Comodule categories and the geometry of the stack of formal groups
Niko Naumann
推广了M. Hovey和N. Strickland关于Landweber正合代数的余模范畴，使用代数栈的形式。
We generalise recent results of M. Hovey and N. Strickland on comodule categories for Landweber exact algebras using the formalism of algebraic stacks.
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In this paper we study quasi-categories of comodules over coalgebras in a stable homotopy theory. We show that the quasi-category of comodules over the coalgebra associated to a Landweber exact S-algebra depends only on the height of the associated formal group. We also show that the quasi-category of E(n)-local spectra is equivalent to the quasi-category of comodules over the coalgebra A\otimes A for any Landweber exact S_(p)-algebra A of height n at a prime p. Furthermore, we show that the category of module objects over a discrete model of the Morava E-theory spectrum in the K(n)-local disc…
Morita theory for Hopf algebroids and presheaves of groupoids
Hopf代数胚上的余模在代数拓扑学中具有重要意义。众所周知，一个Hopf代数胚与Aff上群胚的预层是一回事，Aff是交换环的相反范畴。在本文中，我们表明，余模是相同的东西作为一个拟凝聚层在这个预层的群胚。我们证明的一般定理，内部等价的Grothendieck拓扑上Aff的群胚的预层引起的范畴的层在该拓扑的等价。然后，我们表明使用忠实平坦的下降，内部等价的平坦拓扑产生的等价类的准相干层。相应的声明为Hopf代数胚是弱等价的Hopf代数胚有等价的余模范畴。我们将其应用到形式群的法律，在那里我们得到了相当大的推广的米勒-Ravenel和Hovey-Sadofsky环的变化定理在代数拓扑。
We define a filtration by open substacks on the non-connective spectral moduli stack of formal oriented groups, which simultaneously encodes and relates the chromatic filtration of spectra and the height stratification of the classical moduli stack of formal groups. Using this open filtration, we express various classical constructions in chromatic homotopy theory, such as chromatic localization, the monochromatic layer, and $K(n)$-localization, in terms of restriction and completion of sheaves in non-connective spectral algebraic geometry.
Quasi-coherent sheaves on the moduli stack of formal groups
这本专著的中心目的是提供准相干层的分解结果的模堆叠的一维形式群。这些结果将基于堆栈本身的几何形状，特别是高度过滤和几何点的形式邻域的分析。主要定理是代数色收敛结果和断裂平方分解。在这个故事中有一个主要的技术障碍，因为形式群的模栈不具有通常定义的代数栈所需的有限性属性。这不是一个概念问题，但为了明确这一点并写下一个自成一体的叙述，我已经包括了大量关于堆栈本身几何形状的讨论，并给出了各种等价的描述。
We introduce and study the non-connective spectral stack $\mathcal M_\mathrm{FG}^\mathrm{or}$, the moduli stack of oriented formal groups. We realize some results of chromatic homotopy theory in terms of the geometry of this stack. For instance, we show that its descent spectral sequence recovers the Adams-Novikov spectral sequence. For two $\mathbb E_\infty$-forms of periodic complex bordism $\mathrm{MP}$, the Thom spectrum and Snaith construction model, we describe the universal property of the cover $\mathrm{Spec}(\mathrm{MP})\to\mathcal M_\mathrm{FG}^\mathrm{or}$. We show that Quillen's ce…
We exhibit a relationship between motivic homotopy theory and spectral algebraic geometry, based on the motivic $τ$-deformation picture of Gheorghe, Isaksen, Wang, Xu. More precisely, we identify cellular motivic spectra over $\mathbf C$ with ind-coherent sheaves (in a slightly non-standard sense) on a certain spectral stack $τ_{\ge 0}(\mathcal M_\mathrm{FG}^\mathrm{or})$. The latter is the connective cover of the non-connective spectral stack $\mathcal M_\mathrm{FG}^\mathrm{or}$, the moduli stack of oriented formal groups, which we have introduced previously and studied in connection with chr…
Derived algebras on formal stacks and prismatic gauges
本文研究形式堆栈上导出代数与棱柱标量的关系，证明了相关分类定理，并扩展了导出几何中的结果。
This paper presents a survey on formal moduli problems. It starts with an introduction to pointed formal moduli problems and a sketch of proof of a Theorem (independently proven by Lurie and Pridham) which gives a precise mathematical formulation for Drinfeld's derived deformation theory philosophy, which gives a correspondence between formal moduli problems and differential graded Lie algebras. The second part deals with Lurie's general theory of deformation contexts, which we present in a slightly different way than the original paper, emphasising the (more symmetric) notion of Koszul dualit…
Modeling Group Actions on Stacks (Especially the Lubin-Tate Action)
本文综述了研究投射有限群在形式模栈上作用的两种方法——几何建模与双塔法，并以Lubin-Tate作用为例展示其应用。
Exotic periodic phenomena in the cohomology of the moduli stack of $1$-dimensional formal group laws
本文研究一维形式群律模栈上同调中的新周期结构，区别于$v_n$周期性，通过$\mathbb{C}$-动机Adams谱序列在动机稳定同伦中揭示类似现象。
This paper is the sequel to [PTVV] (IHES Vol. 117, 2013). We develop a general and flexible context for differential calculus in derived geometry, including the de Rham algebra and polyvector fields. We then introduce the formalism of formal derived stacks and prove formal localization and gluing results. These allow us to define shifted Poisson structures on general derived Artin stacks, and prove that the non-degenerate Poisson structures correspond exactly to shifted symplectic forms. Shifted deformation quantization for a derived Artin stack endowed with a shifted Poisson structure is disc…
On representation categories of $A_\infty$-algebras and $A_\infty$-coalgebras
本文借助单子、余单子与 Eilenberg–Moore 范畴构建统一框架，研究 $A_\infty$-代数和余代数的表示范畴，并通过有理配对联系余模与模，进而引入 $A_\infty$-反模。
The notion of a K3 spectrum is introduced in analogy with that of an elliptic spectrum and it is shown that there are "enough" K3 spectra in the sense that for all K3 surfaces X in a suitable moduli stack of K3 surfaces there is a K3 spectrum whose underlying ring is isomorphic to the local ring of the moduli stack in X with respect to the étale topology, and similarly for the ring of formal functions on the formal deformation space.
Algebraic aspects of higher nonabelian Hodge theory
我们更仔细地研究了更高的nonabelian德拉姆上同调的一个光滑的投影品种或家庭的品种，已在以前的一些文件中定义。我们正式使用$n$ -堆栈的概念，形状的基础上，这个nonabel上同调。将de Rham结构推广到任何适当的形式范畴或形式范畴族，产生了de Rham形状的Hodge理论的各种代数方面，例如Hodge过滤，Gauss-Manin连接，Griffiths横截性，Gauss-Manin连接的扩展与奇点上的正则奇点等。我们为$n$ -类别开发了更多的技术，例如$n+1$ -类别$nCAT$的规范替代;我们提出了以下一般类型的问题：非交换上同调$n$ -栈$Hom(X,T)$的性质是什么，作为系数$n$ -栈$T$和域$n$ -栈$X$的性质的函数？
Affineness and reconstruction in complex-periodic geometry
本文探讨了复杂周期几何中仿射性和局部降维的理论基础，通过谱栈构造和染色同调理论，提出了一类0-仿射栈，并展示了可重建栈由其全局截面自然确定。
A note on ideals in derived geometries
本文研究导出几何中理想理论，比较不同完备性概念，构建导出方案理论图像，并探讨非连通性仿射 morphism 的变形空间性质。
Algebraic Geometry over model categories (a general approach to derived algebraic geometry)
对于一个（半）模型范畴M，我们定义了M上的同伦Grothendieck拓扑的概念，以及与之相关的栈模型范畴。我们用它来定义一个对称monoidal基模型范畴上的几何堆栈的概念;几何堆栈是“在模型范畴上做代数几何”的基本对象。我们给出了两个例子，这种形式主义的应用。第一个是DG-方案作为几何叠加在模型范畴的配合物的解释和第二个是一个定义的标准K理论的E_{\infty} -环光谱。这个第一个版本是非常初步的，可能被认为是一个详细的研究公告。一些证据，更多的细节和更多的例子将在即将到来的版本中添加。
Fourier--Mukai equivalences for formal groups and elliptic Hochschild homology
本文比较两种椭圆 Hochschild 同调定义，提出形式群的新 Fourier--Mukai 对偶性以统一扭曲 Hochschild 同调，并引入全局版本暗示了普遍的 TMF-Hochschild 同调形式。
The stable cooperations of Morava $K$-Theory and the fiber product of automorphism groups of formal group laws
本文通过构造形式群律自同构子群纤维积的Hopf代数，建立其与Morava K-理论稳定合作的同构，揭示包括外部代数部分在内的整体结构与形式群律的联系。
We extend a recent result of Pantev-Toen-Vaquie-Vezzosi, who constructed shifted symplectic structures on derived mapping stacks having a Calabi-Yau source and a shifted symplectic target. Their construction gives a clear conceptual framework for the so-called AKSZ formalism. We extend the PTVV construction to derived mapping stacks with boundary conditions, which is required in most applications to quantum field theories (see e.g. the work of Cattaneo-Felder on the Poisson sigma model, and the recent work of Cattaneo-Mnev-Reshetikhin). We provide many examples of Lagrangian and symplectic str…
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