Resolutions and homological dimensions of DG-modules

Minamoto, Hiroyuki

doi:10.48550/arxiv.1802.01994

Cited by 3 publications

(10 citation statements)

References 9 publications

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“…In Section 2, we recall projective dimension and injective dimension for DG-modules defined by Yekutieli [22]. We also recall a sppj resolution and an ifij resolution which are DG-versions of projective and injective resolution from [15].…”

Section: R)mentioning

confidence: 99%

“…The basic properties of P are summarized in the lemma below taken from [15,Lemma 2.8]. We denote by Proj H 0 ⊂ Mod H 0 the full subcategory of projective H 0modules.…”

Section: Projective Dimensionmentioning

confidence: 99%

“…An important feature of sppj resolution is, roughly speaking, that the "length" of sppj resolution of M measures the projective dimension of M. The precise statement below is extracted from [15,Theorem 2.22…”

Section: Projective Dimensionmentioning

confidence: 99%

“…To perform this, we need to replace the projective and injective dimensions for ordinary modules with that for DG-modules introduced by Yekutieli. We also need to replace the projective and injective resolutions for ordinary modules with that for DG-modules, which are called sup-projective (sppj) and inf-injective (ifij) resolutions, introduced in the previous paper [15]. Although DG-versions of the projective and injective resolutions which is called the DG-projective and DG-injective resolutions (or, cofibrant and fibrant replacements, semi-projective and semi-injective resolutions) have been fundamental tools in research of DG-algebras and DG-modules, these are not suitable to measure the projective and injective dimensions.…”

Section: Introductionmentioning

confidence: 99%

“…Although DG-versions of the projective and injective resolutions which is called the DG-projective and DG-injective resolutions (or, cofibrant and fibrant replacements, semi-projective and semi-injective resolutions) have been fundamental tools in research of DG-algebras and DG-modules, these are not suitable to measure the projective and injective dimensions. In [15], the sppj and ifij resolutions are introduced and proved that, roughly speaking, their length measure the projective and injective dimensions respectively (Theorem 2.6 and Theorem 2.13).…”

Section: Introductionmentioning

confidence: 99%

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On commutative differential graded algebras

Minamoto

2019

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In this paper we undertake a basic study on connective commutative differential graded algebras (CDGA), more precisely, piecewise Noetherian CDGA, which is a DG-counter part of commutative Noetherian algebra. We establish basic results for example, Auslaner-Buchsbaum formula and Bass formula without any unnecessary assumptions.The key notion is the sup-projective (sppj) and inf-injective (ifij) resolutions introduced by the author, which are DG-versions of the projective and injective resolution for ordinary modules. These are different from DG-projective and DG-injective resolutions which is known DG-version of the projective and injective resolution. In the paper, we show that sppj and ifij resolutions are powerful tools to study DG-modules. Many classical result about the projective and injective resolutions can be generalized to DG-setting by using sppj and ifij resolutions. . Among other things we prove a DG-version of Bass's structure theorem of a minimal injective resolution holds for a minimal ifij resolution and a DG-version of the Bass numbers introduced by the same formula with the classical case. We also prove a structure theorem of a minimal ifij resolution of a dualizing complex D, which is completely analogues to the structure theorem of a minimal injective resolution of a dualizing complex over an ordinary commutative algebra.Specializing to results about a dualizing complex, we study a Gorenstein CDGA. We generalize a result by Felix-Halperin-Felix-Thomas and Avramov-Foxby which gives conditions that a CDGA R is Gorenstein in terms of its cohomology algebra H(R). CONTENTS 1. Introduction 2. Sup-projective (sppj) resolutions and inf-injective (ifij) resolutions 3. Commutative DG-algebras 4. A dualizing complex and a Gorenstein CDGA Appendix A. DG-projective and DG-injective resolutions References

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Section: R)mentioning

confidence: 99%

“…The basic properties of P are summarized in the lemma below taken from [15,Lemma 2.8]. We denote by Proj H 0 ⊂ Mod H 0 the full subcategory of projective H 0modules.…”

Section: Projective Dimensionmentioning

confidence: 99%

Section: Projective Dimensionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On commutative differential graded algebras

Minamoto

2019

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Koszul complexes over Cohen-Macaulay rings

Shaul

2020

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We prove a Cohen-Macaulay version of a result by Avramov-Golod and Frankild-Jørgensen about Gorenstein rings, showing that if a noetherian ring A is Cohen-Macaulay, and a 1 , . . . , an is any sequence of elements in A, then the Koszul complex K(A; a 1 , . . . , an) is a Cohen-Macaulay DG-ring. We further generalize this result, showing that it also holds for commutative DG-rings. In the process of proving this, we develop a new technique to study the dimension theory of a noetherian ring A, by finding a Cohen-Macaulay DG-ring B such that H 0 (B) = A, and using the Cohen-Macaulay structure of B to deduce results about A. As application, we prove that if f : X → Y is a morphism of schemes, where X is Cohen-Macaulay and Y is nonsingular, then the homotopy fiber of f at every point is Cohen-Macaulay. As another application, we generalize the miracle flatness theorem. Generalizations of these applications to derived algebraic geometry are also given.

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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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Resolutions and homological dimensions of DG-modules

Cited by 3 publications

References 9 publications

On commutative differential graded algebras

On commutative differential graded algebras

Koszul complexes over Cohen-Macaulay rings

Smooth flat maps over commutative DG-rings

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