## 77.5 Group algebraic spaces

Please refer to Groupoids, Section 39.4 for notation.

Definition 77.5.1. Let $B \to S$ as in Section 77.3.

1. A group algebraic space over $B$ is a pair $(G, m)$, where $G$ is an algebraic space over $B$ and $m : G \times _ B G \to G$ is a morphism of algebraic spaces over $B$ with the following property: For every scheme $T$ over $B$ the pair $(G(T), m)$ is a group.

2. A morphism $\psi : (G, m) \to (G', m')$ of group algebraic spaces over $B$ is a morphism $\psi : G \to G'$ of algebraic spaces over $B$ such that for every $T/B$ the induced map $\psi : G(T) \to G'(T)$ is a homomorphism of groups.

Let $(G, m)$ be a group algebraic space over the algebraic space $B$. By the discussion in Groupoids, Section 39.4 we obtain morphisms of algebraic spaces over $B$ (identity) $e : B \to G$ and (inverse) $i : G \to G$ such that for every $T$ the quadruple $(G(T), m, e, i)$ satisfies the axioms of a group.

Let $(G, m)$, $(G', m')$ be group algebraic spaces over $B$. Let $f : G \to G'$ be a morphism of algebraic spaces over $B$. It follows from the definition that $f$ is a morphism of group algebraic spaces over $B$ if and only if the following diagram is commutative:

$\xymatrix{ G \times _ B G \ar[r]_-{f \times f} \ar[d]_ m & G' \times _ B G' \ar[d]^ m \\ G \ar[r]^ f & G' }$

Lemma 77.5.2. Let $B \to S$ as in Section 77.3. Let $(G, m)$ be a group algebraic space over $B$. Let $B' \to B$ be a morphism of algebraic spaces. The pullback $(G_{B'}, m_{B'})$ is a group algebraic space over $B'$.

Proof. Omitted. $\square$

Comment #1333 by Qijun Yan on

In the paragraph below the definition of group algebraic space, the map $i: B\to B$ should be $i: G\to G$.

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