Definable coaisles over rings of weak global dimension at most one

Abstract

In the setting of the unbounded derived category D(R) of a ring R of weak global dimension at most one we consider t-structures with a definable coaisle. The t-structures among these which are stable (that is, the t-structures which consist of a pair of triangulated subcategories) are precisely the ones associated to a smashing localization of the derived category. In this way, our present results generalize those of [8] to the non-stable case. As in the stable case [8], we confine for the most part to the commutative setting, and give a full classification of definable coaisles in the local case, that is, over valuation domains. It turns out that, unlike in the stable case of smashing subcategories, the definable coaisles do not always arise from homological ring epimorphisms. We also consider a non-stable version of the Telescope Conjecture for t-structures and give a ring-theoretic characterization of the commutative rings of weak global dimension at most one for which it is satisfied.

Parametrizing torsion pairs in derived categories

“…Then it is straightforward to check that the t-structure (U p , V p ) is obtained from the chain [20,Theorem 8.7] that the t-structure (U, V) is compactly generated in D(Mod-A) if and only if (U p , V p ) is compactly generated in D(Mod-A p ) for all p ∈ Spec (R). By [20, Theorem 9.4], the t-structure (U p , V p ) is compactly generated in D(Mod-A p ) if and only if the chain…”

“…20 ([14]). A silting module T over a hereditary ring is said to be minimal if there is an exact sequence (4.20.1)A f −→ T 0 → T 1 → 0such that f is a Gen T -envelope and T 1 lies in Add T .…”

Weak global dimension and CM-freeness of rings

“…设 R 的弱整体维数不超过 1. Bazzoni 和 Hrbek [17] 用 R-Mod 中的可定义子范畴刻画了 D(R-Mod) 中可定义子范畴. 下述结论运用这一事实刻画一些同伦子范畴和相对导出范畴的可定义子 范畴.…”

“…尽管 Keller [16] 指出在一般环上 D(R-Mod) 并不满足这一猜想, Neeman、Krause、 St ovícek、 Bazzoni 和 Hrbek 等学者还是运用不同的方法研究了 D(R-Mod) 满足 Telescope 猜想的条件. 事实上, Bazzoni 和 Hrbek [17] 给出了弱整体维数不超过 1 的环上 D(R-Mod) 中可定义子范畴的分类, 并 由此说明在 von Neumann 正则环上 D(R-Mod) 满足 Telescope 猜想. 结合推论 C 和文献 [17] 中的结 论, 我们在 R 弱整体维数不超过 1 时完全分类了推论 C 中任一三角范畴中的可定义子范畴 (参见 推论 4.12); 并说明推论 C 中的三角范畴在 von Neumann 正则环上均满足 Telescope 猜想 (参见 推论 4.13).…”

“…事实上, Bazzoni 和 Hrbek [17] 给出了弱整体维数不超过 1 的环上 D(R-Mod) 中可定义子范畴的分类, 并 由此说明在 von Neumann 正则环上 D(R-Mod) 满足 Telescope 猜想. 结合推论 C 和文献 [17] 中的结 论, 我们在 R 弱整体维数不超过 1 时完全分类了推论 C 中任一三角范畴中的可定义子范畴 (参见 推论 4.12); 并说明推论 C 中的三角范畴在 von Neumann 正则环上均满足 Telescope 猜想 (参见 推论 4.13).…”