AI 中文总结
研究闭层上反常层的不可分解扩张，通过引入\(S -\)小扩张等概念，利用局部系统范畴半单的条件构造加法范畴等价，得到不可分解\(p -\)反常层结构分类，推广了相关描述并产生极大扩张函子。
AI 中文摘要
给定一个拓扑分层空间\(X\)，我们为闭层\(S\)上的反常层扩张建立了一个范畴框架。我们引入了相对于\(X\setminus S\)上固定反常层的\(S -\)小扩张和扩张对的概念。通过用\(S\)上局部系统范畴是半单的范畴条件代替MacPherson和Vilonen工作中的同伦假设\(\pi_1(S)=\pi_2(S)=0\)，我们构造了\(S -\)小扩张和扩张对之间的加法范畴等价。这将MacPherson - Vilonen的描述扩展到更一般的情形，并产生了一个极大扩张函子，推广了Beilinson的构造。结果，我们得到了\(X\)上不可分解\(p -\)反常层的结构分类：每个这样的对象要么是\(S\)上不可分解局部系统的零扩张，要么来自其典范态射不能分解为直和的扩张对。
英文摘要
Given a topologically stratified space $X$, we develop a categorical framework for extensions of perverse sheaves over a closed stratum $S$. We introduce the notion of $S$-small extensions and extension pairs relative to a fixed perverse sheaf on $X\smallsetminus S$. By replacing the homotopical assumptions $π_1(S)=π_2(S)=0$ in the work of MacPherson and Vilonen (Invent. Math., 84(2):403-435, 1986) with the categorical condition that the category of local systems on $S$ is semisimple, we construct an equivalence of additive categories between $S$-small extensions and extension pairs. This extends the MacPherson--Vilonen description to a more general setting and yields a maximal extension functor, generalising Beilinson's construction. As a consequence, we obtain a structural classification of indecomposable $p$-perverse sheaves on $X$: every such object is either an extension by zero of an indecomposable local system on $S$, or arises from an extension pair whose canonical morphism does not decompose as a direct sum.
CommentsAny comments are welcome
