Cohen-Macaulay, Gorenstein, complete intersection and regular defect for the tensor product of algebras

The main goal of this paper is to measure the defect of Cohen-Macaulay, Gorenstein, complete intersection and regularity for the tensor product of algebras over a ring. For this sake, we determine the homological invariants which are inherent to these notions, such as the Krull dimension, depth, injective dimension, type and embedding dimension of the tensor product constructions in terms of those of their components. Our results allow to generalize various theorems in this topic especially [4, Theorem 2.1], [21, Theorem 6] and [14, Theorems 1 and 2] as well as two Grothendieck's theorems on the transfer of Cohen-Macaulayness and regularity to tensor products over a field issued from finite field extensions. To prove our theorems on the defect of complete intersection and regularity, the homology theory introduced by André and Quillen for commutative rings turns out to be an adequate and efficient tool in this respect.