Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence

Positselski, Leonid

doi:10.1090/s0065-9266-2010-00631-8

Cited by 155 publications

(395 citation statements)

References 40 publications

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“…All dg algebras and modules will be cohomologically graded (so the differential will have degree 1). Recall (cf., for example [7,Theorem 8.1a]) that the category of A-modules has the structure of a closed model category where weak equivalences are quasi-isomorphisms and fibrations are the surjective maps. All A-modules are fibrant and cofibrant objects are retracts of cell Amodules; the latter are the A-modules having a filtration whose associated factors are free A-modules.…”

Section: Formulation Of the Main Resultsmentioning

confidence: 99%

“…It has the structure of a closed model category in which the weak equivalences are strictly stronger than quasi-isomorphisms and fibrations are surjections [7,Theorem 8.2a]. All E-modules are fibrant and the cofibrant E-modules are the retracts of free E-modules (disregarding the differential).…”

Section: Formulation Of the Main Resultsmentioning

confidence: 99%

“…Recall from [7] that a curved dg algebra is a graded algebra A together with a 'differential' d : A → A, a derivation of A of degree one such that d 2 (a) = [h, a] where a ∈ A and h ∈ A is an element of degree two such that d(h) = 0, called the curvature of A.…”

Section: Curved Algebras and Modulesmentioning

confidence: 99%

“…From [7,Theorem 6.3a] we know that F, G form a mutually inverse pair of equivalences of categories. This equivalence is then a Quillen equivalence with respect to the closed model category structures on A-modules and B-modules.…”

Section: Koszul Dualitymentioning

confidence: 99%

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Koszul–Morita duality

Chuang

Lazarev

Mannan³

2017

J. Noncommut. Geom.

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Section: Formulation Of the Main Resultsmentioning

confidence: 99%

Section: Formulation Of the Main Resultsmentioning

confidence: 99%

Section: Curved Algebras and Modulesmentioning

confidence: 99%

Section: Koszul Dualitymentioning

confidence: 99%

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Koszul–Morita duality

Chuang

Lazarev

Mannan³

2017

J. Noncommut. Geom.

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“…The assertions of the above theorems can be modified so as to hold for arbitrary augmented DG-algebras and conilpotent DG-coalgebras. One just has to replace the class of quasi-isomorphisms of DG-coalgebras with a finer class of filtered quasi-isomorphisms of conilpotent DG-coalgebras, which are to be inverted in order to obtain a category equivalent to the category of augmented DG-algebras with quasi-isomorphisms inverted (see [15,Section 4] or [36,Section 6.10]). In the form stated above, on the other hand, the assertions of the theorems do not hold already for the negatively cohomologically graded DG-coalgebras that are not simply connected-in fact, this is the class of DG-coalgebras for which the difference between quasi-isomorphisms and filtered quasi-isomorphisms becomes essential.…”

Section: Proof Part (A): Define An Increasing Filtration On a Dg-algmentioning

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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A Microlocal Category Associated to a Symplectic Manifold

Tsygan

2018

Algebraic and Analytic Microlocal Analysis

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For a symplectic manifod subject to certain topological conditions, a category enriched in A ∞ modules over the Novikov ring is constructed. The construction is based on the category of modules over Fedosov's deformation quantization algebra that have an additional structure, namely, an action of the fundamental groupoid up to inner automorphisms. Based in large part on the ideas of Bressler-Soibelman, Feigin, and Karabegov, it is motivated by the theory of Lagrangian distributions and and is related to other microlocal constructions of a category starting from a symplectic manifold, such as those due to Nadler-Zaslow and Tamarkin. In the case when our manifold is a flat two-torus, the answer is very close to both the microlocal category of Tamarkin and the Fukaya category as computed by Polishchuk and Zaslow.

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Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence

Cited by 155 publications

References 40 publications

Koszul–Morita duality

Koszul–Morita duality

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

A Microlocal Category Associated to a Symplectic Manifold

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