Generalized differentials and prolongation spaces
type: Item
instance of: scholarly article
title: Generalized differentials and prolongation spaces
zbMATH Open document ID: 1476.13034
author: Florian Heiderich
published in: Journal of Algebra
publication date: 2021-08-25T00:00:00Z
review text: 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 different…
Mathematics Subject Classification ID: 13N15
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 12H10
Mathematics Subject Classification ID: 16T15
zbMATH DE Number: 7385162
zbMATH Keywords: differential ring
zbMATH Keywords: difference ring
zbMATH Keywords: higher derivation
zbMATH Keywords: prolongation
zbMATH Keywords: jet
zbMATH Keywords: coalgebra
zbMATH Keywords: bialgebra
reviewed by: Alexander B. Levin
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2021.06.026
OpenAlex ID: W3179624280
cites work: Picard-Vessiot extensions of Artinian simple module algebras.
cites work: Hopf Algebraic Approach to Picard-Vessiot Theory
cites work: Néron Models
cites work: A Lindemann-Weierstrass theorem for semi-abelian varieties over function fields
cites work: Geometry of Differential Polynomial Functions, I: Algebraic Groups
cites work: Q4431279
cites work: Difference Galois theory of linear differential equations
cites work: Q3153353
cites work: Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas (Quatrième partie). Rédigé avec la colloboration de J. Dieudonné
cites work: Iterative q-difference Galois theory
cites work: On Hasse-Schmidt rings and module algebras.
cites work: Model theory of endomorphisms of separably closed fields
cites work: Q5767091
cites work: The Nash problem on arc families of singularities
cites work: Order for systems of differential equations and a generalization of the notion of differential ring
cites work: Prolongation of integral domains
cites work: Difference algebra
cites work: Q3882554
cites work: Q3992979
cites work: Galois theory for iterative connections and nonreduced Galois groups
cites work: Jet and prolongation spaces
cites work: Generalized Hasse-Schmidt varieties and their jet spaces
cites work: ×R-BIALGEBRAS ASSOCIATED WITH ITERATIVE q-DIFFERENCE RINGS
cites work: Prolongations in differential algebra
cites work: Q5590089
cites work: A Hopf algebraic approach to the Picard-Vessiot theory
cites work: Twisted Galois stratification
cites work: Galois theory of difference equations
cites work: Q4790565
cites work: Q5457539
cites work: A Chevalley theorem for difference equations
DOI: 10.1016/J.JALGEBRA.2021.06.026
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description: scientific article; zbMATH DE number 7385162