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

Let $A=K< X_1,...,X_n> /< {\cal G}>$ be a $K$-algebra defined by a finite Gr\"obner basis ${\cal G}$. It is shown how to use the Ufnarovski graph $\Gamma ({\bf LM}({\cal G}))$ and the graph of $n$-chains $\Gamma_{\rm C}({\bf LM}({\cal G}))$ to calculate gl.dim$G^{\mathbb{N}}(A)$ and gl.dim$\widetilde{A}$, where $G^{\mathbb{N}}(A)$, respectively $\widetilde{A}$, 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 $K< X_1,...,X_n>$.