Spec of a finitely generated $k$-algebra

Question

Let $R$ be a finitely generated $k$-algebra, with $k$ an algebraically closed field.

I've read that there is a bijective correspondence between points in $Spec(R)$ and $k$-algebra homomorphisms between $R$ and $k$. How is this realized?

Answer 1

More like between points in $\text{Max}(R) \subset {Spec}(R)$ and $k$-algebra homomorphisms from $R$ to $k$.

For every homomorphism $\phi \colon R \to k$ its kernel is a maximal ideal, that is, a point in $\text{Max}(R)$.

Conversely, given a maximal ideal $m\subset R$ the canonical morphism of $k$-algebras $R \to R/m$ gives in fact a morphism from $R$ to $k$. Indeed, the composition $$k \to R \to R/m$$ is an isomorphism, because $k$ is algebraically closed and $R$ is finitely generated ( Hilbert's Nullstellensatz).