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Formal manifolds: foundations
Differential Geometry · Submitted 3 Jan 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We develop the basic theory for formal manifolds, and establish a fully faithful contravariant functor from the category of formal manifolds to the category of topological $\mathbb{C}$-algebras.
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The abstract, sentence by sentence
This is the first paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In this paper, we lay the foundations for this study by introducing the notion of formal manifolds in the context of differential geometry, inspired by the notion of formal schemes in algebraic geometry. We develop the basic theory for formal manifolds, and establish a fully faithful contravariant functor from the category of formal manifolds to the category of topological $\mathbb{C}$-algebras. We also prove the existence of finite products in the category of formal manifolds by studying vector-valued formal functions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This is the first paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. | background 0.75 | 0.43 | 0.07 | asserted 1.00 |
| s1 | In this paper, we lay the foundations for this study by introducing the notion of formal manifolds in the context of differential geometry, inspired by the notion of formal schemes in algebraic geometry. | contribution 0.98 | 0.92 | 0.09 | asserted 0.99 |
| s2 | We develop the basic theory for formal manifolds, and establish a fully faithful contravariant functor from the category of formal manifolds to the category of topological $\mathbb{C}$-algebras. | contribution 0.53 | 0.96 | 0.27 | asserted 1.00 |
| s3 | We also prove the existence of finite products in the category of formal manifolds by studying vector-valued formal functions. | result 0.87 | 0.96 | 0.13 | asserted 1.00 |
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The paper
Original abstract
This is the first paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In this paper, we lay the foundations for this study by introducing the notion of formal manifolds in the context of differential geometry, inspired by the notion of formal schemes in algebraic geometry. We develop the basic theory for formal manifolds, and establish a fully faithful contravariant functor from the category of formal manifolds to the category of topological $\mathbb{C}$-algebras. We also prove the existence of finite products in the category of formal manifolds by studying vector-valued formal functions.
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How the literature uses it
2 arXiv papers cite this in 63 sentences
Papers citing it · 47 sentences without a confident label
Builds on it 2 papers
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This is a sequel to [ref] in a series to study smooth (co)homogogy theory for representations of general Lie pairs.
Function spaces on formal manifolds · Introduction and the main results · 0.94
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This is a sequel to [ref] and [ref] in a series to study the theory of formal manifolds and formal Lie groups, with the ultimate goal of establishing a smooth relative Lie algebra (co)homology theory.
Poincar\'e's lemma for formal manifolds · Introduction · 0.82
Cites it as support 1 paper
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For every $γ∈Γ$ and $(X_1,X_2,…, X_r)∈Der()^r$, we have that [equation] This, together with [ref] , shows that [equation] as desired.
Poincar\'e's lemma for formal manifolds · De Rham complexes I · 0.97
Cites it as background 2 papers
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For every smooth manifold $N$ and $k∈$, denote by $N^(k)$ the locally ringed space $(N, _N^(k))$ over $Spec()$, where [equation] Recall from [ref] that a formal manifold is a locally ringed space $(M, )$ over $Spec()$ such that * the topological space $M$ is paracompact and Hausdorff; and * for every $a∈M$, there is an open neighborhood $U$ of $a$ in $M$ and $n,k∈$ such that $(U, |_U)$ is isomorphic to $(^n)^(k)$ as locally ringed spaces over $Spec()$.
Function spaces on formal manifolds · Notations and conventions · 0.89
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Recall from [ref] that a formal manifold is a locally ringed space $(M, )$ over $Spec()$ such that * the topological space $M$ is paracompact and Hausdorff; and * for every $a∈M$, there is an open neighborhood $U$ of $a$ in $M$ and $n,k∈$ such that $(U, |_U)$ is isomorphic to $(^n)^(k)$ as locally ringed spaces over $Spec()$.
Poincar\'e's lemma for formal manifolds · Notations and conventions · 0.96
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