Noether normalization: what is a finite map?

Question

Let $D$ be a domain, and $R$ a finitely generated $D$ algebra. There exists a nonzero $f \in D$, and a finite injective ring map $D_f[X_1,\dots,X_n] \hookrightarrow R_f$. Here the $X_i$ are indeterminates.

What are $R_f, D_f$, and what is a finite map?

The above is quoted from: MSE answer.

Answer 1

The symbols $R_f$ and $D_f$ are the localizations of $R$ and $D$ at the multiplicative set $\{1,f,f^2,\cdots\}$. Since we're dealing with Domains, we are just inverting $f$ so $R_f=R[\frac{1}{f}]$ and $D_f=D[\frac{1}{f}]$.

A finite map means that $R_f$ is finitely generated as a $D_f[X_1,\cdots X_n]$-module.

Answer 2

If $A$ is a commutative ring and $f\in A$, then $A_f$ denotes the localization of $A$ with respect to $f$, the ring obtained by adjoining an inverse to $A$. You can construct it explicitly as $A[x]/(xf-1)$, where $x$ represents the inverse of $f$.

If $A$ and $B$ are commutative rings, a homomorphism $\varphi:A\to B$ is called finite if $B$ is a finitely generated $A$-module (where $A$ acts on $B$ via $\varphi$).