Gluing of categories and Krull-Schmidt partners (Q2832514)

Let \(\mathcal{A}\) and \(\mathcal{B}\) be DG categories, \(S, T\) be DG \(\mathcal{A}\)-\(\mathcal{B}\)-bimodules. and \(\phi: S\rightarrow T\) be the morphism of DG \(\mathcal{A}\)-\(\mathcal{B}\)-bimodules, and let \(R=\mathrm{Cone}(\phi)\) be the cone of \(\phi\). Any morphism \(\phi: S\rightarrow T\) of \(\mathcal{A}\)-\(\mathcal{B}\)-bimodules induces a \(\mathcal{B}\)-\(\mathcal{C}\)-bimodule. In the paper under review is studying the semi-orthogonal decomposition of upper triangular DG category \(\mathcal{C}=\mathcal{A}\llcorner\mathcal{B}\). The main result is the following:NEWLINENEWLINE(1) Suppose that for any DG \(\mathcal{B}\)-modules \(M\) and \(N\) the following condition holds NEWLINE\[NEWLINE\mathrm{Hom}_{\mathbb D{\mathcal{A}}}(M\otimes^{L}R, N\otimes^{L}T)=0NEWLINE\]NEWLINE Then the derived functor \(-\otimes^{L}\tilde{T}:\mathbb D{\mathcal{A}}\longrightarrow\mathbb D{\mathcal{A}\llcorner\mathcal{B}}\) is fully faithful, and it induces a semi-orthogonal decomposition \(\mathbb D{\mathcal{A}\llcorner\mathcal{B}}=\langle \mathbb D{\mathcal{E}}, \mathbb D{\mathcal{B}}\rangle\) for some small DG category \(\mathcal{E}\).NEWLINENEWLINE(2) If the functors \(-\otimes^{L}\tilde{T}\) and \(\mathrm{RHom}(\tilde{T},-)\) send perfect modules to perfect modules, then there is a semi-orthogonal decomposition of the form \(\mathrm{Perf}{\mathcal{A}\llcorner\mathcal{B}}=\langle \mathrm{Perf}{\mathcal{E}}, \mathrm{Perf}{\mathcal{B}}\rangle\) for some small DG category \(\mathcal{E}\).