I've read an algebraic variety $A$ is finitely presented if there's a coequalizer diagram $$F(m)\rightrightarrows F(n)\twoheadrightarrow A$$ where $F(n)$ is the free model on $n$ generators.
I'm trying to understand the formalism by example and am failing miserably. Let work in the category of $R$-algebras. Suppose $A$ is finitely generated. Equivalence relations are effective so the quotient $R[x_1,\dots ,x_n]$ corresponding to the projection of the equivalence relation is really the coqualizer of its kernel pair, so the $F(m)$ in the diagram above along with its pair of arrows is the kernel pair. This thing is the object of equivalence relations almost by definition, but I'm having trouble with identifying it and writing the maps:
Suppose the equivalence relation correpsonds to a f.g ideal $ I=\langle f_1 ,\dots ,f_m \rangle$. Why does this mean the kernel pair object is $R[x_1,\dots x_m]$? If it isn't, what is?
If $A=R[x_1,\dots,x_n]/I$ with $I=\langle f_1,\dots,f_m\rangle$ then you may take the maps $\varphi,\psi: R[x_1,\dots,x_m]\to R[x_1,\dots,x_n]$ such that $\psi=0$ and $\varphi(x_i)=f_i$.
Then $A$ is indeed the coequalizer of $\psi$ and $\varphi$.