What does a projective resolution mean geometrically?
For R a commutative ring and M an R-module, we can always find a projective resolution of M which replaces M by a sequence of projective R-modules. But as R is commutative, we can consider the affine variety X=Spec R and the sheaf of modules associated to M. What is the projective resolution doing geometrically to this sheaf?
Projectives are locally free sheaves, so if M itself is not projective then it must have some sort of "sharp twisting" or "pinching". In some way a projective resolution is "un-pinching" M. Geometrically, is this the same "un-pinching" that happens in a resolution of a singularity of a variety? Is there an example in low dimensions where one can actually draw a picture of this happening for modules?