Coleff-Herrera currents, duality, and noetherian operators

Mats Andersson

Bulletin de la Société Mathématique de France (2011)

  • Volume: 139, Issue: 4, page 535-554
  • ISSN: 0037-9484

Abstract

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Let ℐ be a coherent subsheaf of a locally free sheaf 𝒪 ( E 0 ) and suppose that ℱ = 𝒪 ( E 0 ) / ℐ has pure codimension. Starting with a residue current R obtained from a locally free resolution of ℱ we construct a vector-valued Coleff-Herrera current μ with support on the variety associated to ℱ such that φ is in ℐ if and only if μ φ = 0 . Such a current μ can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed. By a construction due to Björk one gets Noetherian operators for ℐ from the current μ . The current R also provides an explicit realization of the Dickenstein-Sessa decomposition and other related canonical isomorphisms.

How to cite

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Andersson, Mats. "Coleff-Herrera currents, duality, and noetherian operators." Bulletin de la Société Mathématique de France 139.4 (2011): 535-554. <http://eudml.org/doc/272556>.

@article{Andersson2011,
abstract = {Let $\{\mathcal \{I\}\}$ be a coherent subsheaf of a locally free sheaf $\{\mathcal \{O\}\}(E_0)$ and suppose that $\{\mathcal \{F\}\}=\{\mathcal \{O\}\}(E_0)/\{\mathcal \{I\}\}$ has pure codimension. Starting with a residue current $R$ obtained from a locally free resolution of $\{\mathcal \{F\}\}$ we construct a vector-valued Coleff-Herrera current $\mu $ with support on the variety associated to $\{\mathcal \{F\}\}$ such that $\phi $ is in $\{\mathcal \{I\}\}$ if and only if $\mu \phi =0$. Such a current $\mu $ can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed. By a construction due to Björk one gets Noetherian operators for $\{\mathcal \{I\}\}$ from the current $\mu $. The current $R$ also provides an explicit realization of the Dickenstein-Sessa decomposition and other related canonical isomorphisms.},
author = {Andersson, Mats},
journal = {Bulletin de la Société Mathématique de France},
keywords = {Coleff-Herrera current; duality; noetherian operators; residue current},
language = {eng},
number = {4},
pages = {535-554},
publisher = {Société mathématique de France},
title = {Coleff-Herrera currents, duality, and noetherian operators},
url = {http://eudml.org/doc/272556},
volume = {139},
year = {2011},
}

TY - JOUR
AU - Andersson, Mats
TI - Coleff-Herrera currents, duality, and noetherian operators
JO - Bulletin de la Société Mathématique de France
PY - 2011
PB - Société mathématique de France
VL - 139
IS - 4
SP - 535
EP - 554
AB - Let ${\mathcal {I}}$ be a coherent subsheaf of a locally free sheaf ${\mathcal {O}}(E_0)$ and suppose that ${\mathcal {F}}={\mathcal {O}}(E_0)/{\mathcal {I}}$ has pure codimension. Starting with a residue current $R$ obtained from a locally free resolution of ${\mathcal {F}}$ we construct a vector-valued Coleff-Herrera current $\mu $ with support on the variety associated to ${\mathcal {F}}$ such that $\phi $ is in ${\mathcal {I}}$ if and only if $\mu \phi =0$. Such a current $\mu $ can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed. By a construction due to Björk one gets Noetherian operators for ${\mathcal {I}}$ from the current $\mu $. The current $R$ also provides an explicit realization of the Dickenstein-Sessa decomposition and other related canonical isomorphisms.
LA - eng
KW - Coleff-Herrera current; duality; noetherian operators; residue current
UR - http://eudml.org/doc/272556
ER -

References

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