Homotopy Theory of Commutative Algebras

这个项目的中心焦点是将空间同伦论的方法和技术转换到代数领域，然后将它们应用于空间同伦论中的问题。方法是将代数本身视为单纯代数的特例。在这个更大的范畴中，存在同伦理论和同伦群和同调群的概念，这取决于所考虑的代数的类型，它们是同伦不变量。限制到(单纯)交换代数，所得到的同调理论称为Andre-Quillen同调，它被证明对交换代数和拓扑学都是有用的。特别地，理解这个同调理论何时消失在交换代数和拓扑学中都有意义。在交换代数方面，Quillen的猜想表明Noether止痛环上Noether止痛的Andre-Quillen同调高度消失意味着Noether环上的Andre-Quillen同调一定在二次以上消失。这个项目的目的是解决这个猜想及其对交换代数的影响。在拓扑方面，作为Steenrod代数上的不稳定代数，具有给定模上同调的拓扑空间存在的障碍存在于该不稳定代数的某些Andre-Quillen上同调群中。这个项目还将调查寻找这些障碍消失的条件，以及将安德烈-奎伦上同调的其他方面与该空间的同伦联系联系起来，如果它存在的话。研究拓扑空间的核心是其同伦类型理论，它可以用代数拓扑学提供的方法来理解。这些方法包括给空间分配代数不变量，称为同伦、同调和上同调群，它们的内部结构有助于辨别原始拓扑对象的内部结构。可以发展一种双重的视角，从而可以用同伦理论的方法来研究代数本身的理由。特别地，对于交换代数，有一个称为Andre-Quillen(Co)同调的(Co)同调理论，它已被证明对交换代数和拓扑学都是有用的。本项目的目标是了解这个同调理论在什么条件下局部或整体消失，以及它对交换代数和拓扑学的影响。例如，其中一个重点将是安德烈-奎伦霍姆学在全球范围内高度消失对交换代数的影响。在某些特殊的条件下，我们知道这样的消失意味着赋值交换代数具有一个非常特殊的特征，那就是它是一种称为局部完全交的代数。本项目的一部分涉及在这些特殊条件减弱的情况下确定这种全球消失的影响。在拓扑学方面，空间的上同调可以看作是一个交换代数。由此产生的一个问题是：给定的交换代数何时才能实现为空间的上同调？根据给定交换代数的Andre-Quillen上同调中的某些元素，可以给出一个答案：空间是可实现的当且仅当这些元素消失。本项目的另一个目标将是了解这些阻碍因素及其消失的条件，以及将Andre-Quillen上同调的其他方面与该空间的同伦联系联系起来，如果它存在的话。

9972546Turner The central focus of this project is to translate the methods andtechnology of the homotopy theory of spaces to the realm of algebras andthen apply them back to questions in the homotopy theory of spaces. Theapproach is to view algebras themselves as special cases of simplicialalgebras. In this larger category, there exists a homotopy theory andnotions of homotopy groups and homology groups, depending on the type ofalgebras considered, which are homotopy invariants. Restricting to(simplicial) commutative algebras, the resulting homology theory is calledAndre-Quillen homology, which has proved useful for both commutativealgebra and topology. In particular, understanding when this homologytheory vanishes has implications in both commutative algebra and topology.On the commutative algebra side, conjectures of Quillen indicate that thevanishing in high degrees of the Andre-Quillen homology of Noetherianalgebras, over a Noetherian ring, implies the homology must vanish abovedegree two. It is the intent of this project to resolve this conjectureand its implications for commutative algebra. On the topological side,obstructions to the existence of a topological space with given mod pcohomology, as an unstable algebra over the Steenrod algebra, lie incertain Andre-Quillen cohomology groups of that unstable algebra. Thisproject will also investigate finding conditions for those obstructions tovanish as well as relating other aspects of the Andre-Quillen cohomologyto the homotopy of that space, if it exists. Central to studying topological spaces is the theory of their homotopytype, which can be understood using the methods provided by algebraictopology. These methods involve assigning to spaces algebraic invariants,called homotopy, homology, and cohomology groups, whose internal structureshelp discern the internal structures of the original topological object. Adual perspective can be developed whereby the methods of homotopy theorycan be used to study algebras on their own grounds. In particular, thereis a (co)homology theory, called Andre-Quillen (co)homology, for commutativealgebras that has proved useful to both commutative algebra and topology.The goal of the present project is to understand under what conditions thishomology theory vanishes, either locally or globally, and the implicationsfor commutative algebra and topology. For example, one focus will be on theimplications for commutative algebra of the global vanishing of Andre-Quillenhomology in high degrees. Under certain special conditions, it is known thatsuch a vanishing implies that the imputed commutative algebra has the veryspecial feature of being a type of algebra called a local completeintersection. Part of the present project is concerned with determining theimplications of such global vanishing under a weakening of these specialconditions. On the topological side, the cohomology of a space can be viewedas a commutative algebra. One question that arises is: when can a givencommutative algebra be realized as the cohomology of a space? An answer canbe given in terms of certain elements in the Andre-Quillen cohomology of thegiven commutative algebra: the space is realizable if and only if theseelements vanish. Another goal of the present project will be to understandthese obstruction elements and the conditions under which they vanish as wellas relating other aspects of the Andre-Quillen cohomology to the homotopy ofthat space, if it exists.***