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In this paper, the criteria for minimal Horseshoe Lemma to be true are givenvia quasi-$\delta$-Koszul modules, which are the nongraded version of$\delta$-Koszul modules first introduced by Green and Marcos in 2005. Moreover,some applications of minimal Horseshoe Lemma are also given.

Quasi-Koszulity and minimal Horseshoe Lemma