Open embeddings and pseudoflat epimorphisms

Oleg, Aristov,; Pirkovskii, A. Yu.

doi:10.1016/j.jmaa.2019.123817

Cited by 6 publications

(4 citation statements)

References 33 publications

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“…This strategy has previous been observed in non-commutative geometry by Meyer ([15]) and Connes ([10]), who compute the Hochschild homologies of group convolution algebras and the algebra of smooth functions on the noncommutative 2-torus, using a computable dense subalgebra that is 'isocohomologically' embedded. Similar observations have arisen in the work of Taylor ([20]) and Pirkovskii ( [1]). The main observation of Section 7 is that if A → B is a homotopy epimorphism of simplicial commutative algebras in Ind(Ban R ), then HH(B) ∼ = B ⊗ L A HH(A).…”

supporting

confidence: 89%

“…In analytic geometry, one can no longer reasonably use flat epimorphisms as covers for a Grothendieck topology. Indeed, if one works in the category of Fréchet spaces -which is a large enough category containing algebras of complex analytic functions -then for an open subset U of a Stein space (X, O X ), O X (U ) is usually not topologically flat as a Fréchet O X (X)-module (see [1]). Furthermore, the category of Fréchet spaces is not closed due to a plethora of topologies one can impose on the mapping space between two Fréchet spaces.…”

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confidence: 99%

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Analytification, localization and homotopy epimorphisms

Ben-Bassat,

Mukherjee

2021

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We study the interaction between various analytification functors, and a class of morphisms of rings, called homotopy epimorphisms. An analytification functor assigns to a simplicial commutative algebra over a ring R, along with a choice of Banach structure on R, a commutative monoid in the monoidal model category of simplicial ind-Banach R-modules. We show that several analytifications relevant to analytic geometry -such as Tate, overconvergent, Stein analytification, and formal completion -are homotopy epimorphisms. Another class of examples of homotopy epimorphisms arises from Weierstrass, Laurent and rational localizations in derived analytic geometry. As applications of this result, we prove that Hochschild homology and the cotangent complex are computable for analytic rings, and the computation relies only on known computations of Hochschild homology for polynomial rings. We show that in various senses, Hochschild homology as we define it commutes with localizations, analytifications and completions.

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confidence: 89%

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confidence: 99%

Analytification, localization and homotopy epimorphisms

Ben-Bassat,

Mukherjee

2021

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“…C ∞ -manifolds over R) is an open immersion if and only if it induces a weak homological epimorphism between the associated Fréchet C-algebras by [AP20] Theorem 4.2 (resp. [AP20] Theorem 5.3) is another evidence that a homotopy Zariski open immersion is a reasonable counterpart of an open immersion for the smooth setting.…”

Section: We Show the Relation Between The Multiplications Amentioning

confidence: 95%

Homotopy Epimorphisms and Derived Tate's Acyclicity for Commutative C*-algebras

Bambozzi¹,

Mihara²

2021

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We study homotopy epimorphisms and covers formulated in terms of derived Tate's acyclicity for commutative C * -algebras and their non-Archimedean counterparts. We prove that a homotopy epimorphism between commutative C * -algebras precisely corresponds to a closed immersion between the compact Hausdorff topological spaces associated to them, and a cover of a commutative C * -algebra precisely corresponds to a topological cover of the compact Hausdorff topological space associated to it by closed immersions admitting a finite subcover. This permits us to prove derived and non-derived descent for Banach modules over commutative C * -algebras.

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“…Since then, they were rediscovered several times under different names (see [2,6,8,15,17]), both in the purely algebraic and in the functional analytic contexts. A more detailed historical survey is given in [1,Remark 3.16].…”

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confidence: 99%