Tangent category theory is a categorical framework for differential geometry. In recent work, tangent categories were also employed to study algebraic geometry. In particular, the category of affine schemes is a tangent category whose tangent bundle is given by Kahler differentials.

How special is this tangent structure? Is there any other tangent structure on the category of affine schemes?

In this talk, we classify the representable tangent structures in the category of affine schemes by introducing the new notion of tangentoids. When the base ring R is a principal ideal domain, we show there are only two such tangent structures: the aforementioned one and the trivial one. We also show that when R is not a PID, we have other non-trivial representable tangent structures.