Note
A smooth presentable category that is not compactly generated
Abstract
We construct a dualizable presentable stable -linear category whose coevaluation is an internal left adjoint, but which is not compactly generated. The category is the kernel of extension of scalars along a quotient of a skew Laurent algebra. Its defining ideal is flat and idempotent, and is perfect as a bimodule. A finite resolution obtained from the action on the fractional powers proves the bimodule perfectness. The classical division ring of fractions is a nonzero object of the kernel orthogonal to every compact object. This gives a counterexample to Ramzi’s Conjecture 0.6.