On the modules of \(m\)-integrable derivations in non-zero characteristic
type: Item
instance of: scholarly article
title: On the modules of \(m\)-integrable derivations in non-zero characteristic
zbMATH Open document ID: 1242.14015
published in: Advances in Mathematics
publication date: 2012-04-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1106.1391
reviewed by: Oleksandr Iena
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 13N15
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 6023751
zbMATH Keywords: Derivation
zbMATH Keywords: integrable derivation
zbMATH Keywords: Hasse-Schmidt derivation
zbMATH Keywords: Differential operator
author: Luis Narváez-Macarro
MaRDI profile type: Publication
OpenAlex ID: W1517132175
arXiv ID: 1106.1391
cites work: On the imbedding of derivations of finite rank into derivations of infinite rank
cites work: Determinantal rings
cites work: Hasse-Schmidt derivations and coefficient fields in positive characteristics.
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: Q5767091
cites work: On the Jacobian Ideal of the Module of Differentials
cites work: Integrable derivations
cites work: Q3739243
cites work: Sul modulo delle derivazioni integrabili in caratteristica positiva
cites work: Hasse-Schmidt derivations, divided powers and differential smoothness
cites work: Derivations and integral closure
cites work: Tight closure and differential simplicity
cites work: Localization of the Hasse-Schmidt Algebra
cites work: Q5457539
DOI: 10.1016/J.AIM.2012.01.015
review text: Let \(k\) be a commutative ring, let \(A\) be a commutative \(k\)-algebra, and let \(\delta:A\to A\) be a \(k\)-linear derivation of \(A\). It can be identified with the \(k\)-algebra homomorphism \(\bar\delta: A\to A[\varepsilon]\), \(a\mapsto a+\delta(a)\cdot \varepsilon\), where \(A[\varepsilon]=At/(t^2)\) and \(\varepsilon = t+(t^2)\) is the infinitesimal parameter.NEWLINENEWLINELet \(m\) be a positive integer. The derivation \(\delta\) is called \(m\)-integrable if there is a \(k\)-algebra homomorphism \(\Phi: A\to At/(t^{m+1})\) that lifts the homomorphism \(\bar \delta\). Let \(\bar t\) denote the class of \(t\) in \(At/(t^{m+1})\), then NEWLINE\[NEWLINE \Phi(a)=a+\bar \delta\cdot \bar t+ D_1(a)\cdot \bar {t}^2+\dots+D_{m}\cdot \bar{t}^m, NEWLINE\]NEWLINE where \(D_i: A\to A\) are \(k\)-linear maps satisfying \(D_0=\mathrm{id}_A\), \(D_1=\delta\), and the Leibniz equality NEWLINE\[NEWLINED_{i}(ab)=\sum\limits_{r+s=i}D_r(a)D_s(b).NEWLINE\]NEWLINE Such a sequence \(D_0,\dots D_m\) of \(k\)-linear maps is called a Hasse-Schmidt derivation of length \(m\). The set \(\mathrm{Ider}_k(A; m)\) of all \(m\)-integrable \(k\)-derivations of \(A\) is an \(A\)-submodule of \(\mathrm{Der}_k(A)\).NEWLINENEWLINEThe derivation \(\delta\) is called (\(\infty\)-)integrable if \(\bar \delta\) can be lifted to a \(k\)-algebra homomorphism \(A\to At\), \(a\mapsto a+\delta(a)t+D_2(a)t^2+\dots\), and the sequence of \(k\)-linear operators NEWLINE\[NEWLINE D_0=\mathrm{id}_A,\quad D_1=\delta,\quad D_2, \dots NEWLINE\]NEWLINE satisfies the Leibniz equality and is called a Hasse-Schmidt derivation of infinite length. The set \(\mathrm{Ider}_k(A)=\mathrm{Ider}_k(A; \infty)\) of all integrable \(k\)-derivations of \(A\) is an \(A\)-submodule of \(\mathrm{Der}_k(A)\).NEWLINENEWLINEThere is a sequence of \(A\)-submodules NEWLINE\[NEWLINE \mathrm{Der}_k(A)=\mathrm{Ider}_k(A; 1)\supset \mathrm{Ider}_k(A; 2)\supset \dots \supset \mathrm{Ider}_k(A; \infty)=\mathrm{Ider}_k(A). NE…
description: scientific article; zbMATH DE number 6023751