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I'm reading section 8 Differentials of chapter 2 in Hartshorne. It's is extremely hard to me to understand the nature of the definitions: module of relative differential forms - sheaf of relative differential - tangent sheaf.

It's said that "the sheaf of differential forms is essentially the same as the dual of tangent bundle defined in differential geometry".

I want to understand very explicitly how these constructions come from Diff.Geometry?

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    $\begingroup$ Warner's book "Foundations of differential geometry and Lie groups" covers sheaves from the differential geometry point of view. It is a classic that covers the basics of differential geometry, so it may be a good place to start. $\endgroup$
    – D. Kelleher
    Jun 4, 2013 at 11:46
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    $\begingroup$ For smooth manifolds there are several ways of thinking about (co)tangent spaces and bundles and sheaves. Some, but not all, of these ways of thinking can easily be adapted to algebraic geometry. To get the idea, I suggest thinking about what the definitions that you are reading about in Hartshorne would give if you applied them to $C^\infty$ functions on a manifold. $\endgroup$
    – Tom Goodwillie
    Jun 4, 2013 at 11:50
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    $\begingroup$ Perhaps my answer here will be helpful: mathoverflow.net/questions/88880/…. $\endgroup$
    – Paul Siegel
    Jun 4, 2013 at 12:08

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