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

Let $A=K<X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n>\mathrm{/}<\mathcal{G}>$ be a $K$-algebra defined by a finite Gr"obner basis $\mathcal{G}$. It is shown how to use the Ufnarovski graph $\Gamma ({\bf LM}({\cal G}))$andthegraphof$n$-chains$\Gamma_{\rm C}({\bf LM}({\cal G}))$tocalculategl\mathrm{.}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 $K<X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n>$.