Semibricks and wide subcategories in extended module categories

TL;DR

This paper introduces semibricks and wide subcategories in extended module categories, establishing bijections with simple-minded collections and characterizing silting complex mutations, thus generalizing existing theories in representation theory.

Contribution

It defines semibricks and wide subcategories in extended hearts of triangulated categories and establishes their bijections with simple-minded collections, extending known results to higher dimensions.

Findings

Semibricks biject with finite-length wide subcategories.

Bijections between $(d+1)$-term simple-minded collections and certain subcategories.

Characterization of mutations of silting complexes as $(d+1)$-term.

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 $d\mbox{-}\mathrm{mod}\Lambda$ of a finite-dimensional algebra $\Lambda$ over a field, we define left/right-finite semibricks and left/right-finite wide subcategories in $d\mbox{-}\mathrm{mod}\Lambda$ 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 $\mathrm{mod}\Lambda$. We use a relation between semibricks and silting complexes to characterise which mutations of $(d+1)$-term silting complexes are again $(d+1)$-term.