Parametrizing torsion pairs in derived categories

Abstract

We investigate parametrizations of compactly generated t-structures, or more generally, t-structures with a definable coaisle, in the unbounded derived category D ( M o d - A ) \mathrm {D}({\mathrm {Mod}}\text {-}A) of a ring A A . To this end, we provide a construction of t-structures from chains in the lattice of ring epimorphisms starting in A A , which is a natural extension of the construction of compactly generated t-structures from chains of subsets of the Zariski spectrum known for the commutative noetherian case. We also provide constructions of silting and cosilting objects in D ( M o d - A ) \mathrm {D}({\mathrm {Mod}}\text {-}A) . This leads us to classification results over some classes of commutative rings and over finite dimensional hereditary algebras.

Topological endomorphism rings of tilting complexes

“…If $$C$$ is in addition a cotilting object in $$\mathsf {D}(R\mbox{-}\mathsf {Mod})$$, we call it a cotilting complex. Finally, we say that a cosilting complex is of cofinite type if the associated cosilting t ‐structure is compactly generated (see [4, section 2.6] for details).…”

“…We will also need another duality functor $$(-)^*:= \operatorname{\mathbf {R}\mathsf {Hom}}_{R}(-,R)$$, which induces an equivalence $$(\mathsf {D}{(R\mbox{-}\mathsf {Mod})^{\mathsf {c}})}^\mathsf {op}\xrightarrow {\cong }\mathsf {D}(\mathsf {Mod}\mbox{-}R)^\mathsf {c}$$ on the categories of compact objects. See [4, section 2.2] for details about these dualities, and also the recent preprint [15] where the triangulated character duality is developed in larger generality. By symmetry, we also freely consider both functors going in the opposite direction.…”

“…We recall that $$f^* = \operatorname{\mathbf {R}\mathsf {Hom}}_{R}(f,R)$$ and $$\Phi ^*$$ is a set of maps between compact objects of $$\mathsf {D}(R\mbox{-}\mathsf {Mod})$$. For any map $$f$$ in $$\mathsf {D}(\mathsf {Mod}\mbox{-}R)^\mathsf {c}$$ and any $$X \in \mathsf {D}(\mathsf {Mod}\mbox{-}R)$$, we have an equivalence $$$$\begin{equation*} \operatorname{\mathsf {Hom}}_{\mathsf {D}(\mathsf {Mod}\mbox{-}R)}(f,X) \text{ is zero} \iff \operatorname{\mathsf {Hom}}_{\mathsf {D}(R\mbox{-}\mathsf {Mod})}(f^*,X^+) \text{ is zero}, \end{equation*}$$$$see the proof of [4, Lemma 2.3]. Plugging $$X = Y$$, we obtain that the set $$\Phi ^*$$ of maps in $$\mathsf {D}(R\mbox{-}\mathsf {Mod})^\mathsf {c}$$ defines …”

“…Furthermore, let $$f: R \rightarrow R[y^{-1}]$$ be the localization map. Recall from [6, Proposition 1.3] and [4, Example 4.14(5)] that we can chose $$T = R[y^{-1}] \oplus \mathsf {Cone}(f)$$ (see also [4, Proposition 5.15]). We claim that the map $$g: T^{(\omega)} \rightarrow T^\omega$$ is not a monomorphism in $$\mathcal {H}_T$$, or equivalently, that $$\mathsf {Cone}(g) \not\in \mathcal {H}_T$$.…”

“…By3.3, 2.13, we have for any $$X \in \mathsf {Def}_{\mathsf {D}(\mathsf {Mod}\mbox{-}R)}(T)$$ that $$X^{++} \in \mathcal {H}_T$$. As a consequence, any pure‐injective object from $$\mathsf {Def}_{\mathsf {D}(\mathsf {Mod}\mbox{-}R)}(T)$$ belongs to $$\mathcal {H}_T$$, see [4, Corollary 2.8]. Now let $$X \in \mathsf {Def}_{\mathsf {D}(\mathsf {Mod}\mbox{-}R)}(T)$$ be any object.…”