论文标题
光谱代数几何形状中的双重表示
A Dual Representation in Spectral Algebraic Geometry
论文作者
论文摘要
给定频谱deligne-mumford $ x $，我们将$ x $的感知定义为一定类别的形态$ y \ y \ rightarrow x $。对于SPDM中的仿射形态类别，我们表明，一方面可以从qcoh（$ x $）上提取仿射感知$ \ text {aff} _x _x _x $ $ x $的$ x $，以及一个$ \ infty $的子类别的子类别，表示$ \ infty $ - caltemation $ \ calterge $ \ text {rep text {rep text {pep} _ { $ \ mathfrak {g} _*$与$ x $相关联。对于本地形态的类$ \ text {spét} r \ rightarrow x $，$ x $的本地感知由函数$ \ mathbf {x} = \ text {hom}（\ text {hom}（\ text {spét}（ - ），x），x），x）$表示。如果$ \ mathbf {x} $是几何堆栈，则tannaka二元性使我们能够从$ \ text {qcoh}（\ mathbf {x}）恢复$ \ mathbf {x} $，从中，我们也可以从中获得$ \ text pext {pop} $} $ {g} $ {g} $ {\ g {我们通过考虑函数$ \ mathbf {x}来概括这些结果：\ text {calg}^{\ text {cn}} \ rightarrow \ rightarrow \ mathcal {s} $，这些{s} $可根据lurie的频谱artin artinability of lurie的频谱artinability of lurie的定理表示。
Given a spectral Deligne-Mumford stack $X$, we define a perception of $X$ to be a collection of a certain class of morphisms $Y \rightarrow X$. For the class of affine morphisms in SpDM, we show that from QCoh($X$) on can extract the affine perception $\text{Aff}_X$ of $X$ on the one hand, and a subcategory of an $\infty$-category of representations $\text{Rep}_{\mathfrak{g}_*}$ of a dg Lie algebra $\mathfrak{g}_*$ associated with $X$ on the other. For the class of local morphisms $\text{Spét } R \rightarrow X$, the local perception of $X$ is given by the functor $\mathbf{X} = \text{Hom}(\text{Spét}(-), X)$ it represents. If $\mathbf{X}$ is a geometric stack, Tannaka duality allows us to recover $\mathbf{X}$ from $\text{QCoh}(\mathbf{X})$, from which we can also get, after base change, a subcategory of $\text{Rep}_{\mathfrak{g}_*}$. We generalize those results by considering functors $\mathbf{X}: \text{CAlg}^{\text{cn}} \rightarrow \mathcal{S}$ that are representable in accordance with the spectral Artin representability theorem of Lurie.
