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The cotangent bundle $\Omega_X$ of a smooth scheme $X$ shows up in two places in my understanding of algebraic geometry. The first is deformation theory, where maps out of $\Omega_X$ control the infinitesimal automorphisms, and deformations, of $X$. The second is in Serre duality, where $\det \Omega_X$ serves as the dualizing sheaf and maps into it recover the duals of cohomology groups of coherent sheaves.

The relationship between duality and deformations persists at least as far as the local complete intersection case, where the dualizing sheaf is still the determinant of the cotangent complex.

What do these two interpretations of the cotangent bundle have to do with each other? Can you formulate a relationship that does not require on an explicit construction of the cotangent bundle?

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    $\begingroup$ Verdier's article on coherent duality ("twisted inverse image") in the 1968 Bombay conference proceedings deduces the description of the dualizing sheaf in the smooth case from the lci property of the diagonal, in effect ultimately coming down to the construction of $\Omega^1$ in terms of the 1st-order neighborhood of the diagonal. This is not too far-removed from the reason $\Omega^1$ relates to derivations and hence to 1st-order deformations. So one might try to begin by looking at Verdier's paper. $\endgroup$
    – nfdc23
    Commented Jul 19, 2017 at 2:25

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