The Grothendieck ring of linear representations of a finite category
Let \({\mathcal C}\) be a finite category and k a field of characteristic \(p>0\). Denote by \(G_ 0(k[{\mathcal C}])\) (resp. \(K_ 0(k[{\mathcal C}]))\) the Grothendieck group of the category of finite dimensional (resp. finite dimensional projective) k[\({\mathcal C}]\)-modules. The author examines the Cartan map \(K_ 0(k[{\mathcal C}])\to G_ 0(k[{\mathcal C}])\) to prove a linear version of \textit{T. Yoshida}'s result [Commutative algebra and combinatorics, US-Jap. Joint Semin., Kyoto/Jap. 1985, Adv. Stud. Pure Math. 11, 337-353 (1987; Zbl 0645.20004)].
scientific article; zbMATH DE number 4156724