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 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.
