Infinitesimal 1-parameter subgroups and cohomology

Suslin, Andrei; Friedlander, Eric M.; Bendel, Christopher P.

doi:10.1090/s0894-0347-97-00240-3

Cited by 104 publications

(174 citation statements)

References 9 publications

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“…[19]). In the work of Suslin, Friedlander, and the author [29,30] it was shown that the family of subgroup schemes {G a(r) } plays the role of elementary abelian subgroups in detecting projectivity and nilpotence of cohomology for infinitesimal group schemes. Specifically, [30, [30, proposition 7•6] is an analogue of Chouinard's theorem (Theorem 1•3) about detecting projectivity.…”

Section: Introductionmentioning

confidence: 99%

“…Within the literature, it has been noted that it is often more convenient to work instead with prime ideals, while maintaining the traditional terminology. This study of cohomology via varieties was extended to restricted Lie algebras by Friedlander and Parshall [15,16], as well as by Jantzen [20], and later to arbitrary infinitesimal group schemes by Suslin, Friedlander, and the author [29,30]. Here the detection properties of the family of subgroup schemes {G a(r) } was crucial for identifying the varieties.…”

Section: Introductionmentioning

confidence: 99%

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Cohomology and projectivity of modules for finite group schemes

Bendel¹

2001

Math. Proc. Camb. Phil. Soc.

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Let G be a finite group scheme over a field k, that is, an affine group scheme whose coordinate ring k[G] is finite dimensional. The dual algebra k[G]* ≡ Homk(k[G], k) is then a finite dimensional cocommutative Hopf algebra. Indeed, there is an equivalence of categories between finite group schemes and finite dimensional cocommutative Hopf algebras (cf. [19]). Further the representation theory of G is equivalent to that of k[G]*. Many familiar objects can be considered in this context. For example, any finite group G can be considered as a finite group scheme. In this case, the algebra k[G]* is simply the group algebra kG. Over a field of characteristic p > 0, the restricted enveloping algebra u([gfr ]) of a p-restricted Lie algebra [gfr ] is a finite dimensional cocommutative Hopf algebra. Also, the mod-p Steenrod algebra is graded cocommutative so that some finite dimensional Hopf subalgebras are such algebras.Over the past thirty years, there has been extensive study of the modular representation theory (i.e. over a field of positive characteristic p > 0) of such algebras, particularly in regards to understanding cohomology and determining projectivity of modules. This paper is primarily interested in the following two questions:Questions1·1. Let G be a finite group scheme G over a field k of characteristic p > 0, and let M be a rational G-module.(a) Does there exist a family of subgroup schemes of G which detects whether M is projective?(b) Does there exist a family of subgroup schemes of G which detects whether a cohomology class z ∈ ExtnG(M, M) (for M finite dimensional) is nilpotent?

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Cohomology and projectivity of modules for finite group schemes

Bendel¹

2001

Math. Proc. Camb. Phil. Soc.

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“…Since

V (U) \subseteq V (G)

(cf. [, (1.5)]), the first part of the proof implies that

im α {(u_{r - 1})}_{M}^{ℓ} = V_{ℓ}

for every

ℓ \in {1, \dots, p - 1}

. This shows that the

scriptG

‐module

M

has the equal images property.…”

Section: Modules For Finite Group Schemesmentioning

confidence: 75%

“…General theory then shows that

V false(scriptG false) = {prefixHom}_{Hopf} false(k G_{a false(r false)}, k scriptG false) \subseteq {prefixHom}_{k} false(k G_{a false(r false)}, k scriptG false) : = M

is an affine variety. In fact,

V (G)

is the variety of

k

‐rational points of the scheme of infinitesimal one‐parameter subgroups introduced in .…”

Section: Modules For Infinitesimal Group Schemes and Maps To Grassmanmentioning

confidence: 99%

Representations of finite group schemes and morphisms of projective varieties

Farnsteiner

2017

Proc. London Math. Soc.

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ABSTRACT. Given a finite group scheme G over an algebraically closed field k of characteristic char(k) = p > 0, we introduce new invariants for a G-module M by associating certain morphisms degto M that take values in Grassmannians of M . These maps are studied for two classes of finite algebraic groups, infinitesimal group schemes and elementary abelian group schemes. The maps associated to the so-called modules of constant j-rank have a well-defined degree ranging between 0 and j rk j (M ), where rk j (M ) is the generic j-rank of M . The extreme values are attained when the module M has the equal images property or the equal kernels property. We establish a formula linking the j-degrees of M and its dual M * . For a self-dual module M of constant Jordan type this provides information concerning the indecomposable constituents of the pull-back α * (M ) of M along a p-point α : k[X]/(X p ) −→ kG.

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“…The previous results for p-restricted Lie algebras were extended to infinitesimal group schemes in two papers by A. Suslin, C. Bendel, and the author [30] in 1997. Infinitesimal group schemes are group schemes whose coordinate algebras are finite dimensional, local k-algebras; infinitesimal group schemes of height 1 correspond naturally to p-restricted Lie algebras (see [23]).…”

Section: Extensions and Refinementsmentioning

confidence: 95%

Spectrum of group cohomology and support varieties

Friedlander¹

2013

J. K-Theory

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We give a brief introduction to two fundamental papers by Daniel Quillen appearing in the Annals, 1971. These papers established the foundations of equivariant cohomology and gave a qualitative description of the cohomology of an arbitrary finite group. We briefly describe some of the influence of these seminal papers in the study of cohomology and representations of finite groups, restricted Lie algebras, and related structures.

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Infinitesimal 1-parameter subgroups and cohomology

Cited by 104 publications

References 9 publications

Cohomology and projectivity of modules for finite group schemes

Cohomology and projectivity of modules for finite group schemes

Representations of finite group schemes and morphisms of projective varieties

Spectrum of group cohomology and support varieties

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