Comparing cohomologies of $\mathcal{O}_X$ and $\mathcal{O}_{X_\text{red}}$

Question

Let $X$ be a noetherian scheme. If $X_\text{red}$ is affine, then $H^i(X, \mathcal{O}_{X}) \rightarrow H^i(X,\mathcal{O}_{X_\text{red}})$ is an isomorphism for all $i>0$. Is this true even if $X_\text{red}$ is not affine?

Answer 1

This is in general false. For a simple example, take $Y$, the projective line and consider $R=\mathcal{O}_Y\oplus L$ for any line bundle $L$ on $Y$. Then $R$ can be given an $\mathcal{O}_Y$-algebra structure by putting $L\cdot L=0$. Let $X$ be the spec of $R$. Then, you have $\mathcal{O}_X=R$ and $X_{red}=Y$. By choosing $L=\mathcal{O}_Y(-n)$ for $n\geq 2$, you see that $H^1(\mathcal{O}_Y)=0$, but $H^1(\mathcal{O}_X)\neq 0$.