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For the benefit of myself and other novices in the area, I wanted to add some details to Wilberd's excellent answer, citing Donkin's Invariants of several matrices and Jantzen's book Representations of Algebraic Groups as appropriate.

We work over a base ring $R$.

First, one checks that Donkin's definition of a "good filtration" on a $G$-module $V$ coincides with Jantzens (defined in II, 4.16). The latter uses the isomorphism $H^i(M) := R^i\text{Ind}^G_B(V)\cong H^i(G/B,\mathcal{L}(M))$ (Jantzen I, 5.12), where $B\subset G$ is a Borel, $M$ a $B$-module, and $\mathcal{L}(M)$ denotes the quasi-coherent sheaf on $G/B$ associated to $M$ (I, 5.8). If $M = R$ with trivial $G$-action, then $\mathcal{L}(R)$ is the structure sheaf $\mathcal{O}_{G/B}$.

Now write $R[n] = \bigoplus_{d\ge 0}R[n]_d$, where $R[n]_d$ consists of the polynomials of total degree $d$. One easily checks that each $R[n]_d$ is a $\text{GL}_{n,\mathbb{Z}}$-module. We claim that the $\mathbb{Z}$-finite $G = \text{GL}_{n,\mathbb{Z}}$-modules $\mathbb{Z}[n]_{\le D} := \bigoplus_{0\le d\le D} \mathbb{Z}[n]_d$ have good filtrations for any $D\ge 0$. We use the following Lemma

Lemma (Appendix B.9 of Jantzen) Let $G$ be a split reductive algebraic group, and let $T$ be a maximal torus. Suppose that $R$ is a principal ideal domain. Let $M$ be a $G$-module which is free of finite rank over $R$. As explained in B.4, let $V(\lambda) := H^0(-w_0\lambda)^*)$ be the Weyl module, where $w_0$ is the longest element of the Weyl group (II, 1.5). The following are equivalent:

(i) $M$ has a good filtration.

(ii) $Ext^i_G(V(\lambda),M) = 0$ for all $\lambda\in X(T)_+$ and all $i > 0$

(iii) $Ext^1_G(V(\lambda),M) = 0$ for all $\lambda\in X(T)_+$.

(iv) For each maximal ideal $\mathfrak{m}$ in $k$, the $G_{R/\mathfrak{m}}$-module $M\otimes R/\mathfrak{m}$ has a good filtration.

By (iv), it suffices to check that the $G_{\mathbb{F}_p}$-modules $\mathbb{F}_p[n]_{\le D}$ all have good filtrations. Using flat base change for $G$-module cohomology (see Jantzen I, 4.13), (ii) (or (iii)) implies that it suffices to check the existence of good filtrations over algebraically closed fields $k$. By a result of Donkin (Donkin The normality of closures of conjugacy classes of matrices Proposition 1.2a (i) and (ii)), good filtrations behave well with respect to direct summands, so it suffices to check that $\overline{\mathbb{F}_p}[n]$ have good filtrations. This is done in $\S3$ of Donkin's Invariants of several matrices. Thus we conclude that $\mathbb{Z}[n]_{\le D}$ has a good filtration for any $D\ge 0$.

Let $T$ be a maximal torus of $G = GL_{n,\mathbb{Z}}$, and let $\lambda = 0\in X(T)$, then $\lambda$ is dominant and we have $V(0) = H^0(G/B,\mathcal{L}(0)) = H^0(G/B,\mathcal{O}_{G/B})$. Since $G/B$ is smooth proper over $\mathbb{Z}$ (use Jantzen I, 5.6(9) and I,5.7 to reduce to the classical case over algebraically closed fields), Stein factorization implies that $V(0) = \mathbb{Z}$ (with trivial $G$-action). By (ii) or (iii), we conclude that $Ext^1_G(\mathbb{Z},M) = H^1(G,M) = 0$. By a "universal coefficient theorem" (see Jantzen I, 4.18), vanishing of $H^1(G,M)$ implies that for any ring $R$, $\mathbb{Z}[n]^{GL_{2,\mathbb{Z}}}\otimes R = R[n]^{GL_{2,R}}$, as desired.

For the benefit of myself and other novices in the area, I wanted to add some details to Wilberd's excellent answer, citing Donkin's Invariants of several matrices and Jantzen's book Representations of Algebraic Groups as appropriate.

We work over a base ring $R$.

First, one checks that Donkin's definition of a "good filtration" on a $G$-module $V$ coincides with Jantzens (defined in II, 4.16). The latter uses the isomorphism $H^i(M) := R^i\text{Ind}^G_B(V)\cong H^i(G/B,\mathcal{L}(M))$ (Jantzen I, 5.12), where $B\subset G$ is a Borel, $M$ a $B$-module, and $\mathcal{L}(M)$ denotes the quasi-coherent sheaf on $G/B$ associated to $M$ (I, 5.8). If $M = R$ with trivial $G$-action, then $\mathcal{L}(R)$ is the structure sheaf $\mathcal{O}_{G/B}$.

Now write $R[n] = \bigoplus_{d\ge 0}R[n]_d$, where $R[n]_d$ consists of the polynomials of total degree $d$. One easily checks that each $R[n]_d$ is a $\text{GL}_{n,R}$-module. We claim that the $\mathbb{Z}$-finite $G = \text{GL}_{n,\mathbb{Z}}$-modules $\mathbb{Z}[n]_{\le D} := \bigoplus_{0\le d\le D} \mathbb{Z}[n]_d$ have good filtrations for any $D\ge 0$. We use the following Lemma

Lemma (Appendix B.9 of Jantzen) Let $G$ be a split reductive algebraic group, and let $T$ be a maximal torus. Suppose that $R$ is a principal ideal domain. Let $M$ be a $G$-module which is free of finite rank over $R$. As explained in B.4, let $V(\lambda) := H^0(-w_0\lambda)^*)$ be the Weyl module, where $w_0$ is the longest element of the Weyl group (II, 1.5). The following are equivalent:

(i) $M$ has a good filtration.

(ii) $Ext^i_G(V(\lambda),M) = 0$ for all $\lambda\in X(T)_+$ and all $i > 0$

(iii) $Ext^1_G(V(\lambda),M) = 0$ for all $\lambda\in X(T)_+$.

(iv) For each maximal ideal $\mathfrak{m}$ in $k$, the $G_{R/\mathfrak{m}}$-module $M\otimes R/\mathfrak{m}$ has a good filtration.

By (iv), it suffices to check that the $G_{\mathbb{F}_p}$-modules $\mathbb{F}_p[n]_{\le D}$ all have good filtrations. Using flat base change for $G$-module cohomology (see Jantzen I, 4.13), (ii) (or (iii)) implies that it suffices to check the existence of good filtrations over algebraically closed fields $k$. By a result of Donkin (Donkin The normality of closures of conjugacy classes of matrices Proposition 1.2a (i) and (ii)), good filtrations behave well with respect to direct summands, so it suffices to check that $\overline{\mathbb{F}_p}[n]$ have good filtrations. This is done in $\S3$ of Donkin's Invariants of several matrices. Thus we conclude that $\mathbb{Z}[n]_{\le D}$ has a good filtration for any $D\ge 0$.

Let $T$ be a maximal torus of $G = GL_{n,\mathbb{Z}}$, and let $\lambda = 0\in X(T)$, then $\lambda$ is dominant and we have $V(0) = H^0(G/B,\mathcal{L}(0)) = H^0(G/B,\mathcal{O}_{G/B})$. Since $G/B$ is smooth proper over $\mathbb{Z}$ (use Jantzen I, 5.6(9) and I,5.7 to reduce to the classical case over algebraically closed fields), Stein factorization implies that $V(0) = \mathbb{Z}$ (with trivial $G$-action). By (ii) or (iii), we conclude that $Ext^1_G(\mathbb{Z},M) = H^1(G,M) = 0$. By a "universal coefficient theorem" (see Jantzen I, 4.18), vanishing of $H^1(G,M)$ implies that for any ring $R$, $\mathbb{Z}[n]^{GL_{2,\mathbb{Z}}}\otimes R = R[n]^{GL_{2,R}}$, as desired.

Next, we check that the $G = \text{GL}_{n,\mathbb{Z}}$-module $M = \mathbb{Z}[n]$ has a good filtration. We use the following Lemma

Lemma (Appendix B.9 of Jantzen) Suppose that $R$ is a principal ideal domain. Let $M$ be a $G$-module which is free of finite rank over $R$. As explained in B.4, let $V(\lambda) := H^0(-w_0\lambda)^*)$, where $w_0$ is the longest element of the Weyl group (II, 1.5). The following are equivalent:

We write $M[n] = \bigoplus_{d\ge 0}M[n]_d$, where $M[n]_d$ consists of the polynomials of total degree $d$. One easily checks that each $M[n]_d$ is a $\text{GL}_{n,\mathbb{Z}}$-module. We apply the lemma with $R = \mathbb{Z}$ to the $R$-finite submodules $M[n]_{\le D} := \bigoplus_{0\le d\le D} M[n]_d$ for increasing $D$. Then by (iv), it suffices to check that the $G_{\mathbb{F}_p}$-modules $\mathbb{F}_p[n]_{\le D}$ all have good filtrations. Using flat base change for $G$-module cohomology (see Jantzen I, 4.13), (ii) (or (iii)) implies that it suffices to check the existence of good filtrations over algebraically closed fields $k$. By a result of Donkin (Donkin The normality of closures of conjugacy classes of matrices Proposition 1.2a (i) and (ii)), good filtrations behave well with respect to direct summands, so it suffices to check that $\mathbb{Z}[n]\otimes k$ has a good filtration for any algebraically closed field $k$. This is done in $\S3$ of Donkin's Invariants of several matrices. Thus we conclude that $M = \mathbb{Z}[n]$ has a good filtration.

Now we work over $R = \mathbb{Z}$. Let $\lambda = 0\in X(T)$, then we have $V(0) = H^0(G/B,\mathcal{L}(0)) = H^0(G/B,\mathcal{O}_{G/B})$. Since $G/B$ is smooth proper over $\mathbb{Z}$ (use Jantzen I, 5.6(9) and I,5.7 to reduce to the classical case over algebraically closed fields), Stein factorization implies that $V(0) = \mathbb{Z}$ (with trivial $G$-action), so in particular $\lambda$ is dominant (this is II, 2.6 when $R$ is a field). By (ii) or (iii), we conclude that $Ext^1_G(\mathbb{Z},M) = H^1(G,M) = 0$. By a "universal coefficient theorem" (see Jantzen I, 4.18), vanishing of $H^1(G,M)$ implies that for any ring $R$, $\mathbb{Z}[n]^{GL_{2,\mathbb{Z}}}\otimes R = R[n]^{GL_{2,R}}$, as desired.

Now write $R[n] = \bigoplus_{d\ge 0}R[n]_d$, where $R[n]_d$ consists of the polynomials of total degree $d$. One easily checks that each $R[n]_d$ is a $\text{GL}_{n,\mathbb{Z}}$-module. We claim that the $\mathbb{Z}$-finite $G = \text{GL}_{n,\mathbb{Z}}$-modules $\mathbb{Z}[n]_{\le D} := \bigoplus_{0\le d\le D} \mathbb{Z}[n]_d$ have good filtrations for any $D\ge 0$. We use the following Lemma

Lemma (Appendix B.9 of Jantzen) Let $G$ be a split reductive algebraic group, and let $T$ be a maximal torus. Suppose that $R$ is a principal ideal domain. Let $M$ be a $G$-module which is free of finite rank over $R$. As explained in B.4, let $V(\lambda) := H^0(-w_0\lambda)^*)$ be the Weyl module, where $w_0$ is the longest element of the Weyl group (II, 1.5). The following are equivalent:

By (iv), it suffices to check that the $G_{\mathbb{F}_p}$-modules $\mathbb{F}_p[n]_{\le D}$ all have good filtrations. Using flat base change for $G$-module cohomology (see Jantzen I, 4.13), (ii) (or (iii)) implies that it suffices to check the existence of good filtrations over algebraically closed fields $k$. By a result of Donkin (Donkin The normality of closures of conjugacy classes of matrices Proposition 1.2a (i) and (ii)), good filtrations behave well with respect to direct summands, so it suffices to check that $\overline{\mathbb{F}_p}[n]$ have good filtrations. This is done in $\S3$ of Donkin's Invariants of several matrices. Thus we conclude that $\mathbb{Z}[n]_{\le D}$ has a good filtration for any $D\ge 0$.

Let $T$ be a maximal torus of $G = GL_{n,\mathbb{Z}}$, and let $\lambda = 0\in X(T)$, then $\lambda$ is dominant and we have $V(0) = H^0(G/B,\mathcal{L}(0)) = H^0(G/B,\mathcal{O}_{G/B})$. Since $G/B$ is smooth proper over $\mathbb{Z}$ (use Jantzen I, 5.6(9) and I,5.7 to reduce to the classical case over algebraically closed fields), Stein factorization implies that $V(0) = \mathbb{Z}$ (with trivial $G$-action). By (ii) or (iii), we conclude that $Ext^1_G(\mathbb{Z},M) = H^1(G,M) = 0$. By a "universal coefficient theorem" (see Jantzen I, 4.18), vanishing of $H^1(G,M)$ implies that for any ring $R$, $\mathbb{Z}[n]^{GL_{2,\mathbb{Z}}}\otimes R = R[n]^{GL_{2,R}}$, as desired.

For the benefit of myself and other novices in the area, I wanted to add some details to Wilberd's excellent answer, citing Donkin's Invariants of several matrices and Jantzen's book Representations of Algebraic Groups as appropriate.

We work over a base ring $R$.

First, one checks that Donkin's definition of a "good filtration" on a $G$-module $V$ coincides with Jantzens (defined in II, 4.16). The latter uses the isomorphism $H^i(M) := R^i\text{Ind}^G_B(V)\cong H^i(G/B,\mathcal{L}(M))$ (Jantzen I, 5.12), where $B\subset G$ is a Borel, $M$ a $B$-module, and $\mathcal{L}(M)$ denotes the quasi-coherent sheaf on $G/B$ associated to $M$ (I, 5.8). If $M = R$ with trivial $G$-action, then $\mathcal{L}(R)$ is the structure sheaf $\mathcal{O}_{G/B}$.

Next, we check that the $G = \text{GL}_{n,\mathbb{Z}}$-module $M = \mathbb{Z}[n]$ has a good filtration. We use the following Lemma

Lemma (Appendix B.9 of Jantzen) Suppose that $R$ is a principal ideal domain. Let $M$ be a $G$-module which is free of finite rank over $R$. As explained in B.4, let $V(\lambda) := H^0(-w_0\lambda)^*)$, where $w_0$ is the longest element of the Weyl group (II, 1.5). The following are equivalent:

(i) $M$ has a good filtration.

(ii) $Ext^i_G(V(\lambda),M) = 0$ for all $\lambda\in X(T)_+$ and all $i > 0$

(iii) $Ext^1_G(V(\lambda),M) = 0$ for all $\lambda\in X(T)_+$.

(iv) For each maximal ideal $\mathfrak{m}$ in $k$, the $G_{R/\mathfrak{m}}$-module $M\otimes R/\mathfrak{m}$ has a good filtration.

We write $M[n] = \bigoplus_{d\ge 0}M[n]_d$, where $M[n]_d$ consists of the polynomials of total degree $d$. One easily checks that each $M[n]_d$ is a $\text{GL}_{n,\mathbb{Z}}$-module. We apply the lemma with $R = \mathbb{Z}$ to the $R$-finite submodules $M[n]_{\le D} := \bigoplus_{0\le d\le D} M[n]_d$ for increasing $D$. Then by (iv), it suffices to check that the $G_{\mathbb{F}_p}$-modules $\mathbb{F}_p[n]_{\le D}$ all have good filtrations. Using flat base change for $G$-module cohomology (see Jantzen I, 4.13), (ii) (or (iii)) implies that it suffices to check the existence of good filtrations over algebraically closed fields. This is done in $\S3$ of Donkin's paper. Thus we conclude that $M = \mathbb{Z}[n]$ has a good filtration.

Now we work over $R = \mathbb{Z}$. Let $\lambda = 0\in X(T)$, then we have $V(0) = H^0(G/B,\mathcal{L}(0)) = H^0(G/B,\mathcal{O}_{G/B})$. Since $G/B$ is smooth proper over $\mathbb{Z}$ (use Jantzen I, 5.6(9) and I,5.7 to reduce to the classical case over algebraically closed fields), Stein factorization implies that $V(0) = \mathbb{Z}$ (with trivial $G$-action), so in particular $\lambda$ is dominant (this is II, 2.6 when $R$ is a field). By (ii) or (iii), we conclude that $Ext^1_G(\mathbb{Z},M) = H^1(G,M) = 0$. By a "universal coefficient theorem" (see Jantzen I, 4.18), vanishing of $H^1(G,M)$ implies that for any ring $R$, $\mathbb{Z}[n]^{GL_{2,\mathbb{Z}}}\otimes R = R[n]^{GL_{2,R}}$, as desired.

For the benefit of myself and other novices in the area, I wanted to add some details to Wilberd's excellent answer, citing Donkin's Invariants of several matrices and Jantzen's book Representations of Algebraic Groups as appropriate.

We work over a base ring $R$.

First, one checks that Donkin's definition of a "good filtration" on a $G$-module $V$ coincides with Jantzens (defined in II, 4.16). The latter uses the isomorphism $H^i(M) := R^i\text{Ind}^G_B(V)\cong H^i(G/B,\mathcal{L}(M))$ (Jantzen I, 5.12), where $B\subset G$ is a Borel, $M$ a $B$-module, and $\mathcal{L}(M)$ denotes the quasi-coherent sheaf on $G/B$ associated to $M$ (I, 5.8). If $M = R$ with trivial $G$-action, then $\mathcal{L}(R)$ is the structure sheaf $\mathcal{O}_{G/B}$.

Next, we check that the $G = \text{GL}_{n,\mathbb{Z}}$-module $M = \mathbb{Z}[n]$ has a good filtration. We use the following Lemma

Lemma (Appendix B.9 of Jantzen) Suppose that $R$ is a principal ideal domain. Let $M$ be a $G$-module which is free of finite rank over $R$. As explained in B.4, let $V(\lambda) := H^0(-w_0\lambda)^*)$, where $w_0$ is the longest element of the Weyl group (II, 1.5). The following are equivalent:

(i) $M$ has a good filtration.

(ii) $Ext^i_G(V(\lambda),M) = 0$ for all $\lambda\in X(T)_+$ and all $i > 0$

(iii) $Ext^1_G(V(\lambda),M) = 0$ for all $\lambda\in X(T)_+$.

(iv) For each maximal ideal $\mathfrak{m}$ in $k$, the $G_{R/\mathfrak{m}}$-module $M\otimes R/\mathfrak{m}$ has a good filtration.

We write $M[n] = \bigoplus_{d\ge 0}M[n]_d$, where $M[n]_d$ consists of the polynomials of total degree $d$. One easily checks that each $M[n]_d$ is a $\text{GL}_{n,\mathbb{Z}}$-module. We apply the lemma with $R = \mathbb{Z}$ to the $R$-finite submodules $M[n]_{\le D} := \bigoplus_{0\le d\le D} M[n]_d$ for increasing $D$. Then by (iv), it suffices to check that the $G_{\mathbb{F}_p}$-modules $\mathbb{F}_p[n]_{\le D}$ all have good filtrations. Using flat base change for $G$-module cohomology (see Jantzen I, 4.13), (ii) (or (iii)) implies that it suffices to check the existence of good filtrations over algebraically closed fields $k$. By a result of Donkin (Donkin The normality of closures of conjugacy classes of matrices Proposition 1.2a (i) and (ii)), good filtrations behave well with respect to direct summands, so it suffices to check that $\mathbb{Z}[n]\otimes k$ has a good filtration for any algebraically closed field $k$. This is done in $\S3$ of Donkin's Invariants of several matrices. Thus we conclude that $M = \mathbb{Z}[n]$ has a good filtration.

Now we work over $R = \mathbb{Z}$. Let $\lambda = 0\in X(T)$, then we have $V(0) = H^0(G/B,\mathcal{L}(0)) = H^0(G/B,\mathcal{O}_{G/B})$. Since $G/B$ is smooth proper over $\mathbb{Z}$ (use Jantzen I, 5.6(9) and I,5.7 to reduce to the classical case over algebraically closed fields), Stein factorization implies that $V(0) = \mathbb{Z}$ (with trivial $G$-action), so in particular $\lambda$ is dominant (this is II, 2.6 when $R$ is a field). By (ii) or (iii), we conclude that $Ext^1_G(\mathbb{Z},M) = H^1(G,M) = 0$. By a "universal coefficient theorem" (see Jantzen I, 4.18), vanishing of $H^1(G,M)$ implies that for any ring $R$, $\mathbb{Z}[n]^{GL_{2,\mathbb{Z}}}\otimes R = R[n]^{GL_{2,R}}$, as desired.