{"title":"Slope filtrations: Erratum","authors":"Y. Andre","doi":"10.5802/CML.64","DOIUrl":null,"url":null,"abstract":"Erratum for the paper “Slope Filtrations”, Confluentes Mathematici, Vol. 1, No. 1 (2009) 1–85 (1) Example 1.2.2.(2): actually, the usual orthogonal sum is neither a coproduct (nor a product), and in fact, the category of hermitian spaces does not admit finite coproducts (this is not used elsewhere). (2) Discard Lemma 1.2.8 and the undefined notion of “refinement\", which are used only in Proposition 1.4.18. Replace the given proof of Proposition 1.4.18 by the following: Proof. — It suffices to show (by induction on rk N) that for any strict subobject N ofM , the point (rk N, deg N) lies below NP (M). If i denotes the minimal index such that N is contained in the notch Mi of the flag F(M), this amounts to: deg N 6 degMi−1 + λi(rkN − rkMi−1). Let p : N ↪→ Mi → Mi/Mi−1 be the composed morphism. Since Mi/Mi−1 is semistable of slope λi and N/(N ∩Mi−1)→ Imp is epi-monic, one has μ(N/(N∩Mi−1)) 6 λi. By additivity of rk and deg in the sequence 0→ (N ∩Mi−1)→ N → N/(N ∩Mi−1)→ 0, one gets degN 6 λi(rkN − rk(N ∩Mi−1)) + deg(N ∩Mi−1). On the other hand, by induction, the point (rk (N ∩Mi−1),deg (N ∩Mi−1)) lies below NP (M), which implies that deg(N ∩Mi−1) 6 degMi−1 − λi(rkMi−1 − rk(N ∩Mi−1)). By combining the last two inequalities, one gets deg N 6 degMi−1 + λi(rkN − rkMi−1) as wanted. (3) In Lemma 1.2.18, replace the sum N+P in the sense of Section 1.2.3, which is the image of N ⊕P → Q, by the coimage of this morphism (which is the usual sum of N and P in the abelian envelop A). This lemma is not used elsewhere in the paper.","PeriodicalId":52130,"journal":{"name":"Confluentes Mathematici","volume":"30 1","pages":"147-148"},"PeriodicalIF":0.0000,"publicationDate":"2020-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Confluentes Mathematici","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5802/CML.64","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
Abstract
Erratum for the paper “Slope Filtrations”, Confluentes Mathematici, Vol. 1, No. 1 (2009) 1–85 (1) Example 1.2.2.(2): actually, the usual orthogonal sum is neither a coproduct (nor a product), and in fact, the category of hermitian spaces does not admit finite coproducts (this is not used elsewhere). (2) Discard Lemma 1.2.8 and the undefined notion of “refinement", which are used only in Proposition 1.4.18. Replace the given proof of Proposition 1.4.18 by the following: Proof. — It suffices to show (by induction on rk N) that for any strict subobject N ofM , the point (rk N, deg N) lies below NP (M). If i denotes the minimal index such that N is contained in the notch Mi of the flag F(M), this amounts to: deg N 6 degMi−1 + λi(rkN − rkMi−1). Let p : N ↪→ Mi → Mi/Mi−1 be the composed morphism. Since Mi/Mi−1 is semistable of slope λi and N/(N ∩Mi−1)→ Imp is epi-monic, one has μ(N/(N∩Mi−1)) 6 λi. By additivity of rk and deg in the sequence 0→ (N ∩Mi−1)→ N → N/(N ∩Mi−1)→ 0, one gets degN 6 λi(rkN − rk(N ∩Mi−1)) + deg(N ∩Mi−1). On the other hand, by induction, the point (rk (N ∩Mi−1),deg (N ∩Mi−1)) lies below NP (M), which implies that deg(N ∩Mi−1) 6 degMi−1 − λi(rkMi−1 − rk(N ∩Mi−1)). By combining the last two inequalities, one gets deg N 6 degMi−1 + λi(rkN − rkMi−1) as wanted. (3) In Lemma 1.2.18, replace the sum N+P in the sense of Section 1.2.3, which is the image of N ⊕P → Q, by the coimage of this morphism (which is the usual sum of N and P in the abelian envelop A). This lemma is not used elsewhere in the paper.
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Confluentes Mathematici is a mathematical research journal. Since its creation in 2009 by the Institut Camille Jordan UMR 5208 and the Unité de Mathématiques Pures et Appliquées UMR 5669 of the Université de Lyon, it reflects the wish of the mathematical community of Lyon—Saint-Étienne to participate in the new forms of scientific edittion. The journal is electronic only, fully open acces and without author charges. The journal aims to publish high quality mathematical research articles in English, French or German. All domains of Mathematics (pure and applied) and Mathematical Physics will be considered, as well as the History of Mathematics. Confluentes Mathematici also publishes survey articles. Authors are asked to pay particular attention to the expository style of their article, in order to be understood by all the communities concerned.