Duality pairs, phantom maps, and definability in triangulated categories

Bird, Isaac; Williamson, Jordan

doi:10.48550/arxiv.2202.08113

Cited by 3 publications

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“…Remark 6.15. In other words, the previous result shows that (F n , I n ) is symmetric duality pair in the sense of [4], where F n (resp.,…”

Section: Examplesmentioning

confidence: 82%

Finitistic dimensions over commutative DG-rings

Bird¹,

Shaul²,

Sridhar³

et al. 2022

Preprint

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We study the small and big finitistic projective, injective and flat dimensions over a non-positively graded commutative noetherian DG-ring A with bounded cohomology. Our main results generalize results of Bass and Raynaud-Gruson to this derived setting, showing that any bounded DG-module M of finite flat dimension satisfies proj dim A (M ) ≤ dim(H 0 (A))−inf(M ). We further construct DG-modules of prescribed projective dimension, and deduce that the big finitistic projective dimension satisfies the inequalities dim(HIt is further shown that this result is optimal, in the sense that there are examples that achieve either bound. As an application, new vanishing results for the derived Hochschild (co)homology of homologically smooth algebras are deduced.

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“…Remark 6.15. In other words, the previous result shows that (F n , I n ) is symmetric duality pair in the sense of [4], where F n (resp.,…”

Section: Examplesmentioning

confidence: 82%

Finitistic dimensions over commutative DG-rings

Bird¹,

Shaul²,

Sridhar³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…We will also need another duality functor (−) * ∶= 𝐑𝖧𝗈𝗆 𝑅 (−, 𝑅), which induces an equivalence (𝖣(𝑅-𝖬𝗈𝖽) 𝖼 ) 𝗈𝗉 ≅ → 𝖣(𝖬𝗈𝖽-𝑅) 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.…”

Section: Tilting Complexes and Hearts In The Derived Category Of A Ringmentioning

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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“…on the categories of compact objects. See [AHH21, §2.2] for details about these dualities, and also the recent preprint [BW22] where the triangulated character duality is developed in larger generality. By symmetry, we also freely consider both functors going in the opposite direction.…”

Section: Tilting Complexes and Hearts In The Derived Category Of A Ringmentioning

confidence: 99%

Topological endomorphism rings of tilting complexes

Hrbek¹

2022

Preprint

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In a compactly generated triangulated category, we introduce a class of tilting objects satisfying 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 object endowed with a natural linear topology. This extends the recent result for n-tilting modules of Positselski and Šťovíček. In the setting of the derived category of a ring, we show that the decent tilting complexes are precisely the silting complexes such that their character dual is cotilting. We show that the hearts of cotilting complexes of finite type are equivalent to the category of discrete modules with respect to the same topological ring. Finally, we show that decent tilting complexes parametrize pairs consisting of a tilting and a cotilting derived equivalence as above together with a tensor compatibility condition.

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Duality pairs, phantom maps, and definability in triangulated categories

Cited by 3 publications

References 0 publications

Finitistic dimensions over commutative DG-rings

Finitistic dimensions over commutative DG-rings

Topological endomorphism rings of tilting complexes

Topological endomorphism rings of tilting complexes

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