Hilbert - Samuel multiplicity of $B$ when there is a surjection $A \rightarrow B$

Question

The setting is: Let $A, B$ be commutative, Noetherian, local rings, $\phi:A \rightarrow B$ a surjective homomorphism. Both rings also come with surjections $\lambda_A, \lambda_B$ to a DVR $\mathcal{O}$ which factor as $\lambda_A = \lambda_B \circ \phi$ (if this is helpful at all). Let $e(A)$ denote the Hilbert - Samuel multiplicity of $A$ w.r.t. to its maximal ideal. Considering $B$ as an $A$-module, I'm reading that $e_A(B) \leq e_A$ always holds, but I'm struggling to see why this is true. I went through the related chapters in Matsumura and Bruns - Herzog, but wasn't able to find an explanation, can anyone shed light on this?

Answer 1

There is a short exact sequence $$0 \to \ker \phi \to A \xrightarrow{\phi} B \to 0.$$ Hilbert-Samuel multiplicity is additive across short exact sequences (see Corollary 4.7.7 in Bruns and Herzog), so $e(A)=e_A(B)+e_A(\ker \phi)$, which proves the claim.