Zero-section as homomorphism of rings Let $s : X \to E$ be the zero section of a vector bundle $E$ over a scheme $X$. Zariski-locally this corresponds to a homomorphism $Sym_A(M) \to A$ of $A$-algebras where $M$ is a finitely generated projective $A$-module. What is this homomorphism? By adjunction it corresponds to a homomorphism $M \to A$ of $A$-modules. Is this just the zero morphism $M \to 0 \to A$? Stupid question I know, but I'm not an algebraic geometer, and I couldn't find the answer in any scheme theory book.

Yes, that's correct. This is just the algebra homomorphism $\mathrm{Sym}_A(M) \to \mathrm{Sym}_A(0)$ induced by the module homomorphism $M \to 0$. You can also write down the addition on $E/X$, this corresponds to the algebra homomorphism $\mathrm{Sym}_A(M) \to \mathrm{Sym}_A(M) \otimes_A \mathrm{Sym}_A(M) = \mathrm{Sym}_A(M \oplus M)$ which is induced by the evident module homomorphism $M \to M \oplus M$.