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I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is a part of the paper:

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Definition: Let $f: X \rightarrow Y$ be an irreducible morphism in mod$(A)$, with $X$ or $Y$ indecomposable. The left degree of $f$ is infinite, if for each integer $n \geq 1$, each module $Z \in$ ind$(A)$ and each morphism $g: Z \rightarrow X$ with dp$(g)=n$ we have that $fg \not\in$ rad$^{n+2}$(Z,Y).

Sadly I don't understand the underlines part. How does it follow that the left degree of $g_1$ is infinite? Any help is appreciated!

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1 Answer 1

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The fact you asking for is proved in Lemma 1.2 (and stated explicitly in the corollary to this lemma) in

Liu, S. (1992). Degrees of Irreducible Maps and the Shapes of Auslander-Reiten Quivers. Journal of the London Mathematical Society, s2-45(1), 32–54. https://doi.org/10.1112/jlms/s2-45.1.32

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  • $\begingroup$ Thank you :) I have actually known about this Lemma but I was looking for a more straight forward proof. Liu, S. uses sink maps and I as a someone, who has only had an introductory course in representation theory, don't really understand her proof. $\endgroup$
    – mathStudent
    Dec 29, 2020 at 15:50
  • $\begingroup$ Let Z be an indecomposable module and $h_i \colon X_i \to Z$, $i=1,2,,...,n$ the set of all irreducible maps with codomain $Z$ in AR-quiver. Then the map $h\colon X_1 \oplus X_1 \oplus ... \oplus X_n \to Z$ defined by $h(x_1, \dots, x_n) = h_1(x_1) + \dots + h_n(x_n)$ is a sink map. By very definition of AR-quiver, $h$ is in the radical of the module category, and every map with codomain $Z$ factorizes via $h$. Can you pick up from here? $\endgroup$
    – Ivan Yudin
    Dec 29, 2020 at 16:00

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