Semidualizing DG modules over tensor products (Q2801842)

Let \(k\) be a field, \(A'\), \(A''\) homologically bounded local differential graded \(k\)-algebra and \(M'\) (resp. \(M''\)) differential graded modules over \(A'\) (resp. \(A''\)) such that \(H_i(M')\) (resp. \(H_i(M'')\)) is finitely generated over \(H_0(A')\) (resp. \(H_0(A'')\)) for all \(i\). In the paper under review, as the main result, the author proves thatNEWLINENEWLINE1) \(M'\bigotimes_kM''\) is semidualizing over \(A'\bigotimes_kA''\) if and only if \(M'\) is semidualizing over \(A'\) and \(M''\) is semidualizing over \(A''\);NEWLINENEWLINE2) The map \(\psi:\mathcal{C}(A')\times\mathcal{C}(A'')\rightarrow\mathcal{C}(A'\bigotimes_kA'')\) defined by \(\psi(C',C'')=C'\bigotimes_kC''\) is well defined and injective, where \(\mathcal{C}(A)\) denote the set of shift isomorphism classes of semidualizing differential graded modules over \(A\) in the derived category \(D(A)\).