Consider the definition of a $\mathcal{C}^{\infty}$-scheme given in Dominique Joyce's "Algebraic geometry over C-infty rings". As far as I uderstand (not being an expert either in C-infty rings nor in logic, nor in categories) the setting is the following. You have the Lawvere theory $\mathrm{Euc}$ given by objects $X^n:=\mathbb{R}^n$ where morphisms are dedfined by $\mathrm{Euc}(X^n,X^m):=\mathcal{C}^{\infty}(\mathbb{R}^n,\mathbb{R}^m)$. Define a $\mathcal{C}^{\infty}$-ring to be an algebra for (i.e. a model of) $\mathrm{Euc}$ in $\mathrm{Set}$. A C-infty ring $\mathfrak{C}$ (say $\mathfrak{C}:=F(\mathbb{R})$ for a product-preserving functor $F:\mathrm{Euc}\to\mathrm{Set}$) then acquires the structure of a commutative unital $\mathbb{R}$-algebra, essentially thanks to the fact that each $\mathcal{C}^{\infty}(\mathbb{R}^n,\mathbb{R})$ is, together with many other "smooth" operations coming from compositions of arbitrary smooth functions (i.e. possibly different from $(x,y)\mapsto x+y$ and $(x,y)\mapsto xy$ etc.). One can define local C-infty rings, as well as locally-C-infty ringed spaces and morphisms between such, like in the case of ordinary algebraic geometry. Then there is a notion of spectrum $\mathrm{Spec}(\mathfrak{C})$ of a C-infty ring, hence the notion of C-infty scheme as a locally C-infty ringed space which is locally "affine".

All of this -I think- doesn't really have to do with the fact that we're dealing with smooth functions, but just with the fact that we're given a Lawvere theory $\mathcal{T}$ such that $\mathcal{T}(X^n,X^1)$ is naturally a commutative unital ring.

Suppose we take as $\mathcal{T}$ the category that has objects $X^n:=\mathbb{A}^n$, the affine spaces over a field or ring, and as morphisms the scheme morphisms $\mathcal{T}(X^n,X^m):=\mathrm{Hom}_{\mathrm{Sch}}(\mathbb{A}^n,\mathbb{A}^m)$.

If we defined $\mathcal{T}$-schemes as "spaces with a sheaf of local $\mathcal{T}$-algebras that are locally affine" (in the analogous sense as with Joyce's definition), would we somehow get usual schemes*?

  • (at least in the only-closed-points-allowed definition, à la Serre if I'm not mistaken)

Your proposed Lawvere theory seems to me the usual one associated to commutative algebras over your base ring. Your category $\mathcal T$ is the opposite of the category of polynomial rings in finitely many variables. This in turn is the opposite of the category of free commutative algebras over your base ring. This is the standard Lawvere theory associated to the variety (in the universal algebra sense) of commutative algebras over the base ring. So set theoretic models are commutative algebras over your base ring and so you have recovered affine schemes. Now allowing the locally ringed version recovers schemes.

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