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

By L. Positselski. 2009

271 citations recorded by the source corpus.

This paper can be thought of as an extended introduction to arXiv:0708.3398; nevertheless, most of its results are not covered by loc. cit. We consider the derived categories of DG-modules, DG-comodules, and DG-contramodules, the coderived and contraderived categories of CDG-modules, the coderived categories of CDG-comodules, and the contraderived categories of CDG-contramodules. The equivalence between the latter two categories (the comodule-contramodule correspondence) is established. Nonhomogeneous Koszul duality or "triality" (an equivalence between exotic derived categories corresponding to Koszul dual (C)DG-algebra and CDG-coalgebra) is obtained in the conilpotent and nonconilpotent versions. Various $A_\infty$-structures are considered, and a number of model category structures are described. Homogeneous Koszul duality and $D$-$\Omega$ duality are discussed in the appendices.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d3bca2be35390ffad784aa23198a166d58719c2a

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Citations in the corpus

Calabi-Yau algebras and superpotentials (2010, methodology): “r A. Write PCDGComod(C) and PCDGMod(A) for the corresponding pseudo-compact notions. All these categories are equipped with model structures and connected with each other through Quillen equivalences [16, 18, 24, 28]. See Appendix A for a survey and further details. Let A∈PCAlgc(l). The model struct”
A category of kernels for equivariant factorizations and its implications for Hodge theory (2011, methodology): “rix factorizations on a stack satisfying appropriate conditions and proved that their category of matrix factorizations coincided with the category of singularities of the underlying hypersurface. In [Pos09], L. Positselski, using his work on co- and contra-derived categories of curved dg-modules ov”
A geometric approach to Orlov’s theorem (2010, background): “isomorphic is DQcohΓ(S,F) to a graded Dbrane. The rest of the argument can be adapted mutatis mutandis from the work of Positselski. He proves the analogous statement as Theorem 2 in Section 3.11 of [Pos09]. The arguments given for Theorem 2 in Section 3.11 (including the supporting Theorem 3.6 and ”
Hochschild (co)homology of the second kind I (2010, background): “general formalism of derived categories of the second kind as developed in the second author’s paper [15].”
Global matrix factorizations (2011, methodology): “sQCoh(X,W) is an equivalence of triangulated categories. Therefore the category Inj(X,W) deﬁnes a dg enhancement of DabsQCoh(X,W). Proof. This is a scheme theoretic version of Theorems 3.5 and 3.6 of [Pos1] and is proved in exactly the same way. We give a sketch of the proof. We wish to apply the pr”
Curved Koszul duality theory (2010, background): “When we reduce our bar construction and our cobar construction to semi-augmented algebras and curved coalgebras, we recover the particular case of [31] where the curved coalgebras are coaugmented.”
Hereditary abelian model categories (2015, background): “In this section, we give brief constructions of these categories in terms of abelian model structures recently appearing in [5, 6, 10, 11, 33, 52, 59].”
Singular support of coherent sheaves and the geometric Langlands conjecture (2012, background): “ H.3. IndCoh(Z) as the coderived category. Let us comment on the relation between the results in the main body of the present paper and the non-linear Koszul transform introduced by L. Positselski in [Pos]. H.3.1. The key notion that we need from [Pos] is that of coderived category of modules over a”