It is known that over a smooth variety, $X$, the classical Atiyah class of a coherent sheaf, given by the extension $\text{Ext}^{1}(F, F\otimes \Omega_X)$, satisfies the Leibniz-type property. That is $$At(F\otimes G)= At(F)\otimes 1_G+ 1_{F} \otimes At(G).$$ In the non-smooth case the truncated Atiyah class of a sheaf (defined by Huybrechts and Thomas) is given as a morphism in the derived category$$\text{Ext}^1_{D(X)}(F, F\otimes^{L}\mathbb{L}^{\bullet}_{X})$$where $\mathbb{L}^{\bullet}_{X}$ is defined as a truncation of Illusie's cotangent complex of $X$$$\mathbb{L}^{\bullet}_{X}:= \tau^{\geq -1} L^{\bullet}_X.$$Are there cases where the Leibniz property is satisfied for truncated Atiyah classes?


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