Let $X$ be a scheme and conisder the bounded derived category $D^b(X)=D^b(\textbf{Coh}(X))$.

Assume that $D^b(X)\cong D^b(Y)$ then $\dim X=\dim Y$ by Serre duality when $X$ and $Y$ are smooth projective varieties.

I wonder if we expect this is to be true for more general schemes?