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 $\ell\_S(M\otimes\_R S)=\ell\_S(S/f(\mathfrak{m})S)\cdot\ell\_R(M)$.



Reference for submultiplicativity of length of tensor product
Look at lemmas 45.12 and 45.13 of the following: <http://www.math.columbia.edu/algebraic_geometry/stacks-git/algebra.pdf>


This comes from the Stacks open source textbook project, you can browse their chapters here:
<http://www.math.columbia.edu/algebraic_geometry/stacks-git/browse.html>


