Algebraic properties of tensor product of modules over a field

Let A and B be commutative Noetherian algebras over an arbitrary field such that A ⊗_ B is Noetherian. We consider ideals I and J of A and B, respectively, as well as nonzero finitely generated modules L and N over A and B, respectively. In this paper, we investigate certain algebraic properties of the A ⊗_ B-module L⊗_ N, which are often inherited from the properties of the A-module L and the B-module N. Specifically, we provide characterizations for the Cohen-Macaulayness, generalized Cohen-Macaulayness, and sequentially Cohen-Macaulayness of L⊗_ N with respect to the ideal I ⊗_ B + A ⊗_ J, in terms of the corresponding properties for L and N with respect to I and J, respectively.