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 and so they have integrated proofs for statements which of their reviews are straight forward, these with a purpose to support extra knowing and people that are scarcely on hand in different places. They try to take us as much as the purpose the place we will be able to locate our approach within the unique literature. ..." --The Mathematical Gazette

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Example text

Let 9’ be a finite spectroid and / the ideal of Y which consists of all the radical morphisms v such that pv = 0 whenever p itself is radical. Denote by sir . , M,) and 9’ = {Li}. $ In the particular case where W$ = 0, & is equivalent to the category rep Q described in Example 1, and -11: to the full subcategory formed by the representations without summand of the form s;-. Returning to the general case, we can reduce the investigation of & step by step to the case of a single module: Denote by JV and SC the sequences M,, .

2, IV). As we shall see forthwith, finite posets of width 3 may be finitely represented or not. 3. ’ by deleting a and adding formal suprema p v q for all p, q E PPS (a) such that p # q. Wecall~~thederivativeofBata. If 9% (a) is a chain, 9: coincides with P\(a). It follows that finite posets of width ~2 can be reduced to /zr by repeated derivations. In the case of width 3, the problem seems knottier since some posets of width 3 can be reduced to 0 and some others to posets of width >,4 (Fig. 2)‘.

Suppose that, for some s E 9&, the d-module M contains a submodule S isomorphic to -s and that dim M(s) = 1. Let further 3 denote the ideal of & generated by O,, k? the quotient sJf$, %? the J-module induced by the d-module M/S and N the annihilator of f in M. Then the canonical projection M + MIS induces a quasisurjective functor M&) + Mk. Proof Each M-space (V, x X) is isomorphic to the image of some (V, f, X 0 sd) E MtN,, where the first component I/ + M(X) off induces f and the second V + M(sd) is bijective.

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