Lemma of the day

Let R —> A and R —> B be ring maps. In general there does not exist a functor T : D(B) —> D(B ⊗R A) of triangulated categories such that a B-module M gives an object T(M) of D(B ⊗R A) which maps to M ⊗LR A under the map D(B ⊗R A) —> D(A). See Tag 08J2.