Let $i:X \to \mathbb{P}^n$ be a smooth projective variety defined by the vanishing locus of polynomials $(\underline{f}) = (f_1,\ldots,f_k)$. Recall that we can compute the sheaf cohomology of $\mathcal{O}_X$ using the pushforward $$ H^i(X,\mathcal{O}_X) = H^i(\mathbb{P}^n,i_*(\mathcal{O}_X)) $$ If $X$ is a complete intersection, then this can be computed using the koszul complex $$ H^i(X,\mathcal{O}_X) = H^i(\mathbb{P}^n,K_\bullet(\underline{f})) $$ Does this result extend even to the general case that $X$ is smooth? That is, does $$ H^i(X,\mathcal{O}_X) = \mathbb{H}^i(\mathbb{P}^n, K_\bullet(\underline{f})) $$

Let $i:X \to \mathbb{P}^n$ be a smooth projective variety defined by the vanishing locus of polynomials $(\underline{f}) = (f_1,\ldots,f_k)$ which have degrees $>0$ and are pairwise coprime, meaning $gcd(f_i,f_j) = 1$ for $i \neq j$. Recall that we can compute the sheaf cohomology of $\mathcal{O}_X$ using the pushforward $$ H^i(X,\mathcal{O}_X) = H^i(\mathbb{P}^n,i_*(\mathcal{O}_X)) $$ If $X$ is a complete intersection, then this can be computed using the koszul complex $$ H^i(X,\mathcal{O}_X) = H^i(\mathbb{P}^n,K_\bullet(\underline{f})) $$ Does this result extend even to the general case that $X$ is smooth? That is, does $$ H^i(X,\mathcal{O}_X) = \mathbb{H}^i(\mathbb{P}^n, K_\bullet(\underline{f})) $$