Tensor products of matrix factorizations

Abstract

Abstract.Let K be a field and let f ∈ K[[x1, x2,…,xr]] and g ∈ K[[y1, y2,…,ys]] 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,…,xr, y1, y2,…,ys]], 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.