# Reference for submultiplicativity of length of tensor product

I am looking for a reference, in the form of a textbook, that contains proofs of following statements.

NOTE: I am NOT looking for the proofs, I am looking for a reference! Proofs of these statements are elementary. In fact, I need a reference to avoid writing the proofs. I just want to be able to refer somewhere for the proofs. I am sure these must be somewhere. Surprisingly, I cannot find them in standard textbooks that I know!!

1. Let $f:R\rightarrow S$ be a local homomorphism of Noetherian local rings with residue fields $k_R$ and $k_S$, respectively, and assume that $[^fk_S:k_R]<\infty$. If $N$ is an $S$-module of finite length, then $^fN$ is an $R$-module of finite length, and $\ell_R(^fN)=[^fk_S:k_R]\cdot\ell_S(N)$.

2. Let $f:(R,\mathfrak{m})\rightarrow(S,\mathfrak{n})$ be a homomorphism of Noetherian local rings such that $\ell_S(S/f(\mathfrak{m})S)<\infty$. Let $M$ be an $R$-module of finite length. Then

$\ \ \ \$ a) $M\otimes_RS$ is an $S$-module of finite length and $\ell_S(M\otimes_R S)\leq\ell_S(S/f(\mathfrak{m})S)\cdot\ell_R(M)$.

$\ \ \ \$ b) If in addition $f$ is flat, then $\ell_S(M\otimes_R S)=\ell_S(S/f(\mathfrak{m})S)\cdot\ell_R(M)$.

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