Residue Currents and their Annihilator Ideals
Doktorsavhandling, 2007

This thesis presents results in multidimensional residue theory. From a generically exact complex of locally free analytic sheaves $\mathcal C$ we construct a vector valued residue current $R^\mathcal C$, which in a sense measures the exactness of $\mathcal C$. If $\mathcal C$ is a locally free resolution of the ideal (sheaf) $J$ the annihilator ideal of $R^\mathcal C$ is precisely $J$. This generalizes the Duality Theorem for Coleff-Herrera products of complete intersection ideals and can be used to extend several results, previously known for complete intersections. We compute $R^\mathcal C$ explicitly if $\mathcal C$ is a so called cellular resolution of an Artinian monomial ideal $J$, and relate the structure of $R^\mathcal C$ to irreducible decompositions of $J$. If $\mathcal C$ is the Koszul complex associated with a set of generators $f$ of the ideal $J$ the entries of $R^\mathcal C$ are the residue currents of Bochner-Martinelli type of $f$, which were introduced by Passare, Tsikh and Yger. We compute these in case $J$ is an Artinian monomial ideal and conclude that the corresponding annihilator ideal is strictly included in $J$, unless $J$ is a complete intersection. We also define products of residue currents of Bochner-Martinelli type, generalizing the classical Coleff-Herrera product, and show that if $f$ defines a complete intersection the product of the residue currents of Bochner-Martinelli type of subtuples of $f$ coincides with the residue current of Bochner-Martinelli type of $f$.

coherents sheaves

ideals of holomorphic functions

monomial ideals

Bochner-Martinelli formula

free resolutions of modules

residue currents

cellular resolutions

Euler, Matematiska Vetenskaper, Chalmers Tvärgata 3, Chalmers tekniska högskola
Opponent: Mattias Jonsson, Department of Mathematics, University of Michigan, USA och Institutionen för matematik, KTH

Författare

Elizabeth Wulcan

Chalmers, Matematiska vetenskaper, Matematik

Göteborgs universitet

Residue currents of monomial ideals

Indiana University Mathematics Journal,; Vol. 56(2007)p. 365-388

Artikel i vetenskaplig tidskrift

Products of residue currents of Cauchy-Fantappiè-Leray type

Arkiv for Matematik,; Vol. 45(2007)p. 157-178

Artikel i vetenskaplig tidskrift

Ämneskategorier

Matematik

ISBN

978-91-7291-922-8

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 2603

Euler, Matematiska Vetenskaper, Chalmers Tvärgata 3, Chalmers tekniska högskola

Opponent: Mattias Jonsson, Department of Mathematics, University of Michigan, USA och Institutionen för matematik, KTH