Semibricks and wide subcategories in extended module categories
Esha Gupta, Yu Zhou
TL;DR
This work extends the classical dictionary among bricks, wide subcategories, simple-minded collections, and silting to the higher, $d$-extended setting. By developing the notions of semibricks and (finite-length) wide subcategories inside the $d$-extended heart $ ext{D}^{[-d+1,0]}$, the authors establish bijections with positive torsion classes and with $(d+1)$-term simple-minded collections, generalizing the $d=1$ theory. They introduce and analyze $d$-FAE closed subcategories, torsion pairs, and exact hearts to connect these structures to t-structures, silting theory, and their mutations, culminating in a mutation criterion for $(d+1)$-term silting complexes. In the module setting, left/right-finite semibricks/wide subcategories correspond to $(d+1)$-term simple-minded collections and admit realizations as module categories over endomorphism algebras, unifying higher analogues of known bijections and providing practical tools for mutation and realization.
Abstract
For $d\geq 1$, we define semibricks and wide subcategories in the $d$-extended hearts of bounded $t$-structures on a triangulated category. We show that these semibricks are in bijection with finite-length wide subcategories. When the $d$-extended heart is the $d$-extended module category $d\mbox{-}\mathrm{mod}Λ$ of a finite-dimensional algebra $Λ$ over a field, we define left/right-finite semibricks and left/right-finite wide subcategories in $d\mbox{-}\mathrm{mod}Λ$ and show bijections with $(d+1)$-term simple-minded collections, generalising the bijections between $2$-term simple-minded collections, left/right-finite wide subcategories and left/right-finite semibricks in $\mathrm{mod}Λ$. We use a relation between semibricks and silting complexes to characterise which mutations of $(d+1)$-term silting complexes are again $(d+1)$-term.