Is it true that for any projective variety $X$ that $\dim X = \text{trdeg}_k k(X) = \dim \mathcal O_{X,x}$ for $k$ an algebraically closed field of characteristic zero?
Here we have that $\mathcal O_{X,x}$ is the local ring of $X$ at $x$ defined by $k[X]_{I(x)}$.
I have learned that the equalities hold for standard affine varieties, but am curious if the same holds for projective varieties. Would the proofs be the same / similar?