Relative spectrum of a quasicoherent sheaf of algebras (affine case)

Question

I'm trying to understand the concept of the relative spectrum of a sheaf of quasicoherent algebras. Here is the situation: We are given a scheme $X$ and a quasicoherent sheaf of $\mathscr{O}_X$-algebras (i.e. a sheaf of algebras that is quasicoherent as an $\mathscr{O}_X$-module) $\mathscr{B}$.

Consider the contravariant functor $F = F_{\mathscr{B}, X}$ from schemes to sets that acts on objects by $$W \mapsto \{(\mu : W \to X, \varphi : \mathscr{B} \to \mu_* \mathscr{O}_W )\}$$ where $\mu$ is a morphism of schemes, $\varphi$ is a morphism of $\mathscr{O}_X$-algebras, and the right hand side is the set of all such pairs. The functor $F$ acts on morphisms $a : W' \to W$ by \begin{align} F(a) : F(W) &\to F(W')\\ (\mu, \varphi) &\mapsto (\mu' = \mu \circ a, \varphi' = a^* \varphi), \end{align} where $\varphi' = a^*\varphi : \mathscr{B} \to \mu_* \mathscr{O}_W \to {(\mu')}_* \mathscr{O}_{W'}$ in the composition of $\varphi$ with $\mu_* a^\#$, where $a^\# : \mathscr{O}_W \to a_* \mathscr{O}_{W'}$ (note $\mu_* a_* = (\mu \circ a)_* = (\mu')_*$)

The idea is that we get an affine morphism of schemes $\beta : \underline{\mathrm{Spec}}_X\mathscr{B} \to X$ and morphism of $\mathscr{O}_X$-algebras $\phi : \mathscr{B} \to \beta_* \mathscr{O}_{\underline{\mathrm{Spec}}_X\mathscr{B}}$ such that the pair $(\underline{\mathrm{Spec}}_X\mathscr{B}, (\beta, \phi))$ represents the functor $F$. That is, $F(-) = \mathrm{Hom}_{\mathrm{Sch}_{/X}}(-, \underline{\mathrm{Spec}}_X\mathscr{B})$ and $(\beta, \phi) \in F(\underline{\mathrm{Spec}}_X\mathscr{B})$ corresponds to the identity morphism under the bijection $F(\underline{\mathrm{Spec}}_X\mathscr{B}) \cong \mathrm{Hom}(\underline{\mathrm{Spec}}_X\mathscr{B}, \underline{\mathrm{Spec}}_X\mathscr{B})$. The scheme $\underline{\mathrm{Spec}}_X\mathscr{B}$ is called the relative spectrum of $\mathscr{B}$ over $X$.

How do we get this scheme and universal maps? As usual, we start with the affine case. Suppose $X = \mathrm{Spec}(A)$. Then since $\mathscr{B}$ is quasicoherent, it must be o.t.f. $\mathscr{B} = \tilde{B}$ for some $A$-algebra $B$. This gives a structure morphism $\beta : \mathrm{Spec}(B) \to \mathrm{Spec}(A)$. The multiplication map $B \otimes_A B \to B$ taking $b \otimes b' \mapsto b b'$ induces a morphism of qcoh $\mathscr{O}_{\mathrm{Spec}B}$-algebras $\beta^* \mathscr{B} = \widetilde{B \otimes_A B} \to \tilde{B} = \mathscr{O}_{\mathrm{Spec}B}$, and hence by adjunction a map $\mathscr{B} \to \beta_* \mathscr{O}_{\mathrm{Spec}B}$. We claim that this data represents the functor $F_{\mathscr{B}, \mathrm{Spec}(A)}$. This is proved in https://stacks.math.columbia.edu/tag/01LT, but there are some things I don't understand. The idea is fairly clear, We must show that for all schemes $W$, the map $$ \mathrm{Mor}_{\mathrm{Sch}}(W, \mathrm{Spec} B) \to \{(\mu, \varphi)\} \hspace{10pt} a \mapsto (a^* \beta = \beta \circ a, a^* \varphi) $$ is bijective. Let's call this map $f$. In order to do this, we can define an inverse map $f^{-1}$ and show that for all morphisms $a : W \to \mathrm{Spec}B$, $f^{-1}(f(a)) = a$ and for all pairs $(\mu, \varphi)$, $f(f^{-1}((\mu, \varphi))) = (\mu, \varphi)$. What I don't understand is how to define the inverse map. Specifically, in the lemma I cited above, what is the map $f^*$ ($\mu^*$ in my notation) supposed to be?

As a follow up question, suppose we have a pair $(\mu : W \to \mathrm{Spec} A, \varphi: \mu^* \mathscr{B} \to \mathscr{O}_W)$. Then by adjunction we get $\varphi : \mathscr{B} \to \mu_* \mathscr{O}_W$. Evaluating this map of $\mathscr{O}_{\mathrm{Spec}A}$-algebras on global sections, we have an $A$-algebra homomorphism $B \to \Gamma(W, \mathscr{O}_W)$. This corresponds to a morphism of schemes $W \to \mathrm{Spec}(B)$. If we define the inverse to be this map, does the proof go through? I'm having some trouble showing the maps are mutually inverse.

Answer 1

Your construction looks correct, and it is a natural bijection because it is constructed using adjunctions, which are natural bijective correspondences. Namely, since $\mathscr B=\widetilde B$, it follows from adjunction of $\widetilde{\phantom{B}}$ and $\Gamma$ that morphisms of $\mathscr O_{\mathrm{Spec}A}$-algebras $\mathscr B\to\mu_*\mathscr O_W$ correspond naturally to $A$-algebra morphisms $B\to\Gamma(\mathrm{Spec}A,\mu_*\mathscr O_W)\cong\Gamma(W,\mathscr O_W)$. By adjunction of Spec and $\Gamma$, these correspond naturally to morphisms $W\to\mathrm{Spec}B$ of schemes over $\mathrm{Spec}A$.

Answer 2

Here is an answer to my first question. Suppose $(\mu, \varphi)$ is a pair as above. Taking the construction of https://stacks.math.columbia.edu/tag/01I6, we get a ring homomorphism $$ B = \Gamma(\mathrm{Spec}B, \mathscr{B}) \overset{\mu^*}{\longrightarrow} \Gamma(W, \mu^* \mathscr{B}) \overset{\varphi}{\longrightarrow} \Gamma(W, \mathscr{O}_W) $$ inducing a morphism of schemes $W \to \mathrm{Spec}B$. As mentioned in my question, the map $\varphi$ is clear. I think the map $\mu^*$ is the natural map $\Gamma(Y, \mathscr{G}) \to \Gamma(X, \pi^*\mathscr{G})$ we get for any morphism $\pi : X \to Y$ and qcoh sheaf $\mathscr{G}$ on $Y$ applied to our situation ($\pi = \mu$ and $\mathscr{G} = \mathscr{B}$). The natural map is constructed using adjunction of the identity map on $\mathscr{G}$ to get $\mathscr{G} \to \pi_* \pi^* \mathscr{G}$, applying the global sections functor, and then composing with the identity map $\Gamma(Y, \pi_* \pi^* \mathscr{G}) \to \Gamma(X, \pi^* \mathscr{G})$.