I presume that answers to the following questions are likely to exist in the literature; so this question is mostly a reference request (but failing that, I would be certainly interested in learning a proof which cannot be readily found in published expositions).


Let $R$ be a Noetherian commutative ring (of finite Krull dimension, if it helps), and let $S=R[x\_1,\dotsc,x\_n]$ be the ring of polynomials in $n$ variables with the coefficients in $R$. Hilbert's syzygy theorem claims that if $R$ is a field then the global dimension of $S$ does not exceed (in fact, is equal to) $n$. I am interested in the following relative versions of this classical theorem.


Let $G$ be an $S$-module. Assume that $G$ is flat as an $R$-module. How does one prove that the flat dimension of the $S$-module $G$ does not exceed $n$?


Similarly, let $Q$ be an $S$-module. Assume that $Q$ is projective as an $R$-module. How does one prove that the projective dimension of the $S$-module $Q$ does not exceed $n$?


Most interestingly for me, let $K$ be an $S$-module. Assume that $K$ is injective as an $R$-module. How does one prove that the injective dimension of the $S$-module $K$ does not exceed $n$?


[The context: among other things, I am trying to understand the following more general question, which is relevant for my studies of semi-infinite algebraic geometry. Let $R\longrightarrow S$ be a smooth morphism of Noetherian commutative rings. Let $J^\bullet$ be an unbounded complex of injective $S$-modules. Since $S$ is a flat $R$-module, all injective $S$-modules are also injective as $R$-modules. Assume that, as a complex of $R$-modules, $J^\bullet$ is contractible. I would like to show that $J^\bullet$ is then also contractible as a complex of $S$-modules. Notice that this is certainly not true for an arbitrary flat morphism $R\longrightarrow S$, not even for a flat morphism of finite type (take $R$ to be a field).]



Relative version of Hilbert syzygy theorem
The answers to the first two questions readily follow from Proposition 7.5.2 in [J. C. McConnell and J. C. Robson, "Noncommutative Noetherian rings", AMS, 1987]. Actually, they prove a much more general result which holds for skew polynomial rings and skew Laurent polynomial rings. The base ring $R$ can be arbitrary, not necessarily commutative.


