Enomoto and Sakai introduced wide $ \tau $-tilting modules, which are $ \tau $-tilting modules over functorially finite wide subcategories. They also proved that wide $ \tau $-tilting modules bijection with doubly functorially finite image-cokernel-extension-closed (ICE-closed) subcategories, which extended Adachi-Iyama-Reiten's result. In this paper, we show that this bijection can be restricted to the support sets. As a consequence, we establish bijections between sincere wide $ \tau $-tilting modules, sincere ICE-closed subcategories, and sincere epibricks, and then we show that its number is related to the little Schr$ \ddot{o} $der number for Nakayama algebras.
Citation: Hanpeng Gao, Yunlong Zhou, Yuanfeng Zhang. Sincere wide $ \tau $-tilting modules[J]. Electronic Research Archive, 2025, 33(4): 2275-2284. doi: 10.3934/era.2025099
Enomoto and Sakai introduced wide $ \tau $-tilting modules, which are $ \tau $-tilting modules over functorially finite wide subcategories. They also proved that wide $ \tau $-tilting modules bijection with doubly functorially finite image-cokernel-extension-closed (ICE-closed) subcategories, which extended Adachi-Iyama-Reiten's result. In this paper, we show that this bijection can be restricted to the support sets. As a consequence, we establish bijections between sincere wide $ \tau $-tilting modules, sincere ICE-closed subcategories, and sincere epibricks, and then we show that its number is related to the little Schr$ \ddot{o} $der number for Nakayama algebras.
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Hanpeng Gao, Yunlong Zhou, Yuanfeng Zhang. Sincere wide $ \tau $-tilting modules[J]. Electronic Research Archive, 2025, 33(4): 2275-2284. doi: 10.3934/era.2025099
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