Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|cite|improve this question
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. –  D. Kelleher Jun 4 '13 at 11:46
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. –  Tom Goodwillie Jun 4 '13 at 11:50
Perhaps my answer here will be helpful:…. –  Paul Siegel Jun 4 '13 at 12:08
Thank you very much. –  vdm123 Jun 4 '13 at 14:29

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.