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Galois measurings for noncommutative base change of entwined contramodule and entwined comodule categories

Divya Ahuja, Abhishek Banerjee, Surjeet Kour

Abstract

We study the noncommutative base change of an entwining structure $(A,C,ψ)$ by a Grothendieck category $\mathfrak S$, using two module like categories. These are the categories of entwined comodule objects and entwined contramodule objects in $\mathfrak S$ over the entwining structure $(A,C,ψ)$. We consider criteria for maps between these noncommutative spaces, induced by generalized maps between entwining structures, known as measurings, to behave like Galois extensions. We also study conditions for extensions of these noncommutative spaces, understood as functors between module like categories, to have separability, Frobenius or Maschke type properties.

Galois measurings for noncommutative base change of entwined contramodule and entwined comodule categories

Abstract

We study the noncommutative base change of an entwining structure by a Grothendieck category , using two module like categories. These are the categories of entwined comodule objects and entwined contramodule objects in over the entwining structure . We consider criteria for maps between these noncommutative spaces, induced by generalized maps between entwining structures, known as measurings, to behave like Galois extensions. We also study conditions for extensions of these noncommutative spaces, understood as functors between module like categories, to have separability, Frobenius or Maschke type properties.

Paper Structure

This paper contains 11 sections, 39 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Entwined comodule objects in a Grothendieck category
  3. Entwined contramodule objects in a Grothendieck category
  4. $\mathfrak S$-contra-Galois measurings, $\mathfrak S$-co-Galois measurings and adjoint functors
  5. $\mathfrak S$-contra Galois measurings and entwined contramodule objects
  6. $\mathfrak S$-Galois measurings and entwined comodule objects
  7. Separability properties
  8. Separability of the functors $^{[C,-]}\mathcal{T}$ and $\mathcal{T}^C$
  9. Separability of the functors $^{[C,-]}\mathcal{F}$ and $\mathcal{F}^C$
  10. Frobenius properties
  11. Semisimple and Maschke functors

Key Result

Theorem A

(see Theorem T4.7 and Theorem T4.13) Let $\mathfrak S$ be a $k$-linear Grothendieck category. Let $(\alpha, \gamma)$ be a measuring of entwining structures from $(A',C',\psi')$ to $(A,C,\psi)$. (a) Suppose that the functor $(A',-):\mathfrak S\longrightarrow \mathfrak S$ is exact. For categories of e (b) For categories of entwined comodule objects in $\mathfrak S$, the following are equivalent:

Theorems & Definitions (84)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • ...and 74 more