Behaviour of length function under faithfully flat extension

Question

Let $(R,m)$ and $(S,n)$ be local Noetherian rings such that $S$ is a faithfully flat extension of $R$. Let $J\subsetneq I $ ideals of $R$.

Can we relate $l_R(I/J)$ and $l_S(IS/JS)$?

PS: Here $l(-)$ denotes the length.

Answer 1

Remark. Recall that any finitely generated module $M$ has a filtration $$0 = M_0 \subsetneq M_1 \subsetneq \ldots \subsetneq M_n = M$$ whose successive subquotients $M_i/M_{i-1}$ are isomorphic to $R/\mathfrak p_i$ for some prime ideal $\mathfrak p_i \subseteq R$. If all the $\mathfrak p_i$ are maximal, then the number of terms in such a sequence is independent of the sequence, and we say that $M$ has finite length. We set $\ell_R(M) = n$ in this case.

Exercise. Find an example (where the $\mathfrak p_i$ are not maximal) where there exist two filtrations of the above type with different lengths.

Thus, over a local ring $(R,\mathfrak m)$, the only prime that is allowed to occur is $\mathfrak m$.

Remark. If $R \to S$ is faithfully flat, then it is injective. Indeed, the map $S = R \otimes_R S \to S \otimes_R S$ is split injective, with a splitting given by the multiplication. By faithful flatness, this implies the original map $R \to S$ was injective.

Thus, if $I \subseteq R$ is an ideal, the injection $R \to S$ induces an injection $I \otimes_R S \to S$, which allows us to identify $I \otimes_R S$ with its image $IS$ in $S$. We deduce that $IS/JS = I/J \otimes_R S$.

Answer to your question. Thus, if $I/J$ has finite length, then it has a filtration whose subquotients are $R/\mathfrak m$. Tensoring gives a filtration with subquotients $R/\mathfrak m \otimes_R S = S/\mathfrak mS$. Since length is additive, we deduce that $$\ell_S(IS/JS) = \ell_R(I/J) \cdot \ell_S(S/\mathfrak mS),$$ in the sense that one side is finite if and only if the other is, and then they are equal.

(Note that $S/\mathfrak mS$ need not have finite length, e.g. $S/\mathfrak n$ could be an infinite field extension of $R/\mathfrak m$.)

Reference. For a very quick introduction to lengths, see Appendix A of Fulton's Intersection Theory. It is a shockingly concise treatise of everything you need to know about lengths to do intersection theory. My argument above is very similar to Lemma A.1.3.