Generalized differentials and prolongation spaces
Let \(D\) be a coalgebra over a commutative ring \(k\) and \(A\) and \(B\) are commutative \(k\)-algebras. A \(k\)-homomorphism of algebras \(\psi:D\bigotimes_{k}A\rightarrow B\) is said to be a \textit{\(D\)-measuring} from \(A\) to \(B\) if the image of \(\psi\) under the natural isomorphism of left \(k\)-modules \(_{k}\mathcal{M}(D\bigotimes_{k}A, B)\rightarrow {}_{k}\mathcal{M}(A, {}_{k}\mathcal{M}(D, B))\) is a homomorphism of \(k\)-algebras. (For a \(k\)-coalgebra \(D\) and \(k\)-algebra \(A\), \(_{k}\mathcal{M}(D, A)\) denotes the \(k\)-algebra of all homomorphisms of \(k\)-modules from \(D\) to \(A\)). If \(B\) is a commutative \(A\)-algebra and \(\psi\) is a \(D\)-measuring from \(A\) to \(B\), the authors define a commutative \(A\)-algebra \([D]_{(A, \psi)}B\) that generalizes the symmetric algebra over the module of Kähler differentials \(\mathrm{Sym}_{B}(\Omega^{1}_{B/A})\). It is shown that the spectrum of \([D]_{(A, \psi)}B\) is isomorphic to a prolongation space as defined in [\textit{R. Moosa} and \textit{T. Scanlon}, J. Inst. Math. Jussieu 9, No. 2, 391--430 (2010; Zbl 1196.14008)] and [\textit{R. Moosa} and \textit{T. Scanlon}, Proc. Lond. Math. Soc. (3) 103, No. 2, 197--234 (2011; Zbl 1273.14006)] in terms of Weil restrictions. The paper under review provides a direct construction of such a prolongation in the case of affine schemes and show that they generalize the prolongation spaces defined in [\textit{A. Buium}, Am. J. Math. 115, No. 6, 1385--1444 (1993; Zbl 0797.14016)], [\textit{P. Vojta}, CRM Ser. (N.S.) 4, 335--361 (2007; Zbl 1194.13027)] and [\textit{E. Rosen}, Isr. J. Math. 166, 239--284 (2008; Zbl 1157.12003)]. Furthermore, the universal prolongations of differential and difference kernels can be recovered from generalized differentials introduced in the paper. If \(D\) is a bialgebra, the generalized differentials provide a unified approach to the prolongations of commutative rings unifying the well-known constructions in the differential and difference cases.
scientific article; zbMATH DE number 7385162
scientific article