Filtrations, 1-parameter subgroups, and rational injectivity

Friedlander, Eric M.

doi:10.1016/j.aim.2017.10.023

Cited by 6 publications

(11 citation statements)

References 13 publications

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“…In his final lecture, the author presented his construction M → V (G) M of support varieties for G-modules M , where G is a linear algebraic group "of exponential type". The beginnings of this theory can be found in [21] and some applications in [23]. The theory succeeds in that the support varieties defined here extend those for infinitesimal kernels, have many of the expected properties (see Theorem 4.13), and are formulated intrinsically for those linear algebraic groups for which the theory applies.…”

Section: Lecture Iv: Support Varieties For Linear Algebraic Groupsmentioning

confidence: 87%

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Rational Cohomology and Supports for Linear Algebraic Groups

Friedlander

2018

Springer Proceedings in Mathematics &Amp; Statistics

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Section: Lecture Iv: Support Varieties For Linear Algebraic Groupsmentioning

confidence: 87%

“…In [23], the author shows how to construct mock trivial G-modules for any G which is not unipotent. Definition 4.19.…”

Section: 4mentioning

confidence: 99%

Rational Cohomology and Supports for Linear Algebraic Groups

Friedlander

2018

Springer Proceedings in Mathematics &Amp; Statistics

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“…Among the many important features of support varieties in the finite group scheme setting is their ability to detect the injectivity of a module (which is equivalent to detecting projectivity in this case). A natural question in that is investigated is whether or not the newly introduced rational supports detect injectivity for rational

G

‐modules.…”

Section: Introductionmentioning

confidence: 99%

“…In , it is shown that a

G

‐module

M

has trivial rational support precisely when

M

is injective over every Frobenius kernel of

G

. Such modules are dubbed to be ‘mock injective’.…”

Section: Introductionmentioning

confidence: 99%

On the existence of mock injective modules for algebraic groups

Hardesty

Nakano

Sobaje

2017

Bull. London Math. Soc.

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Let G be an affine algebraic group scheme over an algebraically closed field k of characteristic p > 0, and let Gr denote the rth Frobenius kernel of G. Motivated by recent work of Friedlander, the authors investigate the class of mock injective G-modules, which are defined to be those rational G-modules that are injective on restriction to Gr for all r 1. In this paper, the authors provide necessary and sufficient conditions for the existence of non-injective mock injective G-modules, thereby answering a question raised by Friedlander. Furthermore, the authors investigate the existence of non-injective mock injectives with simple socles. Interesting cases are discovered that show that this can occur for reductive groups, but will not occur for their Borel subgroups.Thus, every central character is of the form χ = λ + ZJ. If we let π : X(T ) → X(Z J ) denote the quotient map, then π(ZΦ) = ZΦ/ZJ ∼ = ZI where I = Δ\J. It follows that any L J -module M , whose weights all lie in ZΦ, has a central character decomposition of the form M = χ∈ZI M χ . This statement particularly applies to k[U J ]. Lemma 3.3.2. The L J -module k[U J ] has a central character decomposition of the formProof. First, recall that k[U J ] is a polynomial ring given bywhere each x γ has weight γ. For any weight λ ∈ X(T ), the weight space k[U J ] λ is spanned by monomials of the form γ∈Φ + \Φ + J x nγ γ , where λ = γ n γ γ with n γ 0 is any expression for λ. Since there are only finitely many ways to express λ as a non-negative integral combination of elements in Φ + \Φ + J , we can conclude that the weight spaces are finite-dimensional. Fix a central character χ ∈ X(Z J ) which acts non-trivially on k[U J ]. By the remark preceding this lemma and the above description of the weight spaces of k[U J ], it follows that there exists a unique weight λ ∈ NI satisfying χ = λ + ZJ.

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“…Support theories have led to the identification and study of various interesting special classes of representations such as "modules of constant Jordan type" [9] and "mock injective" modules [16], as well as invariants given by maximal Jordan types [23]. Support theory has led to the construction of algebraic vector bundles which provide further invariants of representations [22], [6].…”

mentioning

confidence: 99%

Support Varieties and stable categories for algebraic groups

Friedlander¹

2021

Preprint

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We develop a theory of supports M → V (G) M for arbitrary Gmodules M for a linear algebraic group G of exponential type over a field k of positive characteristic p > 0. Our construction is an refinement and a generalization of an earlier approach by the author. We extend our support theory to apply to bounded complexes of G-modules, C • → V (G) C • . We introduce the tensor triangulated category StM od(G), a Verdier quotient of the derived category D b (M od(G)), and we view StM od(G) as the stable module category of G-modules. Our support theory is well defined on StM od(G), satisfies the "standard properties" for a theory of supports, and extends the support theory for the Frobenius kernels G (r) ⊂ G as considered by Suslin, Bendel, and the author. We also consider the stable module category of finite dimensional G-modules, stmod(G).As an application of our support theory, we employ our support varieties to establish the classification of (r)-complete, thick tensor ideals of stmod(G) by locally G-finte subsets of V (G) and the classification of (r)-complete, localizing tensor ideals of the stable module category StM od(G) by G-realizable subsets of V (G).

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Filtrations, 1-parameter subgroups, and rational injectivity

Cited by 6 publications

References 13 publications

Rational Cohomology and Supports for Linear Algebraic Groups

Rational Cohomology and Supports for Linear Algebraic Groups

On the existence of mock injective modules for algebraic groups

Support Varieties and stable categories for algebraic groups

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