Post #3398352

2026-06-25 11:14 UTC

"Derived stacks" are a powerful modern framework for geometry. But what's the idea behind them? Proceeding in steps: A variety or scheme X has a tangent space at each point, and these assemble to form the tangent sheaf TX. If X is smooth is locally free: a vector bundle. But at singularities, the dimension of the tangent space jumps up! For example, the tangent space of this curve y² = x³ has dimension 1 except at the origin, where it jumps up to 2. (1/n)

Replies (1)

@johncarlosbaez@mathstodon.xyz 2026-06-25 11:42

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)

Open ##3398351