The functor $F_n:$Alg${}_R\to$ Sets defined by $X\mapsto \{x\in X: x^n=1\}$.

Question

Let $R$ be a commutative rings and Alg${}_R$ a category of $R$-algebras. I have the following functor $F_n:$Alg${}_R\to$ Sets defined by $X\mapsto \{x\in X: x^n=1\}$. How can I prove that this functor is representable.

Answer 1

Write down an $R$-algebra representing it!

To warm up, let recall how you represent $G:\text{Alg}_R\to\text{Set}$ given by $G(X)=\{x:x\in X\}$. This is represented by the algebra $R[T]$. An $R$-algebra map $R[T]\to X$ is determined by the image of $T$, which can be any element of $X$; a typical one is $\phi_x:f(T)\mapsto f(x)$. One checks that this is a functor, not just a map on objects.

Now we want to restrict to $x\in X$ with $x^n=1$. Then you need $$0=x^n-1=\phi_x(T^n-1).$$ The admissible $\phi_x$ all annihilate $T^n-1$. So, let $S_n=R[T]/(T^n-1)$. The maps from $S_n$ to $X$ are then the $\phi_x:f(T)\mapsto f(x)$ where $x^n=1$, etc.