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Given a smooth variety $X$, one can define the cotangent sheaf $\Omega_X$, the canonical sheaf $\omega_X$ and the deRham complex $\Omega_X^\bullet$. These three object has obvious relations.

For general variety $X$ (say over $\mathbb{C}$), one generalize cotangent sheaf to cotangent complex $L_X$ of Illusie, canonical sheaf to dualization comlex and deRham complex to complex of Deligne-Du Bois (I don't know if this is the standard terminology, which is defined to be the derived pushforward of the deRham complex of a cubical hyperresolution. [c.f. below Theorem 7.22, Mixed Hodge structures by Peters and Steenbrink]).

My question is for general $X$, Is there relation between dualization complex, cotangent complex and Deligne-Du Bois complex?

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Here is a counterexample that shows the relationship doesn't hold in general.

For the variety $X = {\rm Spec}~\mathbb C [x,y]/xy$, the cotangent complex is just the usual module of differentials $$L_X = \frac{\mathcal O_X dx \oplus \mathcal O_X dy}{ y ~dx + x ~dy},$$ but since $X$ is a complete intersection (and therefore Gorenstein), the dualizing complex is a line bundle $\omega_X = \mathcal O_X[1]$

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