Shard Theory for g-Fans

Abstract

For a finite dimensional algebra $A$, the notion of $g$-fan $\Sigma (A)$ is defined from two-term silting complexes of $\textsf{K}^{\textrm{b}}(\textsf{proj} A)$ in the real Grothendieck group $K_{0}(\textsf{proj} A)_{\mathbb{R}}$. In this paper, we discuss the theory of shards to $\Sigma (A)$, which was originally defined for a hyperplane arrangement. We establish a correspondence between the set of join-irreducible elements of torsion classes of $\textsf{mod}A$ and the set of shards of $\Sigma (A)$ for $g$-finite algebra $A$. Moreover, we show that the semistable region of a brick of $\textsf{mod}A$ is exactly given by a shard. We also give a poset isomorphism of shard intersections and wide subcategories of $\textsf{mod}A$.