Is there a "correct" $k$ scheme structure to put on $\coprod_{i=1}^n \operatorname{Spec}(k)$?

Question

Let $k$ be an algebraically closed field, with $n\in\mathbb N\subset k$ invertible. I am trying to prove that if $\mathbb G_m=k[t,t^{-1}]$ is the multiplicative group scheme over $k$, and $\mu_n$ is the kernel of the group map $[n]:\mathbb G_m\rightarrow \mathbb G_m$ defined on all $k$ schemes $S$ by $s\in G(S)\mapsto s^n\in G(S)$, then $\mu_n$ is isomorphic to the group scheme $\coprod_{\alpha\in \mathbb Z/n\mathbb Z}\operatorname{Spec}(k)$.

However, I have just realized that I am not entirely sure how to turn $\coprod_{i=1}^n \operatorname{Spec}(k)$ into a $k$-scheme. Indeed, we can identify $\coprod_{i=1}^n \operatorname{Spec}(k)$ with $\operatorname{Spec}(k^n)$, and then any morphism $k\rightarrow k^n$ turns $\operatorname{Spec}(k^n)$ into a $k$-scheme. Is there any reason I should $x\mapsto (x,0,\dots,0)$ over say $x\mapsto (x,\dots, x)$ or any other such variant?

Answer 1

KReiser's comments and FShrike probably already suffice for you and I'm not saying much more but let me write an answer with a little more detail (completely inside the category of $k$-schemes).

Recall us quickly recall the notion of slice categories from ordinary category theory.

Slice categories. Let $\mathscr{C}$ be a category with an object $c \in \mathscr{C}$. Then, $\mathscr{C}_{/c}$ is the slice category (over $c$) where objects are maps $c' \to c$ and maps $(c' \to c) \to (c'' \to c)$ are maps $c' \to c''$ commuting with the structure maps to $c$. Colimits in slice categories are then computed in the underlying category. Written out: If $(c_i \to c)_{i \in I}$ is a diagram of objects in $\mathscr{C}_{/c}$, then one may check that $$\operatorname{colim}_{i \in I} (c_i \to c)_{i \in I} \cong \left(\operatorname{colim}_{i \in I} c_i \to c \right)$$ in $\mathscr{C}_{/c}$ where the map $\operatorname{colim}_{i \in I} c_i \to c$ is induced by the universal property of the colimit.

Now, we may apply it to the concrete setting of $k$-schemes.

Example ($k$-schemes). By definition, the category of $k$-schemes is the slice category $\mathsf{Sch}_{/\operatorname{Spec}{k}}$, and you are given $n$ copies of $\mathrm{id}_{\operatorname{Spec}{k}}:\operatorname{Spec}{k} \to \operatorname{Spec}{k}$ as $k$-schemes. Taking the coproduct inside the category $\mathsf{Sch}_{/\operatorname{Spec}{k}}$ then yields a $k$-scheme $\coprod_{i=1}^n \operatorname{Spec}{k} \to \operatorname{Spec}{k}$ as above with maps induced by the coproduct, where the map is uniquely determined by the $n$-copies of $\operatorname{Spec}{k}$. In particular, this is the preferred $k$-scheme structure. Concretely, it is $\mathrm{id}_{\operatorname{Spec}{k}}$ on each copy of $\operatorname{Spec}{k}$. You may check that this corresponds to the diagonal map $\Delta : k \to k^n$.

Answer 2

$x\mapsto(x,x,\cdots,x)$ is the correct one, i.e. using the map $\mathsf{Spec}(\Delta)$ where $\Delta:k\to k^n$. Why? Because you have a coproduct of schemes, which are all $k$-schemes; if you want a coproduct of $k$-schemes i.e. the coproduct as taken in the slice category over $\mathsf{Spec}(k)$, which is categorically the best thing to do !, then this is the structural morphism that arises. Caveat: I know this is the coproduct if you restrict to affine ($k$-) schemes. I'm not sure how colimits work for completely general schemes, I'm a newbie.