Do all reductive group schemes over semilocal rings admit finite-dimensional free faithful representations?

Question

The definition of a reductive group scheme is as in SGA III. Frankly, I only know that they exist for the adjoint group (the adjoint representation). In SGA III, I could only find a result for general groups over a regular ring of dimension $\leq 2.$ But since reductive groups are especially nice, maybe they do have such a representation? If not all reductive groups do, which of them do?

Answer 1

In Corollary 3.2 of the paper

R. W. Thomason, Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. Math. 65 (1987), 16–34,

this was proved for semisimple group schemes or, more generally, for reductive group schemes which are either split reductive, or semisimple, or with isotrivial radical and coradical, or over a normal base S.

Answer 2

The usual argument shows that a flat affine group scheme $G$ of finite type over a noetherian ring $k$ has a faithful representation on a finitely generated submodule $M$ of the regular representation. If $M$ is flat over $k$, then it is projective, and hence a direct summand of a free finitely generated $k$-module $L$, and so $G\hookrightarrow\mathrm{GL}_{\mathrm{rank}(L)}$. When $k$ is a Dedekind domain and $G$ is flat, the module $M$ is torsion-free, and hence automatically flat. Thus, every flat affine group scheme of finite type over a Dedekind domain admits an embedding into $\mathrm{GL}_{n}$ for some $n$. As every split reductive group scheme over a ring $k$ arises by base change from a similar group over $\mathbb{Z}$ (Chevalley), such group schemes admit embeddings into $\mathrm{GL}_{n}$. Every reductive group splits over an etale extension of the base ring (SGA 3); when the extension can be taken to be finite, an argument using restriction of scalars proves the statement for the reductive group (cf. question 22078).