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Let $C$ be the category of schemes and $D$ be the category of affine schemes. Then we have the inclusion function $L : C \to D$, and the affinization function $R : D \to C$, where $R(X) = \operatorname{Spec} \Gamma(X, \mathcal{O}_X)$, and for a morphism $f : X \to Y$, $R(f)$ is the morphism $\operatorname{Spec} \Gamma(X, \mathcal{O}_X) \to \operatorname{Spec} \Gamma(Y, \mathcal{O}_Y)$ induced by $f^\#_Y : \Gamma(Y, \mathcal{O}_Y) \to \Gamma(X, \mathcal{O}_X)$.

Various places (here or here) state that these two functors are adjoints, but I am having trouble explicitly writing out what the unit and counit are for this adjunction.

Can anyone say explicitly what the unit and counit do (i.e. what are the topological maps, and the maps of sheaves on an arbitrary scheme/affine scheme)? I think that understanding this explicitly would help me enormously with feeling confident on various affineness criteria.

stillconfused
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1 Answers1

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I'll do this a bit more generally.

Let $\mathbf{LRS}$ be the category of locally ringed spaces and $\mathbf{Aff}$ be the category of affine schemes.

For the unit: Take $(X,\mathcal O_X) \in \mathbf{LRS}$. We want to describe a map $\eta_X:X \to \mathrm{Spec}(\Gamma(X,\mathcal O_X))$. Take any point $x \in X$. Then the stalk $\mathcal O_{X,x}$ is a local ring, so it has a maximal ideal, call it $\mathfrak{m}_{X,x}$. We have a ring homomorphism $i_X:\Gamma(X,\mathcal O_X) \to \mathcal O_{X,x}$. Define $\eta_X(x)$ to be $i_X^{-1}(\mathfrak{m}_{X,x}) \in \mathrm{Spec}(\Gamma(X,\mathcal O_X))$. Varying over $x$, we get a set-function $\eta_X$.

As the next step, we check that $\eta_X$ is continuous. It suffices the preimage of basic opens are open, so let $f \in \Gamma(X,\mathcal O_X)$. Take any $y \in \eta_X^{-1}(D(f))$, then $\eta_X(y) \in D(f)$, so $f \notin i_X^{-1}(\mathfrak{m}_{X,x})$, so $i_X(f)=f_x \notin \mathfrak{m}_{X,x}$. Thus $f_x$ is a unit in $\mathcal O_{X,x}$ (as it's not contained in the maximal ideal of that local ring). Write $1=f_x\cdot g$ for some $g \in \mathcal O_{X,x}$. Now unpack the definition of $\mathcal O_{X,x}$ as a directed colimit. The equation $1=f_x \cdot g$ being true must mean that there's some open subset $U \subset X$ containing $x$ such that $1=f|_U \cdot g$ (where I'm abusing notation with $g$ here). But for all points $y \in U$, we have that $f_y \notin \mathfrak{m}_{X,y}$, so working backwards, we get $U \subset \eta_X^{-1}(D(f))$, showing that $\eta_X^{-1}(D(f))$ is open.

Now we want to upgrade this to a map of locally ringed spaces. This means that , first of all, for all open $U \subset \mathrm{Spec}(\Gamma(X,\mathcal O_X))$, we want a ring map $\Gamma(U,\mathcal O_{\mathrm{Spec}(\Gamma(X,\mathcal O_X))}) \to \Gamma(\eta_X^{-1}(U),\mathcal O_X)$. By a standard gluing argument, it suffices to do this for basic opens. So we want for any $f \in \mathrm{Spec}(\Gamma(X,\mathcal O_X))$, a map $$\Gamma(X,\mathcal O_X)_f \to \Gamma(\eta_X^{-1}(D(f)),\mathcal O_X)$$ What is $\eta_X^{-1}(D(f))?$ We had this guy before, it's $\{x \in X \mid f_x \notin \mathfrak{m}_{X,x}\}:=U_f$. Now as above for any $x \in U_f$ we get that $f_x$ is a unit in $\mathcal O_{X,x}$, which implies (by a small exercise I'll leave open) that $f|_{U_f}$ is a unit in $\Gamma(U_f,\mathcal O_X)$. But this means that our ring map is induces by the restriction map $\Gamma(X,\mathcal O_X) \to \Gamma(U_f,\mathcal O_X)$ and the universal property of localization. To check that this is indeed a map of locally ringed spaces is an exercise in using the uniqueness part of the universal property of localization.

For the counit map: this is pretty easy. You'll find that by definition we have for any ring $A$ a natural isomorphism $\Gamma(\mathrm{Spec}(A),\mathcal O_{\mathrm{Spec}(A)})=A$, so (up to this natural isomorphism) we can take the identity as a counit.

I hope you'll forgive me that I don't check the triangle identities here, as this answer is already quite long despite being basically all formal.

Lukas Heger
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