Conditions under which the tensor product of algebraifolds is an algebraifold

Determine necessary and sufficient conditions on k-algebraifolds A and B ensuring that the algebraic tensor product A \otimes_k B is again a k-algebraifold, i.e., that Der_k(A \otimes_k B) is a finitely generated projective A \otimes_k B-module.

Background

The tensor product A \otimes_k B is a natural candidate for the categorical product in the algebraifold setting. The paper proves that when k is a field and one of A or B is finitely generated as a k-algebra, A \otimes_k B is an algebraifold and has the desired universal property.

Beyond this case, the authors do not know general conditions guaranteeing the tensor product remains an algebraifold.