Some remarks on Beilinson adeles

Let $X$ be a scheme of finite type over a field $k$. Denote by $\cal {A}^{{\textstyle \cdot }}_X$ the sheaf of Beilinson adeles with values in the algebraic De Rham complex $\Omega ^{{\textstyle \cdot }}_{X/k}$. Then $\Omega ^{{\textstyle \cdot }} _{X/k}\rightarrow \cal {A}^{{\textstyle \cdot }}_X$ is a flasque resolution. So if $X$ is smooth, $\cal {A}^{{\textstyle \cdot }}_X$ calculates De Rham cohomology. In this note we rewrite the proof of Deligne-Illusie for the degeneration of the Hodge spectral sequence in terms of adeles. We also give a counterexample to show that the filtration $\cal {A}^{{\textstyle \cdot },\geq q}_X$ does not induce Hodge decomposition.