Chern Classes via Derived Determinant

Motivated by the Chern-Weil theory, we prove that for a given vector bundle $E$ on a smooth scheme $X$ over a field $k$ of any characteristic, the Chern classes of $E$ in the Hodge cohomology can be recovered from the Atiyah class. Although this problem was solved by Illusie in \cite{i}, we present another proof by means of derived algebraic geometry. Also, for a scheme $X$ over a field $k$ of characteristic $p$ with a vector bundle $E$ we construct elements $c^{cris}_n (E, \alpha(E)) \in H_{dR}^{2n} (X)$ using an obstruction $\alpha(E)$ to a lifting of $F^* E$ to a crystal modulo $p^2$ and prove that $c^{cris}_n (E, \alpha(E)) = n! \cdot c_{n}^{dR} (E)$, where $c_{n}^{dR} (E)$ are the Chern classes of $E$ in the de Rham cohomology and $F$ is the Frobenius map.