# Formal base change properties of group schemes

Let $S$ be an arbitrary scheme, and let $X,Y,S'$ be $S$-schemes. EGA 1, Chap 1, 3.3, gives nice properties for products of schemes with respect to base change $S'\to S$. For example (3.3.10): There exists a canonical isomorphism $$\varphi:(X\times_SY)_{S'}\cong X_{S'}\times_{S'}Y_{S'}$$ of $S'$-schemes.

Suppose now that $X$ and $Y$ are $S$-group schemes. Then $X\times_S Y$ has the structure of an $S$-group scheme, and $X_{S'}$ has the structure of an $S'$-group scheme.

Question: Ist the canonical isomorphism $\varphi$ of $S'$-schemes in addition a morphism of $S'$-group schemes? If yes, is there a precise reference for this statement (or more generally for versions of EGA 1, Chap 1, 3.3, for $S$-group schemes)?

• Just use the functor of points. – Daniel Litt Aug 10 '13 at 21:21

To show that the scheme isomorphism is an isomorphism of group schemes, it suffices to check that the multiplication maps coincide, i.e., $$m_{(X \times_S Y)_{S'}} = \phi^{-1} \circ (m_{X_{S'}} \times_{S'} m_{Y_{S'}}) \circ \tau_{23} \circ (\phi \times_{S'} \phi),$$ where $\tau_{23}: X_{S'} \times_{S'} Y_{S'} \times_{S'} X_{S'} \times_{S'} Y_{S'} \to X_{S'} \times_{S'} X_{S'} \times_{S'} Y_{S'} \times_{S'} Y_{S'}$ is the canonical switch isomorphism. The result follows from how we define multiplication on base-changed group schemes (i.e., $m_{(X \times_S Y)_{S'}} = (m_{X \times_S Y} \times_S id_{S'}) \circ \tau_{23}$), and the multiplication law on a direct product of groups (i.e., $m_{X \times_S Y} = (m_X \times_S m_Y) \circ \tau_{23}$).