Suppose $X$ is a projective variety in $\Bbb{P}^n$ of dimension $k$. Everything is over a field. We say $X$ is a scheme-theoretic complete intersection if $X$ can be written as $V_+(f_1) \cap \ldots \cap V_+(f_{n-k})$ hypersurfaces where the intersection here is as schemes. What I mean by this is if $U_i$ are the standard affine covers of $\Bbb{P}^n$, then $X \cap U_i$ should be the scheme $$V\Big( (f_1(x_0,\ldots, \hat{x_i},\ldots,x_n)) + \ldots + (f_{n-k}(x_0,\ldots,\hat{x_i},\ldots,x_n)) \Big).$$
Alternatively we could say $X$ is a ideal theoretic complete intersection if its homogeneous ideal $I(X) := \Gamma_\ast(\mathscr{I}_X)$ is generated by $n-k$ polynomials.
My question is: Are these two concepts equivalent?