Annihilators and kernels in Kato's cohomology in positive characteristic and in Witt groups in characteristic 2

群、环和域是代数中的基本对象。组是由元素组成的对象，可以使用组操作将这些元素组合在一起，以根据严格的规则形成新元素。立方体在空间中的旋转或整数加上通常的加法形成一个群。环由在加法下形成一个群的元素组成，但这些元素也可以彼此相乘，加法和乘法的相互作用受某些规则的支配。例如，用通常的加法和乘法运算的整数组成一个环。域是乘法具有特别好性质的环，例如，总是存在非零元素的倒数。域的例子是有理数和实数。代数学家通常对环或域上定义的对象感兴趣，例如由环或域中系数的一个或多个变量的多项式给出的方程。这样的方程可以根据一定的规则进行变换，一个自然的问题是决定一个方程何时可以变换成另一个方程，即两个方程何时“等价”。要做到这一点，人们试图找到等价方程共享的某些不变量，而这些不变量又可以是群或环中的元素。二次型可以被认为是多个变量的二次多项式，它们已经被广泛研究了几个世纪。在它们的分类中的一个重要结果是证明了Voevodsky的共同称为Milnor猜想(由于这项工作和相关工作在2002年获得了Fields勋章)，其中二次型与所谓的Galois上同调群和Milnor K-群有关，在2不同于0的域的情况下。20世纪80年代，Kazuya Kato使用了我们所称的Kato上同调群，得到了2等于0的域的类似结果。它们在素数p等于0的域的研究中是重要的，并且它们有许多应用，例如在数论(类场论)中。为了更好地理解这些重要的群，我们计划研究两个问题：1.给定Kato上同调中的一个元素，当乘以该给定元素时，Kato上同调中的哪些元素为零，即我们要确定该给定元素的“零化子”。2.当从一个域转移到一个更大的域时，Kato上同调中的哪些元素变为零，即我们要确定那个域扩张的“限制映象的核”。我们还想研究2等于0的域的Witt群的类似问题。这些Witt组基本上分类为二次(分别为双线性)形式，根据Kato的结果，Witt群的问题与Kato上同调的问题密切相关。

Groups, rings and fields are fundamental objects in algebra. Groups are objects consisting of elements that can be combined using a group operation to form new elements according to strict rules. The rotations of a cube in space or the integers together with the usual addition form a group. Rings consist of elements that form a group under an addition, but that can also be multiplied with each other, with the interaction of addition and multiplication being governed by certain rules. E.g., the integers with usual addition and multiplication form a ring. Fields are rings in which multiplication has particularly nice properties, e.g. there always exists the reciprocal of a nonzero element. Examples of fields are the rational and the real numbers. Algebraists are often interested in objects defined over rings or fields, such as equations given by a polynomial in one or more variables with coefficients in a ring or field. Such equations may be transformed according to certain rules, and a natural question is to decide when an equation can be transformed into another, i.e. when two equations are "equivalent". To do so, one tries to find certain invariants that equivalent equations share and these invariants can in turn be elements in a group or ring. Quadratic forms can be considered as polynomial equations of degree 2 in several variables and they have been studied extensively for centuries. An important result in their classification is the proof of the co-called Milnor conjectures by Voevodsky (Fields medal in 2002 for this and related work), where quadratic forms are related to so-called Galois cohomology groups and Milnor K-groups in the case of fields where 2 is different from 0. Analogous results for fields with 2 equal to 0 have been obtained by Kazuya Kato in the 1980s where he used what we call the Kato cohomology groups. These are important in the study of fields in which a prime number p equals 0 and they have many applications, e.g. in number theory (class field theory). To gain a better understanding of these important groups, we plan to study two questions: 1. Given an element in Kato cohomology, which elements in Kato cohomology become zero when multiplied by that given element, i.e. we want to determine the "annihilator" of that given element. 2. Which elements in Kato cohomology become zero when passing from a field to a bigger field, i.e. we want to determine the "kernel of the restriction map" for that field extension. We also want to study analogous questions for the Witt groups of fields in which 2 equals 0. These Witt groups essentially classify quadratic (resp. bilinear) forms over such fields and by Kato's results, the questions for Witt groups are intimately linked to those for Kato's cohomology.