Codimension of a projective-variety $Y\subseteq \mathbb P^n_k$

Question

Let $k$ be an algebraically closed field and $Y\subseteq \mathbb P^n_k$ be a projective variety. Let $I(Y)$ be its homogeneous defining ideal in $k[x_0,...,x_n]$ .

Then is it true that codim$(Y)=ht ( I(Y))$ ?

Here, by codimension of a subspace $Y$ of a Topological space $X$ we mean the supremum of the length $e$ of chain of irreducible closed subsets $Y=Y_0 \subset Y_1 \subset ... \subset Y_e=X$ .