Under what conditions can I expect the restriction of scalars functor to preserve tensor products

Question

Suppose I have the canonical injection $i:H\hookrightarrow G$. Evidently I can induce the map on modules which restricts scalars from $\mathbf{Z}[G]$ to $\mathbf{Z}[H]$; that is, $i^\ast:Mod_{\mathbf{Z}[G]}\to Mod_{\mathbf{Z}[H]}$. I was wondering if there are any conditions upon modules which ensure we have the following isomorphism, $i^\ast(M\otimes_\mathbf{Z} N)\cong i^\ast(M)\otimes_\mathbf{Z} i^\ast(N).$ This seems like it shouldn't be too much of a problem under the right conditions (especially since these modules are free over the integers) but I seem to have a distinct lack of time this week to spend on such a problem.

Answer 1

This is always the case. The obvious isomorphism of $\mathbb{Z}$-modules

$$i^{*}(M \otimes_{\mathbb{Z}} N) \rightarrow i^{*}(M) \otimes_{\mathbb{Z}} i^{*}(N)$$

defined by $i^{*}(m \otimes n) \mapsto i^{*}(m) \otimes i^{*}(n)$ (really the identity map as $\mathbb{Z}$-modules) is always a map of $\mathbb{Z}[H]$-modules. This is true for modules over Hopf algebras in general as long as the algebra map that defines the restriction is a Hopf algebra map.