稳定半正交不可分解簇
Stably semiorthogonally indecomposable varieties
AI总结：
提出非交换稳定半正交不可分解(NSSI)簇的概念，证明到阿贝尔簇有有限态射的簇是NSSI，并应用于证明某些簇中没有幻影子范畴。
摘要：
如果一个三角范畴没有非平凡的半正交分解，则称其为不可分解的。我们引入了非交换稳定半正交不可分解(NSSI)概形的定义。该性质蕴含，每个结构层是完美复形的连通闭子概形具有不可分解的凝聚层导出范畴，并且如果$Y$是NSSI，那么对于任意簇$X$，$X \times Y$的所有半正交分解都是由$X$的分解诱导的。我们证明，任何具有到阿贝尔簇的有限态射的概形都是NSSI，并且以正则NSSI基为底、纤维为NSSI的纤维丛的全空间也是NSSI。我们应用这种不可分解性推断出某些簇中没有幻影子范畴，包括曲面$C \times \mathbb{P}^1$，其中$C$是任意正亏格的光滑真曲线。
英文摘要：
A triangulated category is said to be indecomposable if it admits no nontrivial semiorthogonal decompositions. We introduce a definition of a noncommutatively stably semiorthogonally indecomposable (NSSI) scheme. This property implies, among other things, that each connected closed subscheme whose structure sheaf is a perfect complex has indecomposable derived category of coherent sheaves and that if $Y$ is NSSI, then for any variety $X$ all semiorthogonal decompositions of $X \times Y$ are induced from decompositions of $X$. We prove that any scheme which admits a finite morphism to an abelian variety is NSSI and that the total space of a fibration over a regular NSSI base with NSSI fibers is also NSSI. We apply this indecomposability to deduce that there are no phantom subcategories in some varieties, including surfaces $C \times \mathbb{P}^1$, where $C$ is any smooth proper curve of positive genus.
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