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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 $X: Kmathsf{Alg} to mathsf{Sets}$, 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 $X: Kmathsf{Alg} to mathsf{Sets}$ 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 $mathbb{A}^1_K: Kmathsf{Alg} to mathsf{Sets}$.

So we have a functor $mathsf{Sets}^{Kmathsf{Alg}} to Kmathsf{Alg}$ 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 $X: Kmathsf{Alg} to mathsf{Sets}$? If not, what should we call this?




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

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.

"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).