Derived equivalences via HRS-tilting (Q2324584)

Let \((\mathcal{T}, \mathcal{F})\) be a torsion pair in an abelian category \(\mathcal{A}\) and let \(\mathcal{B}\subseteq \mathbf{D}^b(\mathcal{A})\) be the corresponding Happel-Reiten-Smalø tilt [\textit{D. Happel} et al., Tilting in abelian categories and quasitilted algebras. Providence, RI: American Mathematical Society (AMS) (1996; Zbl 0849.16011)]. In this article, the authors give necessary and sufficient conditions for a realisation functor \(G: \mathbf{D}^b(\mathcal{B}) \to \mathbf{D}^b(\mathcal{A})\) (that is, a triangle functor that restricts to the inclusion on \(\mathcal{B}\) [\textit{A. Beilinson}, Lect. Notes Math. 1289, 27--41 (1987; Zbl 0652.14008)], [A. Beilinson et al., {Faisceaux pervers}, Astérisque 100, (1982; Zbl 0536.14011)]) to be an equivalence. Namely, they show that \(G\) is an equivalence if and only if \(\mathcal{A}\) is in the essential image of \(G\) if and only if the canonical maps \(Yext^2_\mathcal{B}(X, Y) \to \mathbf{D}^b(X, \Sigma^2Y)\) are isomorphisms for all \(X, Y \in \mathcal{B}\) if and only if every object \(A \in \mathcal{A}\) fits into an exact sequence \[ 0 \to F^0 \to F^1 \to A \to T^0 \to T^1 \to 0 \] with \(F^i \in \mathcal{F}, T^i \in \mathcal{T}\) and such that the corresponding class in the Yoneda extension group \(Yext^3_\mathcal{A}(T^1, F^0)\) vanishes (see Theorem 3.4). The final condition is interesting since it does not depend on \(\mathcal{B}\) or \(G\) and is therefore intrinsic. The authors demonstrate their result by providing an example of a torsion pair in an abelian category which is neither splitting nor (co)tiling nor given by a two-term tiling complex that satisfies this final condition (see Example 4.5). The authors also show that the second condition in the above holds, somewhat surprisingly, in more general settings (see Theorem 2.9): Let \(\mathcal{A}\) be the heart of a bounded t-structure in a triangulated category \(\mathcal{D}\) and let \(F: \mathbf{D}^b(\mathcal{A}) \to \mathcal{D}\) be a realisation functor, then \(F\) is an equivalence if and only if it is dense. Interesting intermediate results study when an additive functor \(F: \mathcal{A} \to \mathcal{A}'\) between abelian categories is fully faithful or an equivalence by how \(F\) behaves with respect to the torsion pair \((\mathcal{T}, \mathcal{F})\) (see Lemmas 2.10 and 2.11). In Section 4, examples and applications of the main results relating to splitting torsion pairs, TTF triples and two-term silting complexes are provided.