Let $\mathscr F$ be a quasi-coherent sheaf on $X={\rm Spec}(A)$. How is $\mathscr F |_{D(g)} \cong \widetilde{M \otimes_B A_g}$?

Question

From Hartshorne's Algebraic Geometry page 112:

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Letting $f: B \to A_g$, I see that from Proposition 5.2 we can write $$f^*(\tilde M) \cong \widetilde{M \otimes_B A_g}$$ as sheaves of $\mathcal O_X |_{D(g)}={\rm Spec}(A_g)$-modules, where $$f^*(\tilde M) = \mathcal O_{{\rm Spec}(A_g)} \otimes_{f^{-1}\mathcal O_{{\rm Spec}(B)}} f^{-1}(\tilde M).$$

However, I don't see how $\mathscr F |_{D(g)} \cong \widetilde{M \otimes_B A_g}$.

Answer 1

Restricting a sheaf to an open subscheme $U\subset X$ is the same as taking the pullback along the open immersion $U\to X$. So letting $f:D(g)\to V$ be the open immersion, $\mathcal{F}|_{D(g)}\cong f^*(\widetilde{M})$ and you're done.

Answer 2

Note: Let $A$ be any commutative unital ring and let $E$ be any $A$-module and $R$ any $A$-algebra. Let $X:=Spec(A)$. Define for any open set $U\subseteq X$ the following:

D1. $\mathcal{E}(U):=lim_{D(f)\subseteq U}(A_f \otimes_A E).$

By definition $\mathcal{O}_X(U):=lim_{D(f)\subseteq U}A_f$.

Since the limit commutes with taking tensor product, there is an isomorphism

$\mathcal{E}(U) \cong (lim_{D(f)\subseteq U}A_f)\otimes_A E := \mathcal{O}_X(U)\otimes_A E$.

Definition D1 gives (this is Exercise 4.23 in Atiyah-Macdonald) a sheaf of $\mathcal{O}_X$-modules $\mathcal{E}$ on $X$. This is an equivalent definition to the definition given in Harsthorne.

Define

D2. $\mathcal{R}(U):=\mathcal{O}_X(U)\otimes_A R$.

It follows $\mathcal{R}$ is a sheaf of $\mathcal{O}_X$-algebras on $X$.

With this definition it follows $\mathcal{E}(D(f)):=A_f\otimes_A E \cong E_f$ and $\mathcal{R}(D(f))\cong R_f$.

Doing the exercise in AM will improve your understanding of the construction of $\mathcal{E}$ from $E$. If $A$ is an integral domain it follows

$\mathcal{O}_X(U) \cong \cap_{D(f)\subseteq U}A_f$, since we may view all rings $A_f$ as sub rings of $K(A)$, and then we can take intersections inside this common ring ($K(A)$ is the quotient field of $A$) - "taking the limit" of the rings $A_f$ is a generalization of "taking intersections" $\cap A_f$.

Question: "However, I don't see how $\mathscr F |_{D(g)} \cong \widetilde{M \otimes_B A_g}$."

Answer: If you do the exercise in AM and prove that this is a definition equivalent to the definition in Hartshorne, you will get a better understanding of quasi coherent sheaves. With this new definition $\mathcal{E}(D(f))\cong E_f$ is "by definition".

This is also discussed here

Noetherian $R$-algebra corresponds to a coherent sheaf of rings on $\operatorname{Spec}(R)$