On the Calculation of gl.dim$G^{\mathbb{N}}(A)$ and gl.dim$\tilde{A}$ by Using Gröbner Bases

Abstract

Let $A=K<X_1,...,X_n>/<\mathcal{G}>$ be a $K$-algebra defined by a finite Gröbner basis $\mathcal{G}$. It is shown how to use the Ufnarovski graph $\Gamma (\mathbf{LM}(\mathcal{G}))$ and the graph of $n$-chains ${\Gamma }_{\mathrm{C}}(\mathbf{LM}(\mathcal{G}))$ to calculate gl.dim$G^{\mathbb{N}}(A)$ and gl.dim$\tilde{A}$, where $G^{\mathbb{N}}(A)$, respectively $\tilde{A}$, is the associated $\mathbb{N}$-graded algebra of $A$, respectively the Rees algebra of $A$ with respect to the $\mathbb{N}$-filtration $FA$ of $A$ induced by a weight $\mathbb{N}$-grading filtration of $K<X_1,...,X_n>$.

14 pages