Let d be the left homological dimension or weak dimension. In the present paper we give for a large class of algebras the answer to the question: when the dimension of the tensor product of algebras is the sum of dimensions of the multiples, Tensor d-minimal algebras are introduced and it is proved that a large class of algebras are tensor d-minimal algebras.

We obtain some methods to construct a (strongly) proper resolution (resp. coproper coresolution) of one end term in a short exact sequence from that of the other two terms. By using this method, we prove that for a left and right Noetherian ring R, RR satisfies the Auslander condition if and only if so does every flat left R-module, if and only if the injective dimension of the ith term in a mi...

We provide an example, which shows that studying homological and homotopical properties of cobordisms between graph manifolds is not enough to prove the mu-constant conjecture of Lê Dũng Tráng in complex dimension 2.