From Hag To Dag: Derived Moduli Stacks
From Hag To Dag: Derived Moduli Stacks
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从 Hag 到 Dag：派生模数堆栈
DOI：
10.1007/978-94-007-0948-5_6
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
G. Vezzosi
中科院分区：
文献类型：
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作者：
B. Toen;G. Vezzosi
这些是2002年秋季的一些讲座的扩展笔记，关于同列代数几何，特别强调其在衍生代数几何和衍生变形理论中的应用。我们使用在[HAG-I]中开发的一般框架，特别是模型拓扑，模型位置和它们之上的堆栈的概念，以定义各种派生模函子并研究它们的几何性质。我们首先定义了d -堆的模型范畴，将e -拓扑推广到交换微分梯度代数范畴，并证明了它的同伦范畴包含有趣的对象，如方案、代数堆、高代数堆、dg-堆等。我们定义了几何d -堆的概念，并给出了一些相关的几何结构（o模、完美复合体、k理论、衍生切线堆、余切复合体、各种光滑性概念等）。最后，我们定义并研究了拓扑空间上的局部系统、光滑投影簇上的向量束和a∞-分类结构的衍生模问题。我们给出了这些例子的几何性和平滑性结果。本文结果的证明将主要在[HAG-II]中给出。
These are expanded notes of some talks given during the fall 2002, about homotopical algebraic geometry with special emphasis on its applications to derived algebraic geometry and derived deformation theory. We use the general framework developed in [HAG-I], and in particular the notions of model topology, model sites and stacks over them, in order to define various derived moduli functors and study their geometric properties. We start by defining the model category of D-stacks, with respect to an extension of the etale topology to the category of commutative differential graded algebras, and we show that its homotopy category contains interesting objects, such as schemes, algebraic stacks, higher algebraic stacks, dg-schemes, etc. We define the notion of geometric D-stacks and present some related geometric constructions (O-modules, perfect complexes, K-theory, derived tangent stacks, cotangent complexes, various notions of smoothness, etc.). Finally, we define and study the derived moduli problems classifying local systems on a topological space, vector bundles on a smooth projective variety, and A ∞-categorical structures. We state geometricity and smoothness results for these examples. The proofs of the results presented in this paper will be mainly given in [HAG-II].