Let Ω⊂$R$^{$n$}be an arbitrary open set. In this paper it is shown that if a Sobolev function$f$∈$W$^{1,$p$}(Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, then$f$is weakly zero on ϖΩ in the sense that$f$∈$W$_{0}^{1,$p$}(Ω).
We give bounds for various homological invariants (including Castelnuovo-Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the non-describing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.
F. BAUERMax Planck Institute for Mathematics in the SciencesF. CHUNGUniversity of California, San DiegoYong LinRenmin University of ChinaYuan LiuInstitute of Computational Mathematics and Scientific/Engineering Computing. Chinese Academy of Sciences
CombinatoricsGeometric Analysis and Geometric Topologymathscidoc:1804.06006
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 145, 2033-2042, 2017.1