# When is the kernel pair of a finite presentation of algebraic structures finitely generated?

Fix an algebraic theory; denote its free models by $T^k$.

There are two possible definitions of what it means for a coequalizer $T^m\twoheadrightarrow M$ to be a finite presentation of $M$.

1. $f\colon T^m\twoheadrightarrow M$ is the coequalizer of some pair of morphisms $T^k\rightrightarrows T^m\twoheadrightarrow M$. This is the standard definition in category theory textbooks.
2. If $K$ is a kernel pair $K\rightrightarrows T^m\twoheadrightarrow M$ of $f\colon T^m\twoheadrightarrow M$, then $K$ is finitely generated in the sense that there is some coequalizer $T^k\twoheadrightarrow K$. This is the standard definition in algebra textbooks.

It is easy to see that the second notion implies the first: $T^m\twoheadrightarrow M$ is the coequalizer of the composites $T^k\twoheadrightarrow K\rightrightarrows T^m$. Conversely, the notions are equivalent in the standard examples (groups, $R$-modules, rings). Are the notions always equivalent, or do we need certain conditions on the algebraic theory (e.g. being Mal'cev)?

(Note: the second notion of $M$ being finitely presented is better behaved in the sense that it implies that every coequalizer $T^{m'}\twoheadrightarrow M$ has a finitely generated kernel.)

• since the first condition is equivalent to $\mathrm{Hom}(M,-)$ preserving filitered colimits, and since any $M$ is the colimit of finitely generated free models, it would be sufficient to show that the finitely presented modules in the second sense are stable under coproducts and coequalizers. Commented Sep 18, 2016 at 16:35

The sense in which the kernel pair is finitely generated specified by the question is incorrect. What is actually happening is this.

Given a coequalizer $$X\rightrightarrows Y\to Z$$, we have equalizers $$\mathcal C(X,F(i))\leftleftarrows\mathcal C(Y,F(i)\hookleftarrow\mathcal C(Z,F(i)$$ for each object $$F(i)$$ of a filtered diagram $$F$$ (because representable functors preserve limits). Since filtered colimits commute with equalizers, we get an equalizer $$\varinjlim\mathcal C(X,F(i))\leftleftarrows\varinjlim\mathcal C(Y,F(i))\hookleftarrow\mathcal\varinjlim C(Z,F(i))$$ with comparison maps from the equalizer $$\mathcal C(X,\varinjlim F(i))\leftleftarrows\mathcal C(Y,\varinjlim F(i))\hookleftarrow\mathcal C(Z,\varinjlim F(i))$$.

By https://math.stackexchange.com/a/4408111/400, respectively by definition, a component of the comparison maps is, for all filtered diagrams $$F$$, a monomorphism, respectively isomorphism, if and only if the corresponding object is finitely generated, respectively finitely presentable.

If $$X$$ is finitely generated and $$Y$$ is finitely presentable, then the top parallel pair factors as a monomorphism followed by the bottom parallel pair, whence they have isomorphic equalizers, i.e. $$Z$$ is finitely presentable. It is in this sense that a relation defines a finitely presentable object if it's finitely generated: $$Y\to Z$$ is the coequalizer of $$X\to K\rightrightarrows Y$$ where $$K\rightrightarrows Y$$ is the kernel pair and $$X$$ is finitely generated.

Conversely, if $$X$$ is a filtered colimit of finitely presentable objects, then $$Y$$ and $$Z$$ being finitely presentable imply $$Y\to Z$$ is also the coequalizer of $$W\rightrightarrows Y\to Z$$ for a finitely presentable $$W$$ (with a natural map to $$X$$) (see https://mathoverflow.net/a/382672/75650). Applying this to $$X\rightrightarrows Y$$ the kernel pair of $$Y\to Z$$ shows that any relation defining a finitely presentable object is finitely generated in the above sense.

Note that the kernel pair does not have to be finitely generated. Indeed, we have a cokernel $$\mathbb Z\to F_2\to\mathbb Z^2$$ given by $$1\mapsto xyx^{-1}y^{-1}$$, $$x,y\mapsto(1,0),(0,1)$$ showing $$\mathbb Z^2$$ is finitely presentable, but its kernel, the derived subgroup of $$F_2$$, is not finitely generated. It should also be the case that the kernel pair, which is the semi-direct product of $$F_2$$ and its derived subgroup, is also not finitely generated.