Tensor products of matrix factorizations

Let K be a field and let f is an element of K[[x(1), x(2), ...,x(r)]] and g is an element of K[[y(1), y(2), ..., 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 x over cap Y of f + g over K[[x(1 ), x(2), ...,x(r), y(1), y(2), ..., y(s)]], which we call the tensor produc t of X and Y. After showing several general properties of tenser products, we will prove theorems which give bounds for the number of indecomposable components in t he direct decomposition of X x over cap Y.