An Elimination Lemma for Algebras with PBW Bases

Abstract

Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a finitely generated $K$-algebra with the PBW $K$-basis ${\cal B}=\{a_{1}^{\alpha_1}\cdots a_{n}^{\alpha_n}~|~(\alpha_1,\ldots ,\alpha_n)\in\mathbb{N}^n\}$. It is shown that if $L$ is a nonzero left ideal of $A$ with GK.dim$(A/L)=d<n$ ($=$ the number of generators of $A$), then $L$ has the {\it elimination property} in the sense that ${\bf V}(U)\cap L\ne \{0\}$ for every subset $U=\{ a_{i_1},\ldots ,a_{i_{d+1}}\}\subset\{a_1,\ldots ,a_n\}$ with $i_1<i_2<\cdots <i_{d+1}$, where ${\bf V}(U)=K$-span$\{a_{i_1}^{\alpha_1}\cdots a_{i_{d+1}}^{\alpha_{d+1}}~|~(\alpha_1,\ldots ,\alpha_{d+1})\in\mathbb{N}^{d+1}\}$. In terms of the structural properties of $A$, it is also explored when the condition GK.dim$(A/L)<n$ may hold for a left ideal $L$ of $A$. Moreover, from the viewpoint of realizing the elimination property by means of Gr\"obner bases, it is demonstrated that if $A$ is in the class of binomial skew polynomial rings [G-I2, Serdica Math. J., 30(2004)] or in the class of solvable polynomial algebras [K-RW, J. Symbolic Comput., 9(1990)], then every nonzero left ideal $L$ of $A$ satisfies GK.dim$(A/L)<$ GK.dim$A=n$ ($=$ the number of generators of $A$), thereby $L$ has the elimination property.