Hochschild (co)homology of the second kind I

Polishchuk, Alexander; Positselski, Leonid

doi:10.1090/s0002-9947-2012-05667-4

Cited by 40 publications

(90 citation statements)

References 31 publications

(129 reference statements)

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“…If n ≥ dim X , the module P n is also perfect. It follows that Tot ⊕ P • is perfect and co-isomorphic to M. As before, one can prove that any object of X -perf that is absolutely acyclic with respect to X -coh is absolutely acyclic with respect to X -perf (see [9,Sect. 3…”

Section: Theoremmentioning

confidence: 81%

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DG-modules over de Rham DG-algebra

Rybakov

2014

European Journal of Mathematics

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For a morphism of smooth schemes over a regular affine base we define functors of derived direct image and extraordinary inverse image on coderived categories of DG-modules over de Rham DG-algebras. Positselski proved that for a smooth algebraic variety X over a field k of characteristic zero the coderived category of DGmodules over • X/k is equivalent to the unbounded derived category of quasi-coherent right D X -modules. We prove that our functors correspond to the functors of the same name for D X -modules under Positselski equivalence.

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Section: Theoremmentioning

confidence: 81%

“…We present here a down to earth treatment of the subject [12]. The reader can find the results in full generality in the papers [9][10][11]13]. …”

Section: Positselski Results On Coderived Categoriesmentioning

confidence: 99%

DG-modules over de Rham DG-algebra

Rybakov

2014

European Journal of Mathematics

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“…Note that, in general, (2.5) is not a quasi-isomorphism (although it is a quasi-isomorphism in a number of cases of interest; cf. [12]).…”

Section: The Main Resultmentioning

confidence: 99%

“…As is shown in [12], the canonical morphism (2.5) is a quasi-isomorphism for A := A f . Thus, in the above diagram, we may replace the ordinary cyclic complexes with their completed versions.…”

Section: Proof Of Theorem 31mentioning

confidence: 86%

On a Hodge theoretic property of the Künneth map in periodic cyclic homology

Shklyarov

2016

Journal of Algebra

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We prove that the standard Künneth map in periodic cyclic homology of differential /2graded algebras is compatible with a generalization of the Hodge filtration and explain how this result is related to various Thom-Sebastiani type theorems in singularity theory. DMYTRO SHKLYAROV refer to [8,18] for more details). That the non-commutative Künneth map is compatible with the extensions is an obvious consequence of the explicit formula: roughly, the Künneth map at the level of complexes is regular at u = 0. The non-trivial part of our result is that the map is compatible with the connections.While the compatibility of the classical Künneth map with the mixed Hodge structures on the cohomology of varieties is a good piece of motivation for the subject of the present paper, there are results in geometry of which our theorem is a direct generalization. Namely, our actual aim was to obtain abstract algebraic versions of various Thom-Sebastiani-type results in singularity theory such as, for instance, the Thom-Sebastiani formula for Steenbrink's Hodge filtration on the vanishing cohomology of isolated singularities [15, Sect.8]. The latter calculates the Hodge filtration for the direct sum f ⊕ g of two singularities in terms of the same data for f and g. This result, as well as the original Thom-Sebastiani theorem and some other facts of similar nature, can be deduced from a Künneth property for the Gauss-Manin systems and the Brieskorn lattices associated with the singularities (cf. Lemma 8.7 in loc.cit.) or, equivalently, for the Fourier-Laplace transforms thereof (cf. [14, Sect. 3]). This Künneth property is a special case of our result, as we will explain in Section 3. Conventions.In Section 2, we work over an arbitrary field k of characteristic 0. In Section 3 the ground field is . Given a /2-graded space V , ΠV will stand for V with the reversed /2-grading, the parity of v ∈ V will be denoted by |v|, and the corresponding element of ΠV will be denoted by Πv. We will follow all the standard conventions of super-linear algebra (such as the Koszul rule of signs, etc.). Finally, only unital dg algebras will be considered.Acknowledgement.

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“…We are interested in the cohomology for the mixed differential ∂ W H , which is sensitive to the topology we use. Following [5,6,40], the appropriate complex for Landau-Ginzburg models turns out to be the one of compact type above as induced in [25,36,38]. Definition 2.8.…”

Section: Curved Algebras and Mixed Complexmentioning

confidence: 99%

G-twisted Braces and Orbifold Landau–Ginzburg Models

2020

Commun. Math. Phys.

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Given an algebra with group G-action, we construct brace structures for its G-twisted Hochschild cochains. An an application, we construct G-Frobenius algebras for orbifold Landau-Ginzburg Bmodels and present explicit orbifold cup product formula for all invertible polynomials. be the Hochschild differential. Let HH • (A, A[G]) be the Hochschild cohomology with a natural cup product ∪. We show that ∪ satisfies a G-twisted commutativity relation (see also [46]) and the G-invariant

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Hochschild (co)homology of the second kind I

Cited by 40 publications

References 31 publications

DG-modules over de Rham DG-algebra

DG-modules over de Rham DG-algebra

On a Hodge theoretic property of the Künneth map in periodic cyclic homology

G-twisted Braces and Orbifold Landau–Ginzburg Models

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