C) $$

with the set of $\phi \in \operatorname{Hom}_{\textrm{$R$-alg}}(R[t_1,... , t_n],C)$ such that $\phi(f) = 0$ for all $f \in I$, or equivalently, such that $\phi(f_i) = 0$ for each $1 \leq i \leq r$.

Under the bijection (1), you can check that $\operatorname{Hom}_{\textrm{$R$-alg}}(R[t_1, ... , t_n]/I,C)$ identifies with the set

$$\{ c = (c_1, ... , c_n) \in C^n : f_i(c) = 0, 1 \leq i \leq r\}.$$