  The aim of this work is to give an algebraic weak version of the
Atiyah-Singer index theorem. We compute then a few small examples with the
elliptic differential operator of order $\leq 1$ coming from the Atiyah class
in $\mathrm{Ext}^1_{\mathcal{O}_X}(\mathcal{O}_X,\Omega^1_{X/k})$, where $X
\longrightarrow \mathrm{Spec}(k)$ is a smooth projective scheme over a perfect
field $k$.
