Slope filtrations: Erratum

Q4 Mathematics
Y. Andre
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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.
斜坡过滤:勘误
(1)例1.2.2.(2):实际上,通常的正交和既不是协积(也不是乘积),事实上,厄米空间的范畴不允许有限的协积(这在其他地方没有使用)。(2)抛弃仅在命题1.4.18中使用的引理1.2.8和未定义的“细化”概念。将命题1.4.18给出的证明替换为:证明。-通过归纳rkN足以证明,对于M的任何严格子对象N,点(rkN,度N)位于NP (M)以下。如果i表示使N包含在标志F(M)的陷口Mi内的最小指标,则等于:度N 6 degMi−1 + λi(rkN−rkMi−1)。设p: N“→Mi→Mi/Mi−1”为组合态射。由于Mi/Mi−1是斜率λi的半稳定函数,并且N/(N∩Mi−1)→Imp是外子函数,所以有μ(N/(N∩Mi−1))6 λi。通过rk和deg在序列0→(N∩Mi−1)→N→N/(N∩Mi−1)→0中的可加性,得到degn6 λi(rkN−rk(N∩Mi−1))+ deg(N∩Mi−1)。另一方面,通过归纳,点(rk (N∩Mi−1),deg (N∩Mi−1))位于NP (M)以下,这意味着deg(N∩Mi−1)6 degMi−1−λi(rkMi−1−rk(N∩Mi−1))。通过结合最后两个不等式,我们可以得到den6 degMi−1 + λi(rkN−rkMi−1)(3)在引理1.2.18中,将第1.2.3节意义上的和N+P (N⊕P→Q的像)替换为该态射的共像(通常是在阿贝尔包络A中N与P的和)。此引理在本文其他地方未使用。
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来源期刊
Confluentes Mathematici
Confluentes Mathematici Mathematics-Mathematics (miscellaneous)
CiteScore
0.60
自引率
0.00%
发文量
5
期刊介绍: 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.
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