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Pete I agree that locally free is a good geometric intuition for projective. In my algebra class I gave an exercise for the students to prove that the module of tangent vector fields on a 2-sphere is locally free over the ring of smooth functions, and challenged them to show it is not free.

If you introduce direct limits, maybe the fact you mentioned about flatness being preserved is a way to motivate it. I.e. you want to take direct limits of finite generated projectives, but they lose the property of projectivity. So what property do they keep?

But that may be artificial. it is hard to get away from the points made by Sean Tilson. I guess the difficulty there is that Hom is for many of us more intuitive than tensor. So lifting and extension problems are more intuitive than the distinction between IM and ItensorM.

You might review your reasons why flat modules are in the course. If you have an application for them in mind, maybe the application can motivate the concept.

In the spirit of "flat families", I used to cling to the geometric fact that a surjective morphism of smooth algebraic varieties is flat if and only if the fiber dimension is constant, but I don't know if you can use that. Those "local criteria for flatness" are usually among the more advanced flatness topics. E.g. that is treated in the books of Matsumura near the end.

But this suggests that in some contexts flatness is the algebraic version of locally constant geometry.

I have heard that Serre focused on the concept in some of his work relating algebraic and analytic varieties. Maybe the properties he used are illuminating.

Great question.

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Pete I agree that locally free is a good geometric intuition for projective. In my algebra class I gave an exercise for the students to prove that the module of tangent vector fields on a 2-sphere is locally free over the ring of smooth functions, and challenged them to show it is not free.

If you introduce direct limits, maybe the fact you mentioned about flatness being preserved is a way to motivate it. I.e. you want to take direct limits of finite generated projectives, but they lose the property of projectivity. So what property do they keep?

But that may be artificial. it is hard to get away from the points made by Sean Tilson. I guess the difficulty there is that Hom is for many of us more intuitive than tensor. So lifting and extension problems are more intuitive than the distinction between IM and ItensorM.

You might review your reasons why flat modules are in the course. If you have an application for them in mind, maybe the application can motivate the concept.

In the spirit of "flat families", I used to cling to the geometric fact that a morphism of smooth algebraic varieties is flat if and only if the fiber dimension is constant, but I don't know if you can use that. Those "local criteria for flatness" are usually among the more advanced flatness topics. E.g. that is treated in the books of Matsumura near the end.

But this suggests that in some contexts flatness is the algebraic version of locally constant geometry.

I have heard that Serre focused on the concept in some of his work relating algebraic and analytic varieties. Maybe the properties he used are illuminating.

Great question.