Prolongations in differential algebra (Q948886)

Let \(R\) be a commutative ring with a designated \(\infty\) order higher derivation \(\underline{D}\). An \((R, \underline{D})\) algebra \(B\) over \(R\) is a commutative \(R\) algebra such the kernel of \(R \to B\) is a differential ideal. Let \(B_m=B[t]/(t^{m+1})\) for \(m\) finite and \(B_\infty=B[[t]]\). Let \(\widetilde{B_m}\) denote \(B_m\) made an \(R\) algebra via \(r \mapsto \sum D_i(r)t^i\). Now suppose that both \(A\) and \(B\) denote commutative \((R, \underline{D})\) algebras. The author proves that the functor \(B \mapsto \text{Hom}_R(A, \widetilde{B_m})\) is representable by an algebra \(HS^m_{(A/(R, \underline{D}))}\). This latter is also a differential ring compatible with the higher derivation on \(R\). Now suppose that \(R=K\) is a field and \(X\) a \(K\) scheme. The author patches the spectra of the rings \(HS^m_{(A/(R, \underline{D}))}\) as \(A\) ranges over global sections of affine opens of \(X\) to produce a scheme \(P_m(X/(K, \underline{D}))\), whose structure sheaf consists of differential rings compatible with the higher derivation on \(K\). If \(X\) is extended from the field of constants of \(K\), the author shows that \(P_m(X)\) coincides with the jet scheme \(J_m(X)\) and, in any case, for all \(m\) and \(n\) that \(P_m(J_n(X))\) and \(J_n(P_m(X))\) coincide. His main result is then obtained, namely that \(P_m(X)\) and \(J_m(X)\) are isomorphic via differential polynomial maps.