Cartan matrix

《Cohomologie Locale Des Faisceaux Coherents (Sga 2): Seminaire De Geometrie Algebrique Du Bois Marie 1962 (Documents Mathematiques)》 · 印刷第 131 页 Section 7.4「The Cartan Matrix」，Corollary 7.4.2 前的定义段及该推论完整陈述

Again in the context of a finite-dimensional algebra $A$, we define for each pair of simple $A$-modules $S$ and $T$ the integer $c_{ST}=$ the composition factor multiplicity of $S$ in $P_T$. These are called the Cartan invariants of $A$, and they form a matrix $C=(c_{ST})$ with rows and columns indexed by the isomorphism types of simple $A$-modules, called the Cartan matrix of $A$. Corollary 7.4.2. Let $A$ be a finite-dimensional algebra over a field, let $S$ and $T$ be simple $A$-modules and let $e_S,e_T$ be idempotents so that $P_S=Ae_S$ and $P_T=Ae_T$ are projective covers of $S$ and $T$. Then, $c_{ST}=\frac{\dim\operatorname{Hom}_A(P_S,P_T)}{\dim\operatorname{End}_A(S)}=\frac{\dim e_SAe_T}{\dim\operatorname{End}_A(S)}$. If the ground field $k$ is algebraically closed then $c_{ST}=\dim\operatorname{Hom}_A(P_S,P_T)=\dim e_SAe_T$.