Locally dualizable modules abound

It is proved that given any prime ideal 𝔭 of height at least 2 in a countable commutative noetherian ring A, there are uncountably many more dualizable objects in the 𝔭-local 𝔭-torsion stratum of the derived category of A than those that are obtained as retracts of images of perfect A-complexes. An analogous result is established dealing with the stable module category of the group algebra, over a countable field of positive characteristic p, of an elementary abelian p-group of rank at least 3.