Algebra VIII : representations of finite-dimensional by Кострикин, А. И. Шафаревич, И. Р. ; ; A I Kostrikin; I R

By Кострикин, А. И. Шафаревич, И. Р. ; ; A I Kostrikin; I R Shafarevich

From the stories: "... [Gabriel and Roiter] are pioneers during this topic they usually have integrated proofs for statements which of their critiques are undemanding, these that allows you to support extra knowing and people that are scarcely on hand somewhere else. They try to take us as much as the purpose the place we will be able to locate our method within the unique literature. ..." --The Mathematical Gazette

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Example text

The preceding results draw our attention to the disjoint union ‘Z(M) of the sets GZSformed by the non-zero cyclic &‘(s, s)-submodules of M(s), s E $zz’. This union is finite if and only if 3~~2 is finite and each M(s) uniserial. B and f,4 $ C2B for some f E -ol(s, t). (M) are linearly ordered by this relation. The structure carried by g(M) provides us with an informative combinatorial invariant of the pair (sz’, M). For instance, let us consider two points s,t E YSZI such that M(s), M(t) are uniserial of dimension 2.

We leave it to the reader to convince himself that flu 4 HP can be reduced to a poset of width > 4 unless 4 = 1 or r = 1, q = 2 and p < 4. Y # (a)) < 2. Then there exists a quasi-surjective functor F: rep B --f rep 9: such that the indecomposables annihilated by F are those isomorphic to (a, I), (a, H) or (n, [ 11) where n E 9 # (a). We infer that B is finitely represented if and only if so is pi. If this is the case, the number I(P) of isoclasses of indecomposable representations of 9 is equal to z(9J + 19 # (a)1 + 2, and 9 can be reduced to @ by repeated derivations.

4 Suppose that the total dimension IsoY& dim M(s) and the set A? of maximal submodules of M are finite. Then MS has exactly one non-trivial indecomposable up to isomorphism. 4 the preceding lemma provides us with a workable algorithm: Suppose that M has finite total dimension. If 9 is any finite set of proper submodules, we start with a proper submodule N # 0 which is minimal among those contained in no L E 9’. Y& has one point more than 9~‘: the unique non-trivial indecomposable of Ngnilr. In a second step, we then choose a proper submodule N’ # 0 of fi which is minimal among those contained in no 2 E 9, and so on .

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