Extensions of rings and tilting complexes (Q1910727)

Let a ring \(\Lambda\) be an extension of a subring \(A\), and \(T_A\) be a tilting \(A\)-module, then there are many results studying necessary and sufficient conditions in order that \(T \otimes_A \Lambda\) be a tilting \(\Lambda\)-module, and linking the endomorphism algebras of these two modules [see, for instance, the author, Commun. Algebra 13, 1319-1326 (1985; Zbl 0567.16010)] or the reviewer and \textit{N. Marmaridis} [``Tilting modules over split-by-nilpotent extensions'' (to appear)]. The aim of this paper is to generalise these results to the case of tilting complexes [in the sense of \textit{J. Rickard}, J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989; Zbl 0672.16034)]. The author shows that, if \(\Lambda\) is a split extension of \(A\) and \(T^\bullet\) is an object of \(D^- (\text{Mod }A)\), then \(T^\bullet \otimes^L_A \Lambda\) is a tilting complex for \(\Lambda\) if and only if \(T^\bullet\) is a tilting complex for \(A\), and \(\text{Hom}_{D(\text{Mod }A)} (T^\bullet, T^\bullet \otimes^L_A \Lambda [i])=0\) for all \(i \neq 0\). Moreover, in this case, \(\text{End}_{D(\text{Mod } \Lambda)} (T^\bullet \otimes^L_A \Lambda)\) is a split extension of \(\text{End}_{D(\text{Mod } A)} (T^\bullet)\). In the second part, the author considers Frobenius extensions: a ring \(\Lambda\) is called a Frobenius extension of a subring \(A\) if \(\Lambda_A\) is finitely generated projective, and \(_A \Lambda_\Lambda \cong \text{Hom}_A (_\Lambda \Lambda_A,{_AA_A})\) as \(A\)-\(\Lambda\)-bimodules. He shows that, if \(\Lambda\) is a Frobenius extension of \(A\) such that \(0 \to A \to \Lambda \to M \to 0\) is an exact sequence of \(A\)-\(A\)-bimodules, and \(T^\bullet\) is a tilting complex for \(A\) such that \(\text{Hom}_{D (\text{Mod }A)} (T^\bullet, T^\bullet \otimes^L_A M[i])=0\) for all \(i \leq 0\), then \(T^\bullet \otimes^L_A \Lambda\) is a tilting complex for \(\Lambda\), and \(\Gamma=\text{End}_{D (\text{Mod } \Lambda)} (T^\bullet \otimes^L_A \Lambda)\) is a Frobenius extension of \(B=\text{End}_{D(\text{Mod } A)} (T^\bullet)\) by \(N=\text{Hom}_{D(\text{Mod }\Lambda)} (T^\bullet, T^\bullet \otimes^L_A M)\) such that \(0 \to B \to \Gamma \to N \to 0\) is an exact sequence of \(B\)-\(B\)-bimodules.