Hasse–Schmidt derivations, divided powers and differential smoothness

@article{NarváezMacarro2009,
abstract = {Let $k$ be a commutative ring, $A$ a commutative $k$-algebra and $D$ the filtered ring of $k$-linear differential operators of $A$. We prove that: (1) The graded ring $\rm \{gr\}\,D$ admits a canonical embedding $\theta $ into the graded dual of the symmetric algebra of the module $\Omega _\{A/k\}$ of differentials of $A$ over $k$, which has a canonical divided power structure. (2) There is a canonical morphism $\vartheta $ from the divided power algebra of the module of $k$-linear Hasse–Schmidt integrable derivations of $A$ to $\rm \{gr\}\,D$. (3) Morphisms $\theta $ and $\vartheta $ fit into a canonical commutative diagram.},
affiliation = {Universidad de Sevilla Facultad de Matemáticas Instituto de Matemáticas (IMUS) Departamento de Álgebra P.O. Box 1160 41080 Sevilla (Spain)},
author = {Narváez Macarro, Luis},
journal = {Annales de l’institut Fourier},
keywords = {Derivation; integrable derivation; differential operator; divided powers structure; derivation},
language = {eng},
number = {7},
pages = {2979-3014},
publisher = {Association des Annales de l’institut Fourier},
title = {Hasse–Schmidt derivations, divided powers and differential smoothness},
url = {http://eudml.org/doc/10477},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Narváez Macarro, Luis
TI - Hasse–Schmidt derivations, divided powers and differential smoothness
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 7
SP - 2979
EP - 3014
AB - Let $k$ be a commutative ring, $A$ a commutative $k$-algebra and $D$ the filtered ring of $k$-linear differential operators of $A$. We prove that: (1) The graded ring $\rm {gr}\,D$ admits a canonical embedding $\theta $ into the graded dual of the symmetric algebra of the module $\Omega _{A/k}$ of differentials of $A$ over $k$, which has a canonical divided power structure. (2) There is a canonical morphism $\vartheta $ from the divided power algebra of the module of $k$-linear Hasse–Schmidt integrable derivations of $A$ to $\rm {gr}\,D$. (3) Morphisms $\theta $ and $\vartheta $ fit into a canonical commutative diagram.
LA - eng
KW - Derivation; integrable derivation; differential operator; divided powers structure; derivation
UR - http://eudml.org/doc/10477
ER -