Enhanced Koszul properties in Galois cohomology

Mináč, Ján; Palaisti, Marina; Pasini, Federico W.; Tân, Nguyêñ Duy

doi:10.1007/s40687-020-00208-5

Cited by 10 publications

(13 citation statements)

References 56 publications

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“…(1) A[x] is universally Koszul (2) The direct sum A ⊓ B is universally Koszul Proof. See Proposition 30 in [4].…”

Section: Proofs Of Theorems a And Bmentioning

confidence: 99%

Kurosh theorem for certain Koszul Lie algebras

Blumer¹

2022

Preprint

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The Kurosh theorem for groups provides the structure of any subgroup of a free product of groups and its proof relies on Bass-Serre theory of groups acting on trees. In the case of Lie algebras, such a general theory does not exists and the Kurosh theorem is false in general, as it was first noticed by Shirshov in [6]. However, we prove that, for a class of positively graded Lie algebras satisfying certain local properties in cohomology, such a structure theorem holds true for subalgebras generated in degree 1. Such class consists of Lie algebras, which have all the subalgebras generated in degree 1 that are Koszul.

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“…(1) A[x] is universally Koszul (2) The direct sum A ⊓ B is universally Koszul Proof. See Proposition 30 in [4].…”

Section: Proofs Of Theorems a And Bmentioning

confidence: 99%

Kurosh theorem for certain Koszul Lie algebras

Blumer¹

2022

Preprint

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“…For this reason, some "enhanced forms" of Koszulity -which are stronger than "simple" Koszulity, but at the same time easier to check -have been introduced by several authors. We give now the definition of universal Koszulity as introduced in [10].…”

Section: Koszul Algebras and Universally Koszul Algebras A Quadraticmentioning

confidence: 99%

“…Recently, some stronger versions of the Koszul property were introduced and investigated (see, e.g., [3,5,15,4,10,13]). For example, the universal Koszul property (see Definition 2.6 below), which implies "simple" Koszulity.…”

Section: Introductionmentioning

confidence: 99%

Pro-$p$ groups with few relations and universal Koszulity

Quadrelli¹

2021

Math. Scand.

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“…But how does one look for those properties that distinguish absolute p-Galois groups from the broader class of pro-p groups? There are several research agendas that aim to answer this question using a number of different techniques, including the study of Massey products (see [39,50,60,84,92,93,94,117,118,120]) and the Koszulity of Galois cohomology (see [82,83,102]).…”

Section: Introductionmentioning

confidence: 99%

Galois module structure of square power classes for biquadratic extensions

Chemotti,

Minac,

Schultz

et al. 2021

Preprint

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This paper marks the first time that the Galois module structure of power classes of a field has been fully determined when the modular representation theory allows for an infinite number of indecomposable types. The example and methodology we consider pave the way for a paradigm shift in the study of absolute Galois groups via Galois modules. Whereas generic modules of this type are completely intractable to approach, this work shows that Galois cohomology provides a context where this otherwise impossible problem becomes approachable, interesting, and illuminating.Let V = Z/2Z⊕Z/2Z. For a Galois extension K/F with char(K) = 2 and Gal(K/F ) V , we determine the F 2 [Gal(K/F )]-module structure of K × /K ×2 . Although there are an infinite number of (pairwise non-isomorphic) indecomposable F 2 [V ]-modules, our decomposition includes at most 8 indecomposable types. This is a dramatic development on several fronts. Most conspicuously, it shows that these modules can be computed in settings which might initially seem too forbidding. The elegant nature of this decomposition also points to new structural restrictions on absolute Galois groups. More generally, the techniques used in this paper provide a blueprint for studying a broad class of Galois modules. This opens the door for a novel, systematic study of the Galois theory of maximal p-extensions of fields.Prepare for the unknown by studying how others in the past have coped with the unforeseeable and the unpredictable.

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Enhanced Koszul properties in Galois cohomology

Cited by 10 publications

References 56 publications

Kurosh theorem for certain Koszul Lie algebras

Kurosh theorem for certain Koszul Lie algebras

Pro-$p$ groups with few relations and universal Koszulity

Galois module structure of square power classes for biquadratic extensions

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