Shard Theory for g-Fans

For a finite dimensional algebra A, the notion of g-fan Σ (A) is defined from two-term silting complexes of Kᵇ(proj A) in the real Grothendieck group K₀(proj A)R. In this paper, we discuss the theory of shards to Σ (A), which was originally defined for a hyperplane arrangement. We establish a correspondence between the set of join-irreducible elements of torsion classes of modA and the set of shards of Σ (A) for g-finite algebra A. Moreover, we show that the semistable region of a brick of modA is exactly given by a shard. We also give a poset isomorphism of shard intersections and wide subcategories of modA.