Example 1

\* Let $p$ be a graded prime ideal of $\mathbf{C}[X_0, ..., X_n]$ different from the ideal generated by the $X_i$; let $V$ be the algebraic subvariety of $\mathbf{P}^n(\mathbf{C})$ defined by $p$ and $C$ the algebraic subvariety of $\mathbf{C}^{n+1}$ defined by $p$. Then $C$ is the cone with base $V$, $H = \mathbf{C}[X_0, ..., X_n]/p$ is the affine algebra of $C$ and $S^{-1}H$ the local ring of the cone $C$ at its vertex. \* 2) Let $A$ be a local ring and $a$ an ideal of $A$ distinct from $A$. Then $H = \bigoplus_n a^n/a^{n+1}$ is a graded ring such that $H_0 = A/a$ is local; it is generated by $H_0$ and $H_1$. The ring $S^{-1}H$ is therefore local and the filtration $(S^{-1}H_{\geq n})$ is the $S^{-1}H_{\geq 1}$-adic filtration. It should be noted that in general the rings $A$ and $S^{-1}H$ are not isomorphic. \* In particular an algebraic variety is not in general locally isomorphic in a neighbourhood of a point to its tangent cone at that point. \*