scholarly journals The ℓ-modular representation of reductive groups over finite local rings of length two

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nariel Monteiro
Keyword(s):  
Residue Field ◽  
Reductive Group ◽  
Group Scheme ◽  
Modular Representation ◽  
Group Algebras ◽  
Large Field ◽  
Local Rings ◽  
Reductive Groups ◽  
Good For ◽  
Prime 2

Abstract Let O 2 \mathcal{O}_{2} and O 2 ′ \mathcal{O}^{\prime}_{2} be two distinct finite local rings of length two with residue field of characteristic 𝑝. Let G ⁢ ( O 2 ) \mathbb{G}(\mathcal{O}_{2}) and G ⁢ ( O 2 ′ ) \mathbb{G}(\mathcal{O}^{\prime}_{2}) be the groups of points of any reductive group scheme 𝔾 over ℤ such that 𝑝 is very good for G × F q \mathbb{G}\times\mathbb{F}_{q} or G = GL n \mathbb{G}=\operatorname{GL}_{n} . We prove that there exists an isomorphism of group algebras K ⁢ G ⁢ ( O 2 ) ≅ K ⁢ G ⁢ ( O 2 ′ ) K\mathbb{G}(\mathcal{O}_{2})\cong K\mathbb{G}(\mathcal{O}^{\prime}_{2}) , where 𝐾 is a sufficiently large field of characteristic different from 𝑝.

scholarly journals A Satake isomorphism in characteristic p

2010 ◽  
Vol 147 (1) ◽  
pp. 263-283 ◽  
Author(s):  
Florian Herzig
Keyword(s):  
Galois Representations ◽  
Maximal Compact Subgroup ◽  
Algebraic Closure ◽  
Residue Field ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Reductive Groups ◽  
Characteristic P ◽  
Satake Isomorphism

AbstractSuppose that G is a connected reductive group over a p-adic field F, that K is a hyperspecial maximal compact subgroup of G(F), and that V is an irreducible representation of K over the algebraic closure of the residue field of F. We establish an analogue of the Satake isomorphism for the Hecke algebra of compactly supported,K-biequivariant functions f:G(F)→End   V. These Hecke algebras were first considered by Barthel and Livné for GL 2. They play a role in the recent mod p andp-adic Langlands correspondences for GL 2 (ℚp) , in generalisations of Serre’s conjecture on the modularity of mod p Galois representations, and in the classification of irreducible mod p representations of unramified p-adic reductive groups.

scholarly journals On Grothendieck–Serre’s conjecture concerning principal -bundles over reductive group schemes: I

2014 ◽  
Vol 151 (3) ◽  
pp. 535-567 ◽  
Author(s):  
I. Panin ◽  
A. Stavrova ◽  
N. Vavilov
Keyword(s):  
Reductive Group ◽  
Group Scheme ◽  
Local Rings ◽  
Infinite Field ◽  
Principal Bundles ◽  
Serre's Conjecture ◽  
Irreducible Variety ◽  
Formula Group ◽  
Image Position ◽  
Serre’S Conjecture

AbstractLet$k$be an infinite field. Let$R$be the semi-local ring of a finite family of closed points on a$k$-smooth affine irreducible variety, let$K$be the fraction field of$R$, and let$G$be a reductive simple simply connected$R$-group scheme isotropic over$R$. Our Theorem 1.1 states that for any Noetherian$k$-algebra$A$the kernel of the map$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{1}(R\otimes _{k}A,G)\rightarrow H_{\acute{\text{e}}\text{t}}^{1}(K\otimes _{k}A,G)\end{eqnarray}$$induced by the inclusion of$R$into$K$is trivial. Theorem 1.2 for$A=k$and some other results of the present paper are used significantly in Fedorov and Panin [A proof of Grothendieck–Serre conjecture on principal bundles over a semilocal regular ring containing an infinite field, Preprint (2013),arXiv:1211.2678v2] to prove the Grothendieck–Serre’s conjecture for regular semi-local rings$R$containing an infinite field.

scholarly journals Characterizations of regular local rings via syzygy modules of the residue field

2018 ◽  
Vol 10 (3) ◽  
pp. 327-337
Author(s):  
Dipankar Ghosh ◽  
Anjan Gupta ◽  
Tony J. Puthenpurakal
Keyword(s):  
Residue Field ◽  
Local Rings ◽  
Syzygy Modules ◽  
Regular Local Rings

scholarly journals VANISHING OF (CO)HOMOLOGY OVER DEFORMATIONS OF COHEN-MACAULAY LOCAL RINGS OF MINIMAL MULTIPLICITY

2018 ◽  
Vol 61 (03) ◽  
pp. 705-725
Author(s):  
DIPANKAR GHOSH ◽  
TONY J. PUTHENPURAKAL
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Sufficient Conditions ◽  
Regular Sequence ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Syzygy Modules ◽  
Tor Modules ◽  
Necessary And Sufficient ◽  
Minimal Multiplicity

AbstractLet R be a d-dimensional Cohen–Macaulay (CM) local ring of minimal multiplicity. Set S := R/(f), where f := f1,. . .,fc is an R-regular sequence. Suppose M and N are maximal CM S-modules. It is shown that if ExtSi(M, N) = 0 for some (d + c + 1) consecutive values of i ⩾ 2, then ExtSi(M, N) = 0 for all i ⩾ 1. Moreover, if this holds true, then either projdimR(M) or injdimR(N) is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field.

Reduction numbers and Hilbert polynomials of ideals in higher dimensional CohenMacaulay local rings

1992 ◽  
Vol 111 (1) ◽  
pp. 47-56 ◽  
Author(s):  
Yinghwa Wu
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Primary Ideal ◽  
Local Rings ◽  
Hilbert Polynomials ◽  
Higher Dimensional ◽  
Reduction Number ◽  
Minimal Reduction ◽  
System Of Parameters

Throughout, (R, m) will denote a d-dimensional CohenMacaulay (CM for short) local ring having an infinite residue field and I an m-primary ideal in R. Recall that an ideal J I is said to be a reduction of I if Ir+1 = JIr for some r 0, and a reduction J of I is called a minimal reduction of I if J is generated by a system of parameters. The concepts of reduction and minimal reduction were first introduced by Northcott and Rees12. If J is a reduction of I, define the reduction number of I with respect to J, denoted by rj(I), to be min {r 0 Ir+1 = JIr}. The reduction number of I is defined as r(I) = min {rj(I)J is a minimal reduction of I}. The reduction number r(I) is said to be independent if r(I) = rj(I) for every minimal reduction J of I.

scholarly journals Linearity defect of the residue field of short local rings

2016 ◽  
Vol 16 (09) ◽  
pp. 1750163
Author(s):  
Rasoul Ahangari Maleki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Residue Field ◽  
Positive Answer ◽  
Local Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Linear Resolution

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text] and residue field [Formula: see text]. The linearity defect of a finitely generated [Formula: see text]-module [Formula: see text], which is denoted [Formula: see text], is a numerical measure of how far [Formula: see text] is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether [Formula: see text] implies [Formula: see text], in the case when [Formula: see text].

Central extensions and groups locally of minimal type

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Root System ◽  
Reductive Group ◽  
Reductive Groups ◽  
Central Extensions ◽  
General Lemma ◽  
Quotient Maps ◽  
Central Quotient ◽  
Characteristic 2 ◽  
Minimal Type ◽  
Quadratic Case

This chapter deals with central extensions and groups locally of minimal type. It begins with a discussion of the general lemma on the behavior of the scheme-theoretic center with respect to the formation of central quotient maps between pseudo-reductive groups; this lemma generalizes a familiar fact in the connected reductive case. The chapter then considers four phenomena that go beyond the quadratic case, along with a pseudo-reductive group of minimal type that is locally of minimal type. It shows that the pseudo-split absolutely pseudo-simple k-groups of minimal type with a non-reduced root system are classified over any imperfect field of characteristic 2. In this classification there is no effect if the “minimal type” hypothesis is relaxed to “locally of minimal type.”

Harish-Chandra Modules over ℤ

2019 ◽  
pp. 126-167
Author(s):  
Günter Harder
Keyword(s):  
Fixed Points ◽  
Lie Algebra ◽  
Lie Group ◽  
Maximal Torus ◽  
Reductive Group ◽  
Group Scheme ◽  
Finitely Generated ◽  
Cartan Involution ◽  
Split Maximal Torus ◽  
Definition Of

This chapter shows that certain classes of Harish-Chandra modules have in a natural way a structure over ℤ. The Lie group is replaced by a split reductive group scheme G/ℤ, its Lie algebra is denoted by 𝖌ℤ. On the group scheme G/ℤ there is a Cartan involution 𝚯 that acts by t ↦ t −1 on the split maximal torus. The fixed points of G/ℤ under 𝚯 is a flat group scheme 𝒦/ℤ. A Harish-Chandra module over ℤ is a ℤ-module 𝒱 that comes with an action of the Lie algebra 𝖌ℤ, an action of the group scheme 𝒦, and some compatibility conditions is required between these two actions. Finally, 𝒦-finiteness is also required, which is that 𝒱 is a union of finitely generated ℤ modules 𝒱I that are 𝒦-invariant. The definitions imitate the definition of a Harish-Chandra modules over ℝ or over ℂ.

scholarly journals Representations of reductive groups over finite local rings of length two

2019 ◽  
Vol 525 ◽  
pp. 171-190 ◽  
Author(s):  
Alexander Stasinski ◽  
Andrea Vera-Gajardo
Keyword(s):  
Local Rings ◽  
Reductive Groups

scholarly journals Chevalley groups of polynomial rings over Dedekind domains

2020 ◽  
Vol 23 (1) ◽  
pp. 121-132 ◽  
Author(s):  
Anastasia Stavrova
Keyword(s):  
Reductive Group ◽  
Group Scheme ◽  
Dedekind Domain ◽  
Polynomial Rings ◽  
Higher Dimension ◽  
Orthogonal Groups ◽  
Chevalley Groups ◽  
Dedekind Domains ◽  
Regular Rings

AbstractLet R be a Dedekind domain and G a split reductive group, i.e. a Chevalley–Demazure group scheme, of rank {\geq 2}. We prove thatG(R[x_{1},\ldots,x_{n}])=G(R)E(R[x_{1},\ldots,x_{n}])\quad\text{for any}\ n% \geq 1.In particular, this extends to orthogonal groups the corresponding results of A. Suslin and F. Grunewald, J. Mennicke and L. Vaserstein for {G=\mathrm{SL}_{N},\mathrm{Sp}_{2N}}. We also deduce some corollaries of the above result for regular rings R of higher dimension and discrete Hodge algebras over R.

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