On the cardinality of some quotient rings

Question

Let $R$ be a $2$-dimensional noetherian local domain with finite residue field and maximal ideal $\mathfrak m$. Is it true that for every $k$ we have $|R/\mathfrak m^k|<\infty$ ?

Clearly this is true for $k=1$ and I tried using induction or arguments involving exact sequences but I failed.

Answer 1

Consider first the exact sequence: $$0\to\mathfrak m^{k-1}/\mathfrak m^{k}\to\mathfrak m/\mathfrak m^{k}\to\mathfrak m/\mathfrak m^{k-1}\to 0 $$ to show by induction on $k$ that all $\mathfrak m/\mathfrak m^{k}$ are finite.

Next the exact sequence $$0\to\mathfrak m^{k-1}/\mathfrak m^{k}\to A/\mathfrak m^{k}\to A/\mathfrak m^{k-1}\to 0 $$ and induction again let you conclude.

Answer 2

If $A$ is a finitely generated $R$-module, then $A/\mathfrak{m}A$ is a finite-dimensional vector space over $k=R/\mathfrak{m}$. We are assuming $k$ is finite, so $A/\mathfrak{m}A$ is finite, that is the index $|A:\mathfrak{m}A|$ is finite.

As $R$ is Noetherian, then $\mathfrak{m}^k$ is finitely generated for all $k$. Therefore $|\mathfrak{m}^k:\mathfrak{m}^{k+1}|$ is finite etc.