Coordinate ring of a scheme in functorial algebraic geometry

Question

I will preface this by saying that I am new to algebraic geometry, but I am somewhat experienced with category theory.

I'm just reading the introduction to Milne's notes "Basic Theory of Affine Group Schemes". He uses the functorial point of view here, so I am viewing an affine scheme over $K$ as a representable functor $X: K\mathsf{Alg} \to \mathsf{Sets}$, and a scheme is likewise defined as a functor satisfying appropriate gluing properties. We can think of some general functor as a generalized scheme.

In section I.3 he has a subsection titled "The canonical coordinate ring of an affine group" but I noticed that his construction seems to define a canonical "coordinate ring" for every sort of "generalized scheme", not just affine group schemes. Indeed, if $X: K\mathsf{Alg} \to \mathsf{Sets}$ is a functor then $\mathrm{Nat}(X, \mathbb{A}^1_K)$ is a $K$-algebra (with operations defined pointwise), since the affine line over $K$ is the forgetful functor $\mathbb{A}^1_K: K\mathsf{Alg} \to \mathsf{Sets}$.

So we have a functor $\mathsf{Sets}^{K\mathsf{Alg}} \to K\mathsf{Alg}$ defined by $X \mapsto \mathrm{Nat}(X, \mathbb{A}^1_K)$.

Moreover, we have an obvious natural transformation $\alpha: X \to \mathrm{Spec_K}(\mathrm{Nat}(X, \mathbb{A}^1_K))$, where $\mathrm{Spec}_K$ here is just the contravariant Yoneda embedding (since I am thinking of affine schemes as functors rather than ringed spaces). This natural transformation has components $\alpha_A: X(A) \to \mathrm{Hom}(\mathrm{Nat}(X, \mathbb{A}^1_K), A)$ given by $x \mapsto (f \mapsto f_A(x))$.

My question is:

1. Is it reasonable to call $\mathrm{Nat}(X, \mathbb{A}^1_K), A)$ the coordinate ring for any "generalize scheme" given by a functor $X: K\mathsf{Alg} \to \mathsf{Sets}$? If not, what should we call this?

2. Is the functor $\mathsf{Sets}^{K\mathsf{Alg}} \to K\mathsf{Alg}$ mapping $X$ to $\mathrm{Nat}(X, \mathbb{A}^1_K)$ adjoint (on the left or right) to $\mathrm{Spec}_K: K\mathsf{Alg}^{\mathrm{opp}} \to \mathsf{Sets}^{K\mathsf{Alg}}$? My guess is that it is the left adjoint to $\mathrm{Spec}_K$.

3. Is there a name and interpretation for this natural transformation $\alpha_A: X(A) \to \mathrm{Hom}(\mathrm{Nat}(X, \mathbb{A}^1_K), A)$? I can see that $X$ is an affine scheme over $K$ if and only if this is an isomorphism. But what if $X$ is not affine? How do we interpret this?

Answer 1

1. Yes.

2. The opposites confuse me about which of "left" and "right" I'm supposed to say. It should be left with the correct choice of ops.

3. $\alpha$ deserves to be called "affinization." It's the universal map from a scheme or generalized scheme into an affine scheme; that is, it's the left adjoint of the inclusion of affine schemes into schemes / generalized schemes. As a simple example, the affinization of projective space is a point.

Answer 2

1. Yes. But be careful! There is a size issue. Following Demazure-Gabriel (you will enjoy it if you favor the functorial approach to algebraic geometry), we fix a universe $U$ of small sets and denote the category of small rings by $M$ (modèles). We also choose a universe $V\ni U$ and denote the category of sets in $V$ by $E$ (ensembles). The functor category $\mathrm{Func}(M,E)=:\mathrm{ME}$ is called the category of $\mathbb{Z}$-functors. For a general $X:M\to E$, the coordinate ring $O(X):=\mathrm{ME}(X, \mathbb{A}^1)$ is big and does not belong to $M$. The good news is that for any scheme $X$ (local functor with a small affine open covering), our coordinate ring $O(X)$ is small. The reason is that (i) for any affine scheme $\mathrm{Sp}A:=M(A,-)$, the Yoneda lemma gives $O(A)\simeq A\in M$ (ii) the affine line is a scheme and the locality (see this post, notice that the sheaf condition can be written as $F(U)\simeq\lim F(U_i)$ for a suitable bipartite diagram) implies $\mathrm{ME}(X,\mathbb{A}^1)\simeq\varprojlim\mathrm{ME}(X_i,\mathbb{A}^1)=\varprojlim O(X_i)$, where $\{X_i\hookrightarrow X\}$ is a small affine open covering of $X$ and $\varprojlim O(X_i)\in M$, for $M$ is complete.

2. "Yes". It's quoted for the same reason: the size matters. The isomorphism $\mathrm{ME}(X, \mathrm{Sp}A)\simeq M^\mathrm{op}(O(X),A)$ works for all $X\in \mathrm{ME}$ and $A\in M$. However, $O(X)$ might be too big. You obtain an adjunction when restricted to the category of schemes. Another tricky point comes from the ${}^\mathrm{op}$ operation. We usually say such an adjunction pair is mutually right adjoint.

3. There is an adjunction triangle in algebraic geometry. We have three categories: the models $M$ (small rings), the functorial description $\mathrm{ME}$ and the geometric description $\mathrm{Esg}$ (espace géométrique, the category of locally ringed spaces, as you can easily find in almost all other books about algebraic geometry) and there is an adjunction pair between any two of the three categories. $\mathrm{Sp}(-)\dashv O(-)$ is one of them. The other two are $\mathrm{Spec}(-)\dashv \Gamma(-,O_-)$ (taking affine spectrum / global sections) and $|\cdot|\dashv S$ (geometric realization / functor of points $S(X):=\mathrm{Esg}(\mathrm{Spec}(-),X)$). Two of them are reflective localizations, for $\mathrm{Sp}$ and $\mathrm{Spec}$ are fully faithful. Therefore, the intuition of reflective localization works here. The counit is always an isomorphism and the unit is the "-fication" or associated blabla. For instance, $\mathrm{Sh}(X)\hookrightarrow\mathrm{PSh}(X)$ is reflective and we call the left adjoint "sheafification"/associated sheaf and think the unit as the natural map from a presheaf to its sheafification / the associated sheaf. Similarly, your morphism is the natural morphism into the "affine schemification" or associated affine scheme.

Answer 3

"Indeed, if $X:KAlg→Sets$ is a functor then $Nat(X,\mathbb{A}^1_K)$ is a $K$-algebra (with operations defined pointwise),'' Not quite --- it may be a proper class (i.e., the underlying "set" may not be a set).