The affine line disproves the proper duality formula without properness

The scheme is smooth and pure of dimension one; its local rings are regular and hence CM. It is not proper: after base change to ${\mathbb{A}}_x^1$, the closed subscheme $V(xt-1)\subset {\mathbb{A}}_x^1\times {\mathbb{A}}_t^1$ has image $D(x)$ under projection, which is not closed. Thus the structure map is not universally closed.

By [F1], $H^1(X,F)=0$, whereas ${\mathrm{Ext}}_X^0(F,{\omega }_X)=\mathrm{\Gamma }(X,{\omega }_X)=k[t]dt\mathrm{\ne }0$. The asserted formula in degree $i=1$ would equate these nonzero and zero spaces. This refutes it while retaining smoothness and CM. On an affine nonproper scheme a local dualizing complex still exists; it does not carry the proper global trace-duality representation of the A theorem. The counterexample uses the precise pairing refuted, with ordinary cohomology rather than a compact-support replacement.