论文标题
$ g $ - 粉丝的碎片理论
Shard theory for $g$-fans
论文作者
论文摘要
对于有限的维数代数$ a $，$ g $ -fan $σ（a）$的概念是从真正的grothendieck组$ a $ a $ $ a $ a $ a $ k_0（\ mathsf {proj} a）中定义的。在本文中，我们讨论了$σ（a）$的碎片理论，该理论最初是为超平面布置定义的。我们在$ \ mathrm {mod} a $的扭转类别的poset的一组汇总元素与$ g $ - finite代数$ a $的$σ（a）$的碎片集之间建立了对应关系。此外，我们证明了$ \ mathrm {mod} a $的砖块的半固定区域完全由碎片给出。我们还给出了碎片交叉点和$ \ mathrm {mod} a $的宽子类别的poset同构。
For a finite dimensional algebra $A$, the notion of $g$-fan $Σ(A)$ is defined from two-term silting complexes of $A$ in the real Grothendieck group $K_0(\mathsf{proj} A)_{\mathbb{R}}$. 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 the poset of torsion classes of $\mathrm{mod} A$ and the set of shards of $Σ(A)$ for $g$-finite algebra $A$. Moreover, we show that the semistable region of a brick of $\mathrm{mod} A$ is exactly given by a shard. We also give a poset isomorphism of shard intersections and wide subcategories of $\mathrm{mod} A$.
