Post #3398351

To deal better with non-smooth varieties X, introduce the 'tangent complex', a sheaf of cochain complexes on X. H⁰ of this is the tangent sheaf. H¹ of it is roughly the space of ways of fundamentally different ways of tweaking the shape of X in an infinitesimal neighborhood of a given point. H¹ vanishes at smooth points. But at singularities, it becomes interesting! For example, H¹ is trivial at all points of the curve y² = x³ except the origin, where it has a cusp. There H¹ is 2-dimensional, because you can tweak the curve using two parameters: y² = x³ + ax + b You can do it in other ways, too - but these are enough to give all 'genuinely different' perturbations of the curve in an infinitesimal neighborhood of the origin, where we don't care about changes of coordinates. (2/n)