Equivalences Induced by Infinitely Generated Silting Modules

“…We denote by 𝜏 ⩽0 and 𝜏 >0 the associated smart truncation functors, as well as their obvious shifted variants such as 𝜏 ⩾0 (see, e.g., [43, 7.3.6, 7.3.10]). The heart of this t-structure is identified with 𝖬𝗈𝖽-𝔖 via the zig-zag of dg-ring morphisms 𝔖 ← 𝜏 ⩽0 𝐴 𝑞𝑖𝑠 → 𝐴, see, for example, [13,Remark 1.6]. The following definition was introduced for 1-tilting modules by [7], and then generalized to 𝑛-tilting modules by [14].…”

Section: Good Tilting Complexesmentioning

confidence: 99%

“…The following results are available in [29] (see also [13]), but we gather the relevant parts here in a form directly applicable for our purposes. Theorem 4.4.…”

Section: Which Is Natural In 𝑋 ∈ 𝖣(𝐴-𝖽𝗀𝖬𝗈𝖽) 𝑌 ∈ 𝖣((𝐴 ⊗ ℤ 𝐵 𝗈𝗉 )-𝖬𝗈𝖽) ...mentioning

confidence: 99%

Topological endomorphism rings of tilting complexes

Hrbek

2024

Journal of London Math Soc

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In a compactly generated triangulated category, we introduce a class of tilting objects satisfying a certain purity condition. We call these the decent tilting objects and show that the tilting heart induced by any such object is equivalent to a category of contramodules over the endomorphism ring of the tilting object endowed with a natural linear topology. This extends the recent result for n‐tilting modules by Positselski and Št'ovíček. In the setting of the derived category of modules over a ring, we show that the decent tilting complexes are precisely the silting complexes such that their character dual is cotilting. The hearts of cotilting complexes of cofinite type turn out to be equivalent to the category of discrete modules with respect to the same topological ring. Finally, we provide a kind of Morita theory in this setting: Decent tilting complexes correspond to pairs consisting of a tilting and a cotilting‐derived equivalence as described above tied together by a tensor compatibility condition.

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“…The general case of a silting module

T

over an arbitrary ring

A

is treated by Breaz and Modoi in . Here one has to go a step further and replace

Mod - B

by the heart

H_{σ}

of the silting object

σ

.…”

Section: Silting Complexesmentioning

confidence: 99%

“…The zero cohomologies of such complexes form an interesting class of modules, first studied systematically by Adachi, Iyama and Reiten in , which parametrise torsion pairs in module categories and are also related to ring theoretic localisation . Ongoing work on these modules concerns several representation theoretic and combinatorial aspects, see, for example, . Some of these topics are treated in the surveys .…”

Section: Introductionmentioning

confidence: 99%

Silting Objects

Hügel

2019

Bull. London Math. Soc.

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We give an overview of recent developments in silting theory. After an introduction on torsion pairs in triangulated categories, we discuss and compare different notions of silting and explain the interplay with t-structures and co-t-structures. We then focus on silting and cosilting objects in a triangulated category with coproducts and study the case of the unbounded derived category of a ring. We close the survey with some classification results over commutative Noetherian rings and silting-discrete algebras. Contents

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Equivalences Induced by Infinitely Generated Silting Modules

Abstract: We study equivalences induced by a complex P, consisting of projectives and concentrated in degrees −1 and 0, which is silting in the derived category D(R) of a ring R.

Cited by 5 publications

References 37 publications

The ascent-descent property for $2$-term silting complexes

The ascent-descent property for $2$-term silting complexes

Topological endomorphism rings of tilting complexes

Silting Objects

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