Let $E$ be a holomorphic vector bundle over a compact complex manifold (or projective algebraic variety) $X$.

The Atiyah class of $E$, $a(E)\in Ext^1(T_X,End(E))$, is defined to be the class of the extension $$ 0 \rightarrow End(E) \rightarrow \mathcal{D}(E) \rightarrow T_X \rightarrow 0 $$ where $\mathcal{D}(E)$ is the bundle of differential operators from $E$ to $E$ of order $1$ and scalar symbol, the map to the tangent being the symbol map.

It is a theorem of Atiyah that $a(E)$ generates the characteristic ring of $E$.

My question is: what can be said if $E$ is not a vector bundle, but just a coherent torsion free $\mathcal{O}_X$-module? Could a similar statement be true?

One has anyway the characteristic ring of $E$. To me looks like (although I may be wrong) that one can construct $\mathcal{D}(E)$ that fits the same exact sequence.

The problem is that in Atiyah's theory is essential that $E$ is locally free, since he proves the result through the curvature of connections on $E$, and these does not exist if $E$ is not locally free.

Is there any technique (from K-theory?) that would help? Or my problem is senseless?