Equivariant map

About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.

Equivariant Epsilon Constants, Discriminants and Étale Cohomology

Abstract: Let $L/K$ be a finite Galois extension of number fields. We formulate and study a conjectural equality between an element of the relative algebraic K-group $K_0(\mathbb{Z}[\mathrm{Gal}(L/K)], \mathbb{R})$ which is constructed from the equivariant Artin epsilon constant of $L/K$ and a sum of structural invariants associated to $L/K$. The precise conjecture is motivated by the requirement that a special case of the equivariant refinement of the Tamagawa Number Conjecture of Bloch and Kato (as formulated by Flach and the second-named author) should be compatible with the functional equation of the associated L-function. We show that, more concretely, our conjecture also suggests a completely systematic refinement of the central approach and results of classical Galois module theory. In particular, the evidence for our conjecture that we present here already strongly refines many of the main results of Galois module theory.

Théorie de Lubin-Tate non-abélienne et représentations elliptiques

Abstract: Non abelian Lubin–Tate theory studies the cohomology of some moduli spaces for p-divisible groups, the broadest definition of which is due to Rapoport–Zink, aiming both at providing explicit realizations of local Langlands functoriality and at studying bad reduction of Shimura varieties. In this paper we consider the most famous examples ; the so-called Drinfeld and Lubin–Tate towers. In the Lubin–Tate case, Harris and Taylor proved that the supercuspidal part of the cohomology realizes both the local Langlands and Jacquet–Langlands correspondences, as conjectured by Carayol. Recently, Boyer computed the remaining part of the cohomology and exhibited two defects : first, the representations of GL d which appear are of a very particular and restrictive form ; second, the Langlands correspondence is not realized anymore. In this paper, we study the cohomology complex in a suitable equivariant derived category, and show how it encodes Langlands correspondence for elliptic representations. Then we transfer this result to the Drinfeld tower via an enhancement of a theorem of Faltings due to Fargues. We deduce that Deligne’s weight-monodromy conjecture is true for varieties uniformized by Drinfeld’s coverings of his symmetric spaces. This completes the computation of local L-factors of some unitary Shimura varieties.

Exact cohomology sequences with integral coefficients for torus actions

Abstract: Using methods applied by Atiyah in equivariant K-theory, Bredon obtained exact sequences for the relative cohomologies (with rational coefficients) of the equivariant skeletons of (sufficiently nice) T-spaces X, $T=(S^1)^n,$ with free equivariant cohomology $H_T^*(X;{\mathbb Q})$ over $H_T^*({\rm pt};{\mathbb Q})=H^*(BT;{\mathbb Q}).$ Here we characterise those finite T-CW complexes with connected isotropy groups for which an analogous result holds with integral coefficients.

Auslander-Buchweitz Approximations of Equivariant Modules

Abstract: This book, first published in 2000, focuses on homological aspects of equivariant modules. It presents a homological approximation theory in the category of equivariant modules, unifying the Cohen-Macaulay approximations in commutative ring theory and Ringel's theory of delta-good approximations for quasi-hereditary algebras and reductive groups. The book provides a detailed introduction to homological algebra, commutative ring theory and homological theory of comodules of co-algebras over an arbitrary base. It aims to overcome the difficulty of generalising known homological results in representation theory. This book will be of interest to researchers and graduate students in algebra, specialising in commutative ring theory and representation theory.