$\DeclareMathOperator\GL{GL}$Let $R$ be a commutative ring, let $R[n] := R[M\_d^{\oplus n}]$ be the polynomial ring on $nd^2$ variables corresponding to the coordinates of $n$-many $d\times d$ matrices. Let these matrix variables be $X\_1,\ldots,X\_n$. For an $d\times d$ matrix $A$, let $c\_k(A)$ be the coefficient of $T^k$ in the characteristic polynomial $\det(A-TI)$.


The group scheme $\GL\_{d,R}$ acts on $R[n]$ by simultaneous conjugation on the $d$ matrices. Clearly for any product $X\_{i\_1}X\_{i\_2}\cdots X\_{i\_r}$ (where $i\_j\in[1\ldots n], r\ge 1$), the function $c\_k(X\_{i\_1}\cdots X\_{i\_r})\in R[n]$ is invariant under $\GL\_{d,R}$.


Is it true that for any commutative ring $R$, $R[n]^{\GL\_{d,R}}$ is generated as an $R$-algebra by the functions $c\_k(X\_{i\_1}\cdots X\_{i\_r})$?


Does anyone have a reference for this?


Remark - The statement over $\mathbb{C}$ is a classical result of Sibirski and Procesi. This was later extended to the case $R = \mathbb{Z}$ and $R$ any algebraically closed field by Donkin in [*Invariants of several matrices*](https://www.researchgate.net/profile/Stephen-Donkin/publication/226194761_Invariants_of_several_matrices/links/00b4951a7231dd4ed0000000/Invariants-of-several-matrices.pdf). In Concini-Procesi's [*The invariant theory of matrices*](https://bookstore.ams.org/ulect-69), they also seem to obtain the result when $R$ is an infinite field.



Invariants of matrices (by simultaneous $\mathrm{GL}_n$ conjugation) over arbitrary rings
It is true. The standard reference is the Book by Jantzen, Representations of Algebraic Groups, Second edition. In particular we need the Appendix `Chapter B', and the base change Proposition in part I, 4.18. By Donkin the $G\_{\mathbb Z}$ module $M={\mathbb Z}[n]$ has good filtration. So $H^1(G\_{\mathbb Z},M)$ vanishes. And $H^0(G\_{\mathbb Z},M)$ just means $M^{G\_{\mathbb Z}}$. The base change Proposition tells that the map $M^{G\_{\mathbb Z}}\otimes R\to (M\otimes R)^{G\_{R}}$ is an isomorphism. The result follows.


