An Elimination Lemma for Algebras with PBW Bases

Abstract

Let $K$ be a field, and $A=K[a_1,\dots ,a_n]$ a finitely generated $K$-algebra with the PBW $K$-basis $\mathcal{B}=\{a_1^{{\alpha }_1}\cdots a_n^{{\alpha }_n}|({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{N}}^n\}$. 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 $\mathbf{V}(U)\cap L\ne \{0\}$ for every subset $U=\{a_{i_1},\dots ,a_{i_{d+1}}\}\subset \{a_1,\dots ,a_n\}$ with $i_1<i_2<\cdots <i_{d+1}$, where $\mathbf{V}(U)=K$-span$\{a_{i_1}^{{\alpha }_1}\cdots a_{i_{d+1}}^{{\alpha }_{d+1}}|({\alpha }_1,\dots ,{\alpha }_{d+1})\in {\mathbb{N}}^{d+1}\}$. 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 Gröbner 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.

19 pages. The previously missing references [Zei1], [Zei2], [WZ] are added and some minor corrections are made

References in corpus (2)

• A Constructive Characterization of Solvable Polynomial Algebras
• Looking for Groebner Basis Theory for (Almost) Skew 2-Nomial Algebras