Narrow Results

Let $M$ be a finitely generated module of dimension $d$ over a Noetherian local ring $(A,𝔪)$ and $I$ an $𝔪$-primary ideal. Let be a pair of good $I$-filtrations $𝔽$ and ${𝔽}^{\prime }$ of $M$. We show that the Hilbert coefficients $e_i(𝔽)$ are bounded below and above in terms of $i$, $e_0({𝔽}^{\prime }),\dots ,e_i({𝔽}^{\prime })$, and reduction numbers of $𝔽$ and ${𝔽}^{\prime }$, for all $i\ge 1$.

This paper studies certain relations among vertex algebras, vertex Lie algebras and vertex Poisson algebras. In this paper, the notions of vertex Lie algebra (conformal algebra) and vertex Poisson algebra are revisited and certain general construction theorems of vertex Poisson algebras are given. A notion of filtered vertex algebra is formulated in terms of a notion of good filtration and it is proved that the associated graded vector space of a filtered vertex algebra is naturally a vertex Poisson algebra. For any vertex algebra V, a general construction and a classification of good filtrations are given. To each ℕ-graded vertex algebra V=∐_n∈ℕV_(n) with , a canonical (good) filtration is associated and certain results about generating subspaces of certain types of V are also obtained. Furthermore, a notion of formal deformation of a vertex (Poisson) algebra is formulated and a formal deformation of vertex Poisson algebras associated with vertex Lie algebras is constructed.

This paper shows that mixed multiplicities and the multiplicity of Rees modules of good filtrations and that of their reductions are the same. As an application of this result, we obtain interesting results on mixed multiplicities and the multiplicity of Rees modules.