Are finitely generated projective models of an algebraic theory always finitely presented?

Question

I know that for modules over rings, a finitely generated projective module is finitely presented. I was wondering whether this holds in full generality for algebraic theories, and if not, which parts of the abelian structure of the category of modules are needed...

Answer 1

Suppose $P$ is a finitely generated projective $\mathbb{T}$-algebra. Then there is a finitely generated free $\mathbb{T}$-algebra $F$ and a surjective homomorphism $r : F \to P$, and since $P$ is projective, there is also a homomorphism $s : P \to F$ such that $r \circ s = \mathrm{id}_P$. Then, $r : F \to P$ is the coequaliser of $\mathrm{id}_F$ and $s \circ r$, so $P$ is also a finitely presented $\mathbb{T}$-algebra.

More generally, if you have an $\mathcal{E}$-projective object $P$ in a locally finitely presentable category and an $\mathcal{E}$-morphism $F \to P$ where $F$ is a finitely presentable object, then $P$ is also a finitely presentable object.