The twisted inverse image pseudofunctor over commutative DG rings and perfect base change

Shaul, Liran

doi:10.1016/j.aim.2017.08.041

Cited by 12 publications

(7 citation statements)

References 18 publications

(29 reference statements)

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“…Proof. This was shown in [25, Proposition 2.2( 1)] (see also [26,Proposition 6.7]). There, we assumed the stronger condition N ∈ D b (A) and the (possible) weaker condition that bfl dim A (K) < ∞.…”

Section: The Telescope Complex and The Telescope Dg-modulesupporting

confidence: 57%

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Injective DG-modules over non-positive DG-rings

Shaul

2017

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Let A be an associative non-positive differential graded ring. In this paper we make a detailed study of a category Inj(A) of left DG-modules over A which generalizes the category of injective modules over a ring. We give many characterizations of this category, generalizing the theory of injective modules, and prove a derived version of the Bass-Papp theorem: the category Inj(A) is closed in the derived category D(A) under arbitrary direct sums if and only if the ring H 0 (A) is left noetherian and for every i < 0 the left H 0 (A)-module H i (A) is finitely generated. Specializing further to the case of commutative noetherian DG-rings, we generalize the Matlis structure theory of injectives to this context. As an application, we obtain a concrete version of Grothendieck's local duality theorem over commutative noetherian local DG-rings.

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Section: The Telescope Complex and The Telescope Dg-modulesupporting

confidence: 57%

“…Under the assumption that A has bounded cohomology, it follows by [25, Corollary 4.6] that bfl dim A ( A) = 0. Hence, by [26,Lemma 6.3], we have that…”

Section: In D(a)mentioning

confidence: 92%

Injective DG-modules over non-positive DG-rings

Shaul

2017

Preprint

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“…Thus, the above corollary states that in the above smooth situation, there exist a tilting DG-module over B which is rigid over B relative to A. In this smooth context, one may view this result as a generalization of similar results from [2,11,17,19].…”

Section: Homological Smoothness and Flatness Over A Base Ringmentioning

confidence: 64%

“…Given a homomorphism A → B of flat dimension 0, it follows from the definition that, considered as an object of D(A), the DG-module B belongs to the class F , in the sense of [8,Section 4.2]. In particular, by [8,Lemma 4.6(4)], for any M ∈ D(A) and any n ∈ Z, there is a natural isomorphism [11,Lemma 6.3] and [14, Section 2.2.2] for discussions about these facts.…”

Section: Preliminariesmentioning

confidence: 99%

Smooth flat maps over commutative DG-rings

Shaul

2020

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We study smooth maps that arise in derived algebraic geometry. Given a map A → B between non-positive commutative noetherian DG-rings which is of flat dimension 0, we show that it is smooth in the sense of Toën-Vezzosi if and only if it is homologically smooth in the sense of Kontsevich. We then show that B, being a perfect DG-module over B ⊗ L A B has, locally, an explicit semi-free resolution as a Koszul complex. As an application we show that a strong form of Van den Bergh duality between (derived) Hochschild homology and cohomology holds in this setting.

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“…Let R be a dualizing DGmodule over A. By [8,Corollary 6.11], the localization Rp is a dualizing DG-module over Ap. According to [13,Corollary 7.16], the uniqueness of dualizing DG-modules over local DG-rings implies that there exists some n ∈ Z such that Rp[n] ∼ = Ap.…”

Section: The Regular and The Gorenstein Locimentioning

confidence: 99%

Open loci results for commutative DG-rings

Shaul

2020

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Given a commutative noetherian non-positive DG-ring with bounded cohomology which has a dualizing DG-module, we study its regular, Gorenstein and Cohen-Macaulay loci. We give a sufficient condition for the regular locus to be open, and show that the Gorenstein locus is always open. However, both of these loci are often empty: we show that no matter how nice H 0 (A) is, there are examples where the Gorenstein locus of A is empty. We then show that the Cohen-Macaulay locus of a commutative noetherian DG-ring with bounded cohomology which has a dualizing DG-module always contains a dense open set. Our results imply that under mild hypothesis, eventually coconnective locally noetherian derived schemes are generically Cohen-Macaulay, but that even in very nice cases, they need not be generically Gorenstein.

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The twisted inverse image pseudofunctor over commutative DG rings and perfect base change

Cited by 12 publications

References 18 publications

Injective DG-modules over non-positive DG-rings

Injective DG-modules over non-positive DG-rings

Smooth flat maps over commutative DG-rings

Open loci results for commutative DG-rings

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