Let $A$ be a graded, noetherian ring and $\mathfrak p$ a minimal (minimal in the set of all prime ideals) homogeneous ideal. Is it true that the rings $A_{\mathfrak p}$ and $A_{(\mathfrak p)}$ have the same length?
(Here, $A_{(\mathfrak p)}$ is the set of elements of degree $0$ of $T^{-1} A$, where $T$ is the set of homogeneous elements of $A$ not contained in $\mathfrak p$).
Also, is it true for finitely generated graded modules over $A$?
To give some context, I am doing exercises in algebraic geometry and if this was true then my life would be much easier :)
