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I am having some trouble following the argument in EGA (II, 1.3.1). Let $S$ be a prescheme and $\mathcal{B}$ a quasi-coherent $\mathscr{O}_S$-algebra. Define for an affine open subset $U \subset S$ the $S$-prescheme $X_U = \operatorname{Spec}(\Gamma(U, \mathcal{B}))$. Then he says it follows from (I, 1.3.7), (I, 1.3.13) and (I, 1.6.3) that $\mathcal{A}(X_U) = \mathcal{B}|_U$, where $\mathcal{A}(X_U)$ is the direct image by $X_U \to U$, viewed as an $\mathscr{O}_S|_U$-module.

Here is what I can see: Let $\Psi = (^a \phi, \tilde{\phi}) : (X_U, \mathscr{O}_{X_U}) \to (U, \mathscr{O}_S|_U)$ be the morphism induced by the homomorphism $\phi : A' = \Gamma(U, \mathscr{O}_S) \to A = \Gamma(U, \mathcal{B})$. Now we have

$\mathcal{A}(X_U) = \Psi_\ast(\mathscr{O}_{\operatorname{Spec}(A)}) = \Psi_\ast(\tilde{A}).$

By (I, 1.6.3), we have a canonical isomorphism $\widetilde{A_{[\phi]}} = \Psi_{\ast}(\tilde{A})$, where $A_{[\phi]}$ is $A$ viewed as an $A'$-module via $\phi$. (I, 1.3.13) gives $\widetilde{A_{[\phi]}}$ an $\mathscr{O}_S|_U$-module structure. We can look at the sections over some $D(f) \subset U$ ($f \in A'$):

$\Gamma(D(f), \mathcal{A}(X_U)) = \Gamma(D(f), \widetilde{A_{[\phi]}}) = (A_{[\phi]})_f = A_{\phi(f)}$

by (I, 1.3.7). On the other hand, since $\mathcal{B}$ is quasi-coherent, we have $B|_U = \tilde{M}$ for some $A'$-module $M$. Therefore

$\Gamma(D(f), \mathcal{B}) = \Gamma(D(f), \tilde{M}) = M_f$

again by (I, 1.3.7).

Now I don't see how to identify $A_{\phi(f)}$ and $M_f$. Am I missing something about quasi-coherent $\mathscr{O}_S$-algebras?

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    Matt: Yes, sorry!2012-08-02

0 Answers 0