Cohomology of unipotent and prounipotent groups

Lubotsky, Alexander; Magid, Andy R.

doi:10.1016/0021-8693(82)90006-0

Cited by 25 publications

(28 citation statements)

References 10 publications

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“…If G is prounipotent, then cd(G) ≤ n if and only if H n+1 (G, k a ) = 0, since k a is the only simple G-module. Since the following statement is important for the succeeding exposition and for demonstration how Pontryagin duality helps one to transfer arguments from [17] into our situation, let us give this statement with a full proof. Denote by Hom(G, k a ) the set of affine group scheme homomorphisms from G to k a .…”

Section: Proposition 8 [17 112] Let G Be a Prounipotent Group And mentioning

confidence: 97%

“…Proposition 12. [17,Th. 2.4] Let G be a prounipotent group, then the following conditions are equivalent:…”

Section: Presentations Of Prounipotent Groupsmentioning

confidence: 99%

“…For example, the image of a closed subgroup under a homomorphism of prounipotent groups will be always a closed subgroup. The main theorems on the structure of normal series, nontriviality of the center of a unipotent group [11,VII,17]…”

Section: Prounipotent Groupsmentioning

confidence: 99%

“…Cohomology theory of prounipotent groups over the algebraically closed field k has been developed by A. Lubotzky and A. Magid [17], when k has zero characteristics. This theory closely parallels that of the cohomology of pro-p-groups [34, I,4]: the free prounipotent groups turn out to be those of cohomological dimension ones, the dimension of the first and second cohomology groups give numbers of generators and defining relations.…”

Section: Cohomology and Presentationsmentioning

confidence: 99%

“…Indeed, let k be a non algebraically closed field of zero characteristics with algebraic closure k and consider unipotent group G k over k. We already know that Below we introduce necessary notions for defining Hochschild cohomology groups of affine group schemes in the modern setting [13]. In the case of algebraically closed field they coincide [13, p. 28] with the ones defined in [17].…”

Section: Cohomology and Presentationsmentioning

confidence: 99%

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Identity Theorem for Pro-p-groups

Mikhovich

2019

Knots, Low-Dimensional Topology and Applications

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The concept of schematization consists in replacing simplicial groups by simplicial affine group schemes. In the case when the coefficient field has a zero characteristic, there is a prominent theory of simplicial prounipotent groups, the origins of which lead to the rational homotopy theory of D. Quillen. The specificity of a prime finite field F p is that the Zariski topology on the n-dimensional affine space F n p turns out to be discrete. Therefore, although finite p-groups are F p -points of F p -unipotent affine group schemes, this observation is not actually used. Nevertheless, schematization reveals the profound properties of F p -prounipotent groups, especially in connection with prounipotent groups in the zero characteristic and in the study of quasirationality. In this paper, using results on representations and cohomology of prounipotent groups in characteristic 0, we prove an analogue of Lyndon Identity theorem for one-relator pro-p-groups (question posed by J.P. Serre) and demonstrate the application to one more problem of J.-P. Serre concerning onerelator pro-p-groups of cohomological dimension 2. Schematic approach makes it possible to consider the problems of pro-p-groups theory through the prism of Tannaka duality, concentrating on the category of representations. In particular we attach special importance to the existence of identities in free pro-p-groups ("conjurings").

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Section: Proposition 8 [17 112] Let G Be a Prounipotent Group And mentioning

confidence: 97%

“…Proposition 12. [17,Th. 2.4] Let G be a prounipotent group, then the following conditions are equivalent:…”

Section: Presentations Of Prounipotent Groupsmentioning

confidence: 99%

Section: Prounipotent Groupsmentioning

confidence: 99%

Section: Cohomology and Presentationsmentioning

confidence: 99%

Section: Cohomology and Presentationsmentioning

confidence: 99%

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Identity Theorem for Pro-p-groups

Mikhovich

2019

Knots, Low-Dimensional Topology and Applications

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The Picard–Vessiot Antiderivative Closure

Magid

2001

Journal of Algebra

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F is a differential field of characteristic zero with algebraically closed field of constants C. A Picard-Vessiot antiderivative closure of F is a differential field extension E ⊃ F which is a union of Picard-Vessiot extensions of F , each obtained by iterated adjunction of antiderivatives, and such that every such Picard-Vessiot extension of F has an isomorphic copy in E. The group G of differential automorphisms of E over F is shown to be prounipotent. When C is the complex numbers and F = C(t) the rational functions in one variable, G is shown to be free prounipotent. 1991 Mathematics Subject Classification. 12H05.

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Proalgebraic Crossed Modules of Quasirational Presentations

Mikhovich¹

2016

Trends in Mathematics

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For every quasirational (pro-p)relation module R we construct the so called p-adic rationalization, which is the pro-fd-module R ⊗Q p = limstands for the rational points of the abelianization of the continuous p-adic Malcev completion of R. We show how R ∧ w embeds into a sequence which arises from a certain prounipotent crossed module. The latter can be seen as concrete examples of proalgebraic homotopy types.We provide the Identity Theorem for pro-p-groups, giving a positive feedback to the question of Serre.

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Cohomology of unipotent and prounipotent groups

Cited by 25 publications

References 10 publications

Identity Theorem for Pro-p-groups

Identity Theorem for Pro-p-groups

The Picard–Vessiot Antiderivative Closure

Proalgebraic Crossed Modules of Quasirational Presentations

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