Functor from the category of rings to the category of affine schemes

Question

We have shown in an exercise that $$A\mapsto Spec(A),\ \ (\varphi:A\to B) \mapsto\ (^a\varphi:Spec(B)\to Spec(A), \ ^a\varphi(\mathfrak{p}):=\varphi^-1(\mathfrak{p}))$$ defines a contravariant functor from the category of rings to the category of topological spaces.

Now we are supposed to show how this functor can be extended to a functor $\mathcal{F}$ from the category of rings to the category of affine schemes. Following our definitions I tried to define this by $$\mathcal{F}(A) = (Spec(A),\mathcal{O}_{Spec(A)}),\ \mathcal{F}(\varphi:A\to B) = (^a\varphi, (^a\varphi)^b),$$ where $^a\varphi$ is defined just like before, and $(^a\varphi)^b:\mathcal{O}_{Spec(A)}\to ^a\varphi_*\mathcal{O}_{Spec(B)}$ should be some morphism of sheafs of rings on $Spec(A)$.

Basically everything here should be fine, except I do not know how to define the morphism of sheafs $(^a\varphi)^b$ exactly. I guess this must be somehow induced by the ring homomorphism $\varphi$, or just by the continuous function $^a\varphi$, but neither do I see any natural way to define it, nor have we had any example of how this can be done. As a morphism of sheafs of rings, $(^a\varphi)^b$ should be a family of ring homomorphisms, so for any $U\subset Spec(A)$ open, I should define a ring homomorphism $$(^a\varphi)^b_U:\mathcal{O}_{Spec(A)}(U) \to\ ^a\varphi_*\mathcal{O}_{Spec(B)}(U)=\mathcal{O}_{Spec(B)}(^a\varphi^{-1}(U)).$$ Maybe my biggest issue is that I have no information on the above rings, or on how they are connected to each other or to $A$ and $B$.

Thanks for your help.

Answer 1

Hint: It should suffice to define the map of sheaves on basic open sets, i.e., those of the form $$ \newcommand{\p}{\mathfrak{p}} \newcommand{\O}{\mathcal{O}} \DeclareMathOperator{\Spec}{Spec} X_f = \{\p \in \Spec(A) : f \notin \p\} \, . $$ Let $X = \Spec(A)$ and $Y = \Spec(B)$, let $\psi = {^a \varphi}: Y \to X$ and let $\psi^\sharp: \O_X \to \psi_*\O_Y$ be the map of sheaves ($({^a \varphi})^b$ in your notation). Recall that $\O_X(X_f) = A[1/f]$. Do you see how to define $\psi^\sharp$ on $X_f$? Spoiler below.

\begin{align*} \psi^\sharp_{X_f}: \O_X(X_f) = A[1/f] &\longrightarrow \O_Y(\psi^{-1}(X_f)) = B[1/\varphi(f)]\\ \frac{a}{f^n} &\longmapsto \frac{\varphi(a)}{\varphi(f)^n} \end{align*}

Then check that your definition agrees on overlaps: if $X_{f} \cap X_{g} \neq \varnothing$, make sure that $\psi^\sharp_{X_f}|_{X_{f} \cap X_{g}} = \psi^\sharp_{X_g}|_{X_{f} \cap X_{g}}$.

Alternatively, you could use the definition of the structure sheaf as given in Hartshorne (p. 70), where he defines $\O_X(U)$ as the set of functions $s: U \to \coprod_{\p \in U} A_\p$ with some properties. (These are the sections of the espace étalé of $\O_X$.)