Deriving DG categories

Keller, Bernhard

doi:10.24033/asens.1689

Cited by 705 publications

(842 citation statements)

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“…Assume that k is a field and A/AeA is finite-dimensional over k. Let K = (A/AeA)/ rad(A/AeA). Assume that K is a direct product of finitely many (say, r) copies of k. We claim that B has the structure of an augmented dg algebra over K up to quasi-equivalence in the sense of [54,Section 7]. Since Hom (a) The functor i * induces an equivalence of triangulated categories The next result is used in the proof of Theorem 5.5 and is also interesting in itself.…”

Section: )mentioning

confidence: 94%

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Relative singularity categories I: Auslander resolutions

Kalck

Yang

2016

Advances in Mathematics

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Abstract. Let R be an isolated Gorenstein singularity with a non-commutative resolution A = End R (R ⊕ M ). In this paper, we show that the relative singularity category ∆ R (A) of A has a number of pleasant properties, such as being Hom-finite. Moreover, it determines the classical singularity category D sg (R) of Buchweitz and Orlov as a certain canonical quotient category. If R has finite CM type, which includes for example Kleinian singularities, then we show the much more surprising result that D sg (R) determines ∆ R (Aus(R)), where Aus(R) is the corresponding Auslander algebra. The proofs of these results use dg algebras, A ∞ Koszul duality, and the new concept of dg Auslander algebras, which may be of independent interest.

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Section: )mentioning

confidence: 94%

“…Let A be a dg k-algebra. Consider the derived category D(A) of dg A-modules, see [54]. It is a triangulated category with shift functor being the shift of complexes [1].…”

Section: 4mentioning

confidence: 99%

Relative singularity categories I: Auslander resolutions

Kalck

Yang

2016

Advances in Mathematics

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“…By inverting quasi-isomorphisms in it, one obtains its derived category, which we will denote by D(A). See [13] for details about this construction. We denote by D + (A), D − (A) and D b (A) the full triangulated subcategories of D(A) composed of DG-modules whose cohomologies are bounded below, bounded above or bounded.…”

Section: Preliminariesmentioning

confidence: 99%

“…We will mostly follow the notations of [23] concerning DG-rings and their derived categories. See [1,6,13] for more background on commutative DG-rings. Below is a short summary of the notation we use here.…”

Section: Preliminariesmentioning

confidence: 99%

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

Shaul

2017

Advances in Mathematics

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ABSTRACT. Let K be a Gorenstein noetherian ring of finite Krull dimension, and consider the category of cohomologically noetherian commutative differential graded rings A over K, such that H 0 (A) is essentially of finite type over K, and A has finite flat dimension over K. We extend Grothendieck's twisted inverse image pseudofunctor to this category by generalizing the theory of rigid dualizing complexes to this setup. We prove functoriality results with respect to cohomologically finite and cohomologically essentially smooth maps, and prove a perfect base change result for f ! in this setting. As application, we deduce a perfect derived base change result for the twisted inverse image of a map between ordinary commutative noetherian rings. Our results generalize and solve some recent conjectures of Yekutieli.

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“…This is the derived functor of tensor product of DG-modules over A • defined on the "conventional" derived categories of DG-modules (obtained by inverting the DG-module morphisms inducing isomorphisms of the cohomology modules); see [21], [14], or [36, Section 1]. The spectral sequence E pq r is called the algebraic Eilenberg-Moore spectral sequence associated with a DG-algebra A • [9,12,13].…”

Section: Koszulity Implies Quasi-formalitymentioning

confidence: 99%

Koszulity of cohomology = K(π,1)-ness + quasi-formality

Positselski

2017

Journal of Algebra

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Abstract. This paper is a greatly expanded version of [37, Section 9.11]. A series of definitions and results illustrating the thesis in the title (where quasi-formality means vanishing of a certain kind of Massey multiplications in the cohomology) is presented. In particular, we include a categorical interpretation of the "Koszulity implies K(π, 1)" claim, discuss the differences between two versions of Massey operations, and apply the derived nonhomogeneous Koszul duality theory in order to deduce the main theorem. In the end we demonstrate a counterexample providing a negative answer to a question of Hopkins and Wickelgren about formality of the cochain DG-algebras of absolute Galois groups, thus showing that quasi-formality cannot be strengthened to formality in the title assertion.

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Deriving DG categories

Abstract: We investigate the (unbounded) derived category of a di erential Z-graded category

Cited by 705 publications

References 7 publications

Relative singularity categories I: Auslander resolutions

Relative singularity categories I: Auslander resolutions

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

Koszulity of cohomology = K(π,1)-ness + quasi-formality

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