# Affine Group Scheme Definition

From what I understand, a affine group scheme $$G$$ should be an affine scheme on which there exists a group structure in the sense that $$\phi: G \times_k G \to G$$ is also a morphism of groups.

What does "morphism of groups" means for schemes?

For a Lie group $$G$$, it implies that the topological space has the structure of a group, i.e $$\phi(xy)=\phi(x)\phi(y)$$ for all $$x,y \in G$$, etc.

However, for schemes such a definition is confusing since we don't know what "type" of points such a definition would be referring to.

Is an affine group scheme a scheme whose underlying topological space has the structure of a group? (i.e the morphism $$\phi$$ is defined on $$k$$-points of $$G$$)

Or is it more general, i.e for all schemes $$S$$, must the $$S$$-points of $$G$$ also have the structure of a group?

• It's the latter thing you said. For all $k$-schemes $S$ the set $G(S)$ must be equipped with a group structure functorial in $S$ Apr 11, 2019 at 20:40
• Ok that makes more sense. Thank yoU! Apr 11, 2019 at 20:43

Just to clarify, the real subtle point here is the following. For a $$k$$-scheme $$X$$ let us denote by $$|X|$$ the underlying topological space. Then, the behind-the-scenes complication here is that $$|X\times_k X|\ne |X|\times |X|$$. In fact, there is a continuous surjection $$|X\times_k X|\to |X|\times |X|$$ which is, in general, not an isomorphism (exercise!).
So, if we have a map $$\phi:G\times_k G\to G$$ and we have $$(g,h)\in |G|$$ there is no way to make sense of $$\phi(g,h)$$--one would have to try and lift them to $$|G\times_k G|$$ and apply $$\phi$$ there, but this is not well-defined (exercise!).
To convince yourself of why this is not so weird, consider $$G=\mathrm{GL}_2$$. Note then that we have that $$H=\mathrm{SL}_2$$ is a closed subscheme of $$G$$. Let $$\eta$$ denote the genric point of $$G$$ and $$\eta'$$ the generic point of $$H$$. Then, we have that $$k(\eta)=k(x,y,z,w)$$ and $$k(\eta')=\mathrm{Frac}(k[x,y,z,w]/(xy-zw-1))$$. The maps $$\mathrm{Spec}(k(\eta))\hookrightarrow G$$ and $$\mathrm{Spec}(k(\eta'))\hookrightarrow G$$ correspond to the matrices $$\begin{pmatrix}x & y\\ z & w\end{pmatrix}$$ interpreted in $$k(\eta)$$ and $$k(\eta')$$ respectively. How do you 'multiply' those?
The point though is that while $$|G\times_k G|\ne |G|\times |G|$$ for every $$k$$-scheme $$S$$ we have that $$(G\times_k G)(S)=G(S)\times G(S)$$. Thus, for every $$k$$-scheme $$S$$ the map $$\phi:(G\times_k G)(S)\to G(S)$$ is a map $$G(S)\times G(S)\to G(S)$$ which is actually the multiplication for a group structure on $$G(S)$$.