AI 中文总结
本文发展等变局部系理论以范畴化真正等变上同调，引入全局、真正及调和局部系范畴，并证明调和局部系与拟凝聚层等价及smashing局部化性质。
AI 中文摘要
我们发展了一种等变局部系理论，它范畴化了真正等变上同调理论——例如Atiyah–Segal $K$-理论、Lurie的调和上同调，以及Grojnowski、Greenlees和Gepner–Meier的等变椭圆上同调——类似于普通局部系的范畴化奇异上同调的方式。更精确地说，对于全局空间$X$和系数系统$\mathscr{A} \colon \mathrm{Orb}^{\mathrm{op}} \to \mathrm{Pr}^{\mathrm{L}}$，我们引入了全局等变局部系的范畴$\mathrm{LS}^{\mathrm{glo}}(X,\mathscr{A})$和真正局部系的范畴$\mathrm{LS}^{\mathrm{gen}}(X,\mathscr{A})$，后者将紧李群$G$的真正稳定范畴$\mathrm{Sp}^G$推广到非常数系数。当系数来自局部复周期基上的定向阿贝尔群栈$A$时，我们还构造了调和局部系的范畴$\mathrm{LS}^{\mathrm{temp}}(X,A)$，将Lurie的理论推广到有限群之外；其定义条件由Atiyah–Segal完备化定理的调和形式所证明。我们证明全局截面诱导等价$\mathrm{LS}^{\mathrm{temp}}(BU(1),A) \simeq \mathrm{QCoh}(A)$，并且证明如果$X$是轨道空间，则$\mathrm{LS}^{\mathrm{temp}}(X,A)$是$\mathrm{LS}^{\mathrm{gen}}(X,A)$的smashing局部化。
英文摘要
We develop a theory of equivariant local systems which categorifies genuine equivariant cohomology theories -- such as Atiyah--Segal $K$-theory, Lurie's tempered cohomology, and the equivariant elliptic cohomology of Grojnowski, Greenlees, and Gepner--Meier -- analogously to how the category of ordinary local systems categorifies singular cohomology. More precisely, we introduce, for a global space $X$ and a coefficient system $\mathscr{A} \colon \mathrm{Orb}^{\mathrm{op}} \to \mathrm{Pr}^{\mathrm{L}}$, the categories $\mathrm{LS}^{\mathrm{glo}}(X,\mathscr{A})$ and $\mathrm{LS}^{\mathrm{gen}}(X,\mathscr{A})$ of globally equivariant and genuine local systems, the latter generalizing the genuine stable category $\mathrm{Sp}^G$ of a compact Lie group $G$ to non-constant coefficients. When the coefficients come from an oriented abelian group stack $A$ over a locally complex periodic base, we also construct the category $\mathrm{LS}^\mathrm{temp}(X,A)$ of tempered local systems, extending Lurie's theory beyond finite groups; the defining condition is justified by a tempered form of the Atiyah--Segal completion theorem. We show that global sections induce an equivalence $\mathrm{LS}^{\mathrm{temp}}(BU(1),A) \simeq \mathrm{QCoh}(A)$ and we prove that $\mathrm{LS}^\mathrm{temp}(X,A)$ is a smashing localization of $\mathrm{LS}^\mathrm{gen}(X,A)$ if $X$ is an orbispace.
Comments129 pages. Comments very welcome!
