Tensor products of matrix factorizations

Let K be a field and let f ∈ K[[x₁, x₂,. . . , x_r]] and g ∈ K[[y₁, y₂, . . . , y_s]] be non-zero and non-invertible elements. If X (resp. Y) is a matrix factorization of f (resp. g), then we can construct the matrix factorization X ⊗ Y of f + g over K[[x₁, x₂, . . . , x_r, y₁, y₂, . . . , y_s]], which we call the tensor product of X and Y. After showing several general properties of tensor products, we will prove theorems which give bounds for the number of indecomposable components in the direct decomposition of X ⊗ Y.