Given a coalgebra D, a commutative algebra A, a commutative A-algebra B and a measuring $\psi :D\otimes A\to B$, we define an algebra ${[D]}_{(A,\psi )}B$ that generalizes the symmetric algebra over the module of Kähler differentials ${\mathrm{Sym}}_B({\mathrm{\Omega }}_{B/A}^1)$. We show that the spectrum of ${[D]}_{(A,\psi )}B$ is isomorphic to a prolongation space as defined by Moosa and Scanlon, providing a direct construction of the latter. These prolongation spaces generalize those of Gillet, Rosen and Vojta. The universal prolongations of differential and difference kernels can also be recovered from our generalized differentials. When D is moreover a bialgebra, our generalized differentials provide a unified approach to the prolongations of commutative rings, unifying the well-known constructions in the differential and difference case.

给定一个余代数D、一个交换代数A、一个交换A -代数B和一个测量$\psi ：D\otimes \mathrm{一种}\to 乙$，我们定义一个代数 ${[D]}_{(\mathrm{一种},\psi )}乙$ 将对称代数推广到 Kähler 微分的模上 ${\mathrm{符号}}_{乙}({\mathrm{\Omega }}_{乙/\mathrm{一种}}^1)$. 我们证明了${[D]}_{(\mathrm{一种},\psi )}乙$与 Moosa 和 Scanlon 定义的延伸空间同构，提供后者的直接构造。这些延长空间概括了 Gillet、Rosen 和 Vojta 的延长空间。差分核和差分核的普遍扩展也可以从我们的广义差分中恢复。此外，当D是双代数时，我们的广义微分为交换环的扩展提供了统一的方法，统一了微分和差分情况下的众所周知的构造。