By Кострикин, А. И. Шафаревич, И. Р. ; ; A I Kostrikin; I R Shafarevich
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Extra info for Algebra VIII : representations of finite-dimensional algebras
Annihilating the entries 4. Finitely iI Spaced i2 Modules ‘r-1 41 ‘r Fig. 5 of the matrices of &‘(s, t) which lie above the stairs, we obtain a linear map a: JZZ’(S,t) + S which is compatible with the filtrations and induces bijections between their quotients (of dimension < 1). Therefore, a itself is bijective, and the “canonical” basis of S is the image of a basis (bj’) of &‘(s, t) such that bj/ = 1, whereas the other entries ek vanish if u < j or if (u, u) is “located below the stairs” (the indexing pair (j, i) runs through the “staircase”).
Ahence f $ 4, the functor F: mod A + Mk is an equivalence. Geometry here intrudes on algebra: If 9 is a spectroid of A, the simple A-module e then corresponds to some Y-module SC, where s E Y. Our assumptions imply that s is a sink of 9’ (Y(s, s) = kll, and 9’(s, t) = 0 if t # s) and that the ideal fS of Y associated with ,$ satisfies fS(t, p) = 0 if p # s and yS(t, s) = Y(t, s) for all t E 9. The full subcategory Y of 9’ formed by the points #s is identified with a spectroid of A. The natural embedding mod 2 + mod A corresponds to the extension n H no of Smodules to 9’ by 0, and Ext$(s-, no) is identified with Hom(s” 15, n) =: N(n).
Besides general statements, we present a method which reduces representations of algebras to such problems. 4). 34 4. Finitely Spaced Modules In all sections except Sect 9, we henceforth suppose for simplicity’s closed field. 1. As we have seen earlier, modules over aggregates are natural generalizations of modules over algebras. On the other hand, they also look in some sense like generalizations of posets. Let us explain this point of view, which is essential in the present section. Consider an aggregate d.
Algebra VIII : representations of finite-dimensional algebras by Кострикин, А. И. Шафаревич, И. Р. ; ; A I Kostrikin; I R Shafarevich