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

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 $e:R\to R{\otimes }_k\mathcal{D}$ 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.

32 pages. This version contains a different proof of the main theorem (section 5). A sketch of the original proof now appears in the appendix