Consider a composition of morphisms of schemes, $$Z \stackrel{j}{\longrightarrow} X \stackrel{f}{\longrightarrow} Y,$$ where $j: Z \rightarrow X$ is a closed immersion with sheaf of ideals $\mathscr{I}$. The conormal exact sequence for the sheaves of differentials should read, $$ j^{*}(\mathscr{I} / \mathscr{I}^{2}) \longrightarrow j^{*} \Omega_{X / Y} \longrightarrow \Omega_{Z/Y} \longrightarrow 0. $$ I am trying to prove this via the corresponding conormal sequence for modules. So suppose the composition of morphisms of schemes above are of affine schemes. Say $X = \text{spec}B$, $Y = \text{spec}A$, and $Z = \text{spec}C$, with $C \simeq B / I$ for some ideal $I$ of $B$. Then the conormal exact sequence of modules reads, $$ I / I^{2} \longrightarrow \Omega_{B/A} \otimes_{B} C \longrightarrow \Omega_{C/A} \longrightarrow 0, $$ where this is an exact sequence of $C$-modules. Now we just have to take the corresponding sequence of quasicoherent $\mathcal{O}_{Z}$-modules. We are viewing $I/I^{2}$ as a $C$-module via the isomorphism, $$ I \otimes_{B} C \simeq I \otimes_{B} B / I \simeq I / I^{2}. $$ But then $I/I^{2}$ is just the $B$-module $I$ with the change of ring $- \otimes_{B} C$ applied. In that case, the exact sequence of $\mathcal{O}_{Z}$-modules should just be $$ j^{*}\mathscr{I} \longrightarrow j^{*} \Omega_{X / Y} \longrightarrow \Omega_{Z/Y} \longrightarrow 0. $$ So where is my flaw in reasoning? Are $j^{*}\mathscr{I}$ and $j^{*}(\mathscr{I} / \mathscr{I}^{2})$ somehow canonically isomorphic or what?


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