Yu Xie
$j$-multiplicity and depth of associated graded modules.
Thu, 01/13/2011 - 16:01 — SomNambulAuthors: Yu Xie, Claudia Polini
Subjects: Commutative Algebra
AbstractLet $R$ be a Noetherian local ring. We define the minimal $j$-multiplicity
and almost minimal $j$-multiplicity of an arbitrary $R$-ideal on any finite
$R$-module. For any ideal $I$ with minimal $j$-multiplicity or almost minimal
$j$-multiplicity on a Cohen-Macaulay module $M$, we prove that under some
residual assumptions, the associated graded module ${\rm gr}_I(M)$ is
Cohen-Macaulay or almost Cohen-Macaulay, respectively.Formulas for the multiplicity of graded algebras.
Thu, 01/13/2011 - 16:01 — SomNambulAuthors: Yu Xie
Subjects: Commutative Algebra
AbstractLet $R$ be a standard graded Noetherian algebra over an Artinian local ring.
Motivated by the work of Achilles and Manaresi in intersection theory, we first
express the multiplicity of $R$ by means of local $j$-multiplicities of various
hyperplane sections. When applied to a homogeneous inclusion $A\subseteq B$ of
standard graded Noetherian algebras over an Artinian local ring, this formula
yields the multiplicity of $A$ in terms of that of $B$ and of local
$j$-multiplicities of hyperplane sections along ${\rm Proj}\,(B)$.
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