scholarly journals DISPLAYED EQUATIONS FOR GALOIS REPRESENTATIONS

2018 ◽  
Vol 235 ◽  
pp. 86-114 ◽  
Author(s):  
EIKE LAU
Keyword(s):  
Local Ring ◽  
Galois Representations ◽  
Residue Field ◽  
Group Scheme ◽  
Galois Representation ◽  
Divisible Group ◽  
Noetherian Local Ring ◽  
Formula Group

The Galois representation associated to a $p$ -divisible group over a normal complete noetherian local ring with perfect residue field is described in terms of its Dieudonné display. As a consequence, the Kisin module associated to a commutative finite flat $p$ -group scheme via Dieudonné displays is related to its Galois representation in the expected way.

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].

scholarly journals Images of Galois representations in mod p Hecke algebras

2020 ◽  
pp. 1-21
Author(s):  
Laia Amorós
Keyword(s):  
Finite Field ◽  
Commutative Algebra ◽  
Galois Representations ◽  
Residue Field ◽  
Representation Formula ◽  
Galois Representation ◽  
Number Fields ◽  
Abelian Extensions ◽  
Hecke Eigenform ◽  
Coefficient Ring

Let [Formula: see text] denote the mod [Formula: see text] local Hecke algebra attached to a normalized Hecke eigenform [Formula: see text], which is a commutative algebra over some finite field [Formula: see text] of characteristic [Formula: see text] and with residue field [Formula: see text]. By a result of Carayol we know that, if the residual Galois representation [Formula: see text] is absolutely irreducible, then one can attach to this algebra a Galois representation [Formula: see text] that is a lift of [Formula: see text]. We will show how one can determine the image of [Formula: see text] under the assumptions that (i) the image of the residual representation contains [Formula: see text], (ii) [Formula: see text] and (iii) the coefficient ring is generated by the traces. As an application we will see that the methods that we use allow to deduce the existence of certain [Formula: see text]-elementary abelian extensions of big non-solvable number fields.

On Special Fiber Rings of Modules

2019 ◽  
Vol 72 (1) ◽  
pp. 225-242
Author(s):  
Cleto B. Miranda-Neto
Keyword(s):  
Local Ring ◽  
Arbitrary Dimension ◽  
Residue Field ◽  
Independent Interest ◽  
General Situation ◽  
Finitely Generated ◽  
Fiber Ring ◽  
Noetherian Local Ring ◽  
Reduction Number

AbstractWe prove results concerning the multiplicity as well as the Cohen–Macaulay and Gorenstein properties of the special fiber ring $\mathscr{F}(E)$ of a finitely generated $R$-module $E\subsetneq R^{e}$ over a Noetherian local ring $R$ with infinite residue field. Assuming that $R$ is Cohen–Macaulay of dimension 1 and that $E$ has finite colength in $R^{e}$, our main result establishes an asymptotic length formula for the multiplicity of $\mathscr{F}(E)$, which, in addition to being of independent interest, allows us to derive a Cohen–Macaulayness criterion and to detect a curious relation to the Buchsbaum–Rim multiplicity of $E$ in this setting. Further, we provide a Gorensteinness characterization for $\mathscr{F}(E)$ in the more general situation where $R$ is Cohen–Macaulay of arbitrary dimension and $E$ is not necessarily of finite colength, and we notice a constraint in terms of the second analytic deviation of the module $E$ if its reduction number is at least three.

scholarly journals Hilbert-Samuel function and Grothendieck group

2000 ◽  
Vol 43 (1) ◽  
pp. 73-94
Author(s):  
Koji Nishida
Keyword(s):  
Local Ring ◽  
Grothendieck Group ◽  
Residue Field ◽  
Primary Ideal ◽  
Hilbert Polynomial ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Graded Module ◽  
Primary Ideals

AbstractLet (A, m) be a Noetherian local ring such that the residue field A/m is infinite. Let I be arbitrary ideal in A, and M a finitely generated A-module. We denote by ℓ(I, M) the Krull dimension of the graded module ⊕n≥0InM/mInM over the associated graded ring of I. Notice that ℓ(I, A) is just the analytic spread of I. In this paper, we define, for 0 ≤ i ≤ ℓ = ℓ(I, M), certain elements ei(I, M) in the Grothendieck group K0(A/I) that suitably generalize the notion of the coefficients of Hilbert polynomial for m-primary ideals. In particular, we show that the top term eℓ (I, M), which is denoted by eI(M), enjoys the same properties as the ordinary multiplicity of M with respect to an m-primary ideal.

scholarly journals The rank of syzygies under the action by a finite group

1978 ◽  
Vol 71 ◽  
pp. 1-12 ◽  
Author(s):  
Shiro Goto
Keyword(s):  
Finite Group ◽  
Local Ring ◽  
Group Ring ◽  
Group Algebra ◽  
Maximal Ideal ◽  
Residue Field ◽  
Noetherian Local Ring ◽  
Trace Ideal ◽  
A Finite Group ◽  
Twisted Group Ring

Let S be a Noetherian local ring with maximal ideal J and k the residue field of S. Let G be a finite group of order n and suppose that G acts on S as automorphisms. Let R = SG and I = JG. We denote by S[G] (resp. R[G]) the twisted group ring of G over S (resp. the group algebra of G over R). Recall that the multiplication of S[G] is defined as follows : sg · th = sg(t) · gh for s, t ∊ S and g, h ∊ G. Let tG(S) = {Σg ∈ Gg(s)/s ∈ S} and call it the trace ideal of S. Note that tG(S) = R if n is a unit of S. We say that S has a normal basis if S ≅ R[G] as R[G]-modules. This condition says that there is an element s of S so that {g(s)}g ∈ G forms an R-free basis of S.

Cofiniteness and vanishing of local cohomology modules

1991 ◽  
Vol 110 (3) ◽  
pp. 421-429 ◽  
Author(s):  
Craig Huneke ◽  
Jee Koh
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Residue Field ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let R be a noetherian local ring with maximal ideal m and residue field k. If M is a finitely generated R-module then the local cohomology modules are known to be Artinian. Grothendieck [3], exposé 13, 1·2 made the following conjecture:If I is an ideal of R and M is a finitely generated R-module, then HomR (R/I, ) is finitely generated.

scholarly journals The Hilbert Coefficients of the Fiber Cone and the a-Invariant of the Associated Graded Ring

2009 ◽  
Vol 61 (4) ◽  
pp. 762-778 ◽  
Author(s):  
Clare D'Cruz ◽  
Tony J. Puthenpurakal
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Hilbert Coefficients ◽  
Fiber Cone

Abstract.Let (A,m) be a Noetherian local ring with infinite residue field and let I be an ideal in A and let be the fiber cone of I. We prove certain relations among the Hilbert coefficients f0(I), f1(I), f2(I) of F(I) when the a-invariant of the associated graded ring G(I) is negative.

KUMMER COVERINGS AND SPECIALISATION

2021 ◽  
pp. 1-37
Author(s):  
Martin Olsson
Keyword(s):  
Local Ring ◽  
Fundamental Groups ◽  
Technical Result ◽  
Noetherian Local Ring ◽  
Log Schemes ◽  
Log Scheme

Abstract We prove versions of various classical results on specialisation of fundamental groups in the context of log schemes in the sense of Fontaine and Illusie, generalising earlier results of Hoshi, Lepage and Orgogozo. The key technical result relates the category of finite Kummer étale covers of an fs log scheme over a complete Noetherian local ring to the Kummer étale coverings of its reduction.

scholarly journals Maximal depth property of finitely generated modules

2018 ◽  
Vol 17 (11) ◽  
pp. 1850202 ◽  
Author(s):  
Ahad Rahimi
Keyword(s):  
Local Ring ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Maximal Depth ◽  
Finitely Generated Modules ◽  
Associated Prime

Let [Formula: see text] be a Noetherian local ring and [Formula: see text] a finitely generated [Formula: see text]-module. We say [Formula: see text] has maximal depth if there is an associated prime [Formula: see text] of [Formula: see text] such that depth [Formula: see text]. In this paper, we study finitely generated modules with maximal depth. It is shown that the maximal depth property is preserved under some important module operations. Generalized Cohen–Macaulay modules with maximal depth are classified. Finally, the attached primes of [Formula: see text] are considered for [Formula: see text].

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

2021 ◽  
Vol 28 (01) ◽  
pp. 13-32
Author(s):  
Nguyen Tien Manh
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Graded Modules ◽  
Fiber Cone

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

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