Homotopy Theory of Commutative Algebras

9972546Turner本课题的重点是将空间同伦理论的方法和技术应用于代数领域，然后再将其应用于空间同伦理论中的问题。这种方法是把代数本身看作简单代数的特殊情况。在这个更大的范畴中，存在同伦理论和同伦群和同伦群的概念，这取决于所考虑的代数的类型，它们是同伦不变量。对于（简单的）交换代数，所得到的同调理论被称为andre - quillen同调，它已被证明对交换代数和拓扑都很有用。特别是，理解这种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Noetherian代数在Noetherian环上的Andre-Quillen同调在高次上的消失，意味着该同调必须在2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在于该不稳定代数的不确定Andre-Quillen上同调群。该项目还将研究寻找这些障碍消失的条件，以及将Andre-Quillen上同伦的其他方面与该空间的同伦联系起来，如果它存在的话。研究拓扑空间的核心是它们的同伦型理论，这可以用代数拓扑提供的方法来理解。这些方法包括赋予空间代数不变量，称为同伦，同调和上同调群，其内部结构有助于识别原始拓扑对象的内部结构。对偶视角可以发展，在此基础上，同伦理论的方法可以用来研究代数。特别地，对于交换代数有一个（co）同调理论，称为Andre-Quillen （co）同调，它已被证明对交换代数和拓扑都很有用。本项目的目标是了解在什么条件下这个同调理论消失，无论是局部的还是全局的，以及对交换代数和拓扑的含义。例如，一个重点将是对交换代数的含义的整体消失的Andre-Quillenhomology在高度。在某些特殊条件下，我们知道这样的消失意味着输入交换代数具有一种称为局部完全交的代数的非常特殊的特征。本项目的部分内容是在这些特殊条件减弱的情况下确定这种全球消失的含义。在拓扑方面，空间的上同调可以看作是交换代数。出现的一个问题是：给定的交换代数何时可以被实现为空间的上同调？可以用给定交换代数的Andre-Quillen上同调中的某些元素给出答案：当且仅当这些元素消失时，空间是可实现的。本项目的另一个目标是理解这些阻碍元素和它们消失的条件，以及将Andre-Quillen上同伦的其他方面与该空间的同伦联系起来，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.***