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. 



Behaviour of length function under faithfully flat extension
**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**.


