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We define a tensor product of linear sites, and a resulting tensor product ofGrothendieck categories based upon their representations as categories oflinear sheaves. We show that our tensor product is a special case of the tensorproduct of locally presentable linear categories, and that the tensor productof locally coherent Grothendieck categories is locally coherent if and only ifthe Deligne tensor product of their abelian categories of finitely presentedobjects exists. We describe the tensor product of non-commutative projectiveschemes in terms of Z-algebras, and show that for projective schemes our tensorproduct corresponds to the usual product scheme.

On the tensor product of linear sites and Grothendieck categories