The Grothendieck group of finitely copresented comodules over incidence coalgebras
Volume 176 / 2024
Colloquium Mathematicum 176 (2024), 247-276
MSC: Primary 16G20; Secondary 16G60, 16T15
DOI: 10.4064/cm9495-11-2024
Published online: 12 December 2024
Abstract
The Grothendieck group ${\bf K}_0({\rm Rep}_{\rm fc}(I))$ of the category ${\rm Rep}_{\rm fc}(I)$ of finitely copresented representations of an interval finite poset $I$ of finite left width is free. It admits a $\mathbb Z$-basis $\{b_i: i\in I\}$, formed by some $b_i \in \{[S(i)], [E(i)]\}$, $i\in I$, where $E(i)$ are injective envelopes of simple representations $S(i)$. In consequence, the analogous result holds for the Grothendieck group ${\bf K}_0({K^\Box I}$-${\rm Comod}_{{\rm fc}})$ of the category ${K^\Box I}$-${\rm Comod}_{{\rm fc}}$ of finitely copresented left comodules over the incidence coalgebra ${K^\Box I}$ of the poset $I$ as above, in particular, for a large class of the coalgebras ${K^\Box I}$ of tame comodule type.
