Let $X,Y$ be $S$-schemes. Define the sheaf $\underline{\hom}_S(Y,X)$ on $S$-schemes by $\underline{\hom}_S(Y,X)(T) := \hom_S(T \times_S Y,X)$. In other words, we have
$\hom_S(T,\underline{\hom}_S(Y,X))\cong \hom_S(T \times_S Y,X)$,
which explains the name: If this sheaf is representable, it is the internal hom object. It coincides with the Weil restriction $\mathfrak{R}_{Y/S}(X \times_S Y)$ (see Néron models by BLR). Grothendieck mentions in FGA 5, C.2, that $\underline{\hom}_S(Y,X)$ has only a chance to be representable when $Y$ is flat and proper over $S$. So my first question is: where Grothendieck actually proved this result?
I claim that $\underline{\hom}_S(Y,X)$ is representable whenever $Y \to S$ is finite locally free and a universal homeomorphism: one just has to look at the proof for Weil restrictions in Theorem 7.6/4 in BLR; the condition on the fibers of $X$ is not needed. In particular this holds for $Y=S[\varepsilon]/\varepsilon^{n+1}$ for some $n \geq 0$, which is the example which interests me.
For example, we have $\underline{\hom}(S,X) = X$ and $\underline{\hom}(S[\varepsilon]/\varepsilon^2,X) = T(X/S) = \mathbb{V}(\Omega^1_{X/S})$ is the tangent bundle of $X/S$. This is just a reformulation of the universal properties, but there is a sort of global description of $\Omega^1_{X/S}$, namely as $I / I^2$, where $I$ is the kernel of the multiplication $\mathcal{O}_X \otimes_{\mathcal{O}_S} \mathcal{O}_X \to \mathcal{O}_X$. My main question is: Is there any such global description for $$T^n(X/S) := \underline{\hom}(S[\varepsilon]/\varepsilon^{n+1},X)$$? Probably this is what people call the $n$-th jet bundle, but I could not find any concise description in the literature. Remark that we have a seqeunce of morphisms $$\dotsc \to T^n(X/S) \to T^{n-1}(X/S) \to \dotsc \to T(X) \to X.$$ The proof of the Theorem in BLR works roughly as follows: Everything is local on $S$, to suppose it is affine. If $X$ is some affine space, you can find explicitly a representing scheme. More generally, for every quasi-coherent module $M$ on $S$ and $\mathcal{A}$ the quasi-coherent algebra on $S$ corresponding to $Y \to S$ which is finite locally free, we have $\underline{\hom}_S(Y,\mathbb{V}(M)) = \mathbb{V}(\underline{\hom}_{\mathcal{O}_S}(\mathcal{A},M))$. This implies $T^n(\mathbb{A}^m/S) \cong \mathbb{A}^{m(n+1)}$. If $X$ is affine, embed it into some affine space. If $X$ is arbitrary, glue everything. So this description is not really global and even depends on affine charts on $X$ and choices of generators and relations on the affine charts.
This can be done here even more explicit: Assume that $S=\mathrm{Spec}(R)$ and $X=\mathrm{Spec}(A)$. Let $A^{(n)}$ be the $A$-algebra generated by symbols $h_i(a)$ for every $0 \leq i \leq n$ and $a \in A$, subject to the following relations:
- $h_0(a)=a \cdot 1$
- $h_i(1)=0$ for $i \geq 1$
- $h_i(a+b)=h_i(a)+h_i(b)$
- $h_i(ab)=\sum_{p+q=i} h_p(a) h_q(b)$
Then $T^n(X/S) = \mathrm{Spec}(A^{(n)})$. The morphism $T^n(X/S) \to T^{n-1}(X/S)$ is induced by the obvious homomorphism $A^{(n-1)} \to A^{(n)}$ which inserts the generators. This description is well-known for $n=1$, where $h_1$ is just a derivation with respect to $h_0$.
I am not looking for a reference about jet spaces or alike. I would like to know if we can construct some sort of "concrete" $A$-algebra out of the $R$-algebra $A$ which turns out to be isomorphic to $A^{(n)}$. For example, when $n=1$, the well-known answer is the symmetric algebra of $I/I^2$, where $I = \ker(A \otimes_R A \to A)$, with $h_1(a)=a \otimes 1 - 1 \otimes a$.
