The Weil descent functor in the category of algebras with free operators

TL;DR

This paper establishes a version of Weil descent in the category of $ ext{D}$-algebras, including difference algebras, under mild conditions, expanding the theoretical framework of algebraic descent methods.

Contribution

It introduces a Weil descent functor for $ ext{D}$-algebras, including difference algebras, which was previously not available in the literature.

Findings

Weil descent exists in the category of $ ext{D}$-algebras.

The construction applies under mild assumptions on associated endomorphisms.

This extends the Weil descent framework to difference algebra contexts.

Abstract

We prove that there exists a version of Weil descent, or Weil restriction, in the category of $\mathcal{D}$-algebras. The objects of this category are $k$-algebras $R$ equipped with a homomorphism $e \colon R \to R \otimes_k \mathcal{D}$ for some fixed field $k$ and finite-dimensional $k$-algebra $\mathcal{D}$. We do this under a mild assumption on the so-called associated endomorphisms. In particular, this yields the existence of the Weil descent functor in the category of difference algebras, which, to our knowledge, does not appear elsewhere.