##
On the modules of m-integrable derivations in non-zero characteristic

Author | Narváez Macarro, Luis |

Department | Universidad de Sevilla. Departamento de álgebra |

Date | 2012-03-20 |

Published in | Advances in Mathematics, 229 (5), 2712-2740. |

Document type | Article |

Abstract | Let k be a commutative ring and A a commutative k-algebra. Given
a positive integer m, or m = ∞, we say that a k-linear derivation δ of
A is m-integrable if it extends up to a Hasse–Schmidt derivation D =
(Id, D1 = δ, ... Let k be a commutative ring and A a commutative k-algebra. Given a positive integer m, or m = ∞, we say that a k-linear derivation δ of A is m-integrable if it extends up to a Hasse–Schmidt derivation D = (Id, D1 = δ, D2, . . . , Dm) of A over k of length m. This condition is automatically satisfied for any m under one of the following orthogonal hypotheses: (1) k contains the rational numbers and A is arbitrary, since we can take Di = δ i i! ; (2) k is arbitrary and A is a smooth k-algebra. The set of m-integrable derivations of A over k is an A-module which will be denoted by Iderk(A; m). In this paper we prove that, if A is a finitely presented k-algebra and m is a positive integer, then a k-linear derivation δ of A is m-integrable if and only if the induced derivation δp : Ap → Ap is m-integrable for each prime ideal p ⊂ A. In particular, for any locally finitely presented morphism of schemes f : X → S and any positive integer m, the S-derivations of X which are locally mintegrable form a quasi-coherent submodule Ider S(OX; m) ⊂ Der S(OX) such that, for any affine open sets U = Spec A ⊂ X and V = Spec k ⊂ S, with f(U) ⊂ V , we have Γ(U,Ider S(OX; m)) = Iderk(A; m) and Ider S(OX; m)p = IderOS,f(p) (OX,p; m) for each p ∈ X. We also give, for each positive integer m, an algorithm to decide whether all derivations are m-integrable or not. |

Cite | Narváez Macarro, L. (2012). On the modules of m-integrable derivations in non-zero characteristic. Advances in Mathematics, 229 (5), 2712-2740. |

##### Impact

10.1016/j.aim.2012.01.015

WoS
//WoS
var doi = document.getElementById("doi").innerHTML;
var url_wos = 'https://ahus.us.es/idus/wos/wos.php?doi=' + doi;
var xhttp_wos = new XMLHttpRequest();
xhttp_wos.onreadystatechange = function() {
if (xhttp_wos.status == 200) {
if (xhttp_wos.responseText !== '0') {
var info_wos = xhttp_wos.responseText.split("|");
if(xhttp_wos.responseText !== '') {
if (info_wos[1]!=0) {
$('#Citas-WoS').attr('href',info_wos[0]);
$('#Citas-WoS .badge').html(info_wos[1]);
$('#Citas-WoS').removeClass('hidden');
}
}
}
}
};
xhttp_wos.open("GET", url_wos, true);
xhttp_wos.send();

##### Statistics

##### Share

##### Metadata

Show full item recordFiles | Size | Format | View | Description |
---|---|---|---|---|

On the modules of m-integrable ... | 357.7Kb | [PDF] | View/ | |