The untwisted Hall algebra of a finitary abelian category

Let $\mathcal{A}$ be a small finitary abelian category in the sense of Small finitary abelian categories for Hall multiplication, and let $R$ be a commutative ring with unit. Define the free $R$-module $H_{\mathcal{A},R}:={⨁}_{[M]\in \mathrm{Iso}(\mathcal{A})}R[M],$ where $[M]$ is a basis symbol for the isomorphism class of $M$. For objects $N,L$, let ${\mathrm{Mon}}_{\mathcal{A}}(N,L):=\{f\in {\mathrm{Hom}}_{\mathcal{A}}(N,L):f\text{ is monic}\}\mathrm{.}$ This is a finite set by the Hom condition in Small finitary abelian categories for Hall multiplication. For $f,g$ in this set, write $f{\sim }_{N}g$ when $g=f\circ u$ for an isomorphism $u:N\to N$; this is an equivalence relation by Isomorphism, groupoid, and connected category. The quotient set ${\mathrm{Mon}}_{\mathcal{A}}(N,L)\mathrm{/}{\sim }_N$ represents exactly the subobjects of $L$ isomorphic to $N$: each class gives the usual subobject class from Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms, and the mutual- factorisation theorem says two such monomorphisms give the same subobject exactly when they differ by such an isomorphism (Mutual factorisation is an equivalence relation on monomorphisms into an object and dually on epimorphisms out of it). Every subobject class with domain isomorphic to $N$ is represented by composing its monomorphism with an isomorphism from $N$, one class at a time. The cokernel quotient is independent of that representative by The quotient by a subobject is independent of the chosen representing monomorphism and The quotient of an object by a subobject. Define $g_{M,N}^L$ to be the number of classes $[f{]}_{{\sim }_N}$ for which $\mathrm{coker}(f)\cong M$. The finiteness lemma below also proves that the resulting product has finite support. Extend the operation $[M]\ast [N]:={\sum }_{[L]\in \mathrm{Iso}(\mathcal{A})}{g}_{M,N}^L[L]$ $R$-bilinearly, mapping each natural number coefficient to its multiple of $1_R$. The order is fixed: the counted short exact sequence is $0⟶N⟶L⟶M⟶0.$

This is the untwisted Hall product. It does not use an Euler form, a square root of a field size, or division by automorphism cardinalities. The finite coefficient and finite-support claims needed for the operation to define a bilinear product are established in Untwisted Hall structure constants and products are finite ↗.