Galois Cohomology and Hochschild Cohomology over a Base Topos.

差代数差环是具有单位元的交换环，具有自同态。自然地，我们定义了与环结构和自同态交换的态射，并考虑了差环范畴；我们还定义了模结构与自同态交换的差模，并考虑了差模范畴。在此之前，模型论方法已经成功地建立了交换代数、伽罗瓦理论和布尔逻辑在不同情况下的类似。然而，在上同调的情况下，这些方法被证明是不成功的，托马西奇博士的项目的目的是从范式论和拓扑论的角度继续哈基姆的专著。这有可能将许多关于代数、相对方案和相对代数群的上同调理论联系起来。在环模的背景下，差上同调上同调代数依赖于这样一个简单的事实：对于给定的环R，给定R-mod的两个对象，hom-集合本身就是R-mod中的对象。正是在这里，差类比是不足的；一般来说，给定两个差集，它们的HOM-集是光集，即它不一定被赋予差自同态。同样，给定两个不同的R-模，它们的HOM-集一般不是不同的R-模，因此不可能在不同的R-模范畴中进行明显的上同调类比。丰富范畴理论托马西奇博士转而考虑丰富范畴中的差异代数，他通过其中一个对象与自然数的差集的直和来定义任何两个差集的内部主对象，并配备了预期的移位自同态。因此，上面定义的不同对象的类别是这些丰富的类别的基本类别。这不仅产生了一个本身是不同对象的HOM-集合，而且它获得了涉及两个对象的张量的HOM-集合的“Currying”同构，这是开发上同调框架所需的。Topos理论Tomasic博士指出，这样一个丰富的范畴等价于具有加法作用的自然数么半群的集合范畴，即Grothendieck Topos，称为自然数的分类Topos。因此，我们寻求在任意基Topos上发展我们的上同调理论。通过研究一个群甚至一个群胚的分类拓扑点，就可以得到相应的群等变代数几何。因此，具体地说，差集的拓扑点产生了各自的差代数几何。Galois上同调和Hochschild上同调我正在研究的具体主题是在任意基拓扑域上发展Galois上同调和Hochschild上同调，同时注意在差集的拓扑域上的发展。这项研究将是我的硕士论文的自然延续，我的硕士论文是对Ambrus Pal博士即将发表的一篇论文中的一个分级Hochschild上同调消失结果的阐述。特别地，我将研究差代数的Galois上同调和Hochschild上同调。这一主题是与托马西奇博士商定的，它将成为该计划的一个内在组成部分。Tomasic博士的项目已经建立了丰富的/内同调代数的基础，进一步探索应用于Galois上同调和Hochschild上同调是这项工作的引人注目的继续。

Difference algebra A difference ring is a commutative ring with identity, equipped with an endomorphism. Naturally, we define morphisms which commute with the ring structure and the endomorphism, and consider the category of difference rings; we also define difference modules such that the module structure commutes with the endomorphism, with analogous modules, and consider the category of difference modules. Previously, model-theoretic approaches have been successful in establishing the analogues of commutative algebra, Galois theory and Boolean logic in the difference case. However, in the cohomological case, these methods have proved unsuccessful, and the aim of Dr Tomasic's project is to pursue a continuation of Hakim's monograph from a category-theoretic and topos-theoretic viewpoint. This has the potential to connect numerous cohomology theories for algebras, relative schemes and relative algebraic groups. Difference cohomology Cohomological algebra in the context of ring modules relies on the simple fact that given two objects of R-mod for a given ring R, the hom-set is itself an object in R-mod. It is here that the difference analogue falls short; in general, given two difference sets, their hom-set is a bare set, that is, it is not necessarily endowed with a difference endomorphism. By the same token, given two difference R-modules, their hom-set is not in general a difference R-module, and therefore the obvious analogy of cohomology in the category of difference R-modules is impossible. Enriched category theory Dr Tomasic instead considers difference algebra within an enriched category, by defining the internal hom-object of any two difference sets via the direct sum of one of the objects with the difference set of natural numbers, equipped with the expected shift endomorphism. The categories of difference objects defined above are then the underlying categories of these enriched categories. Not only does this produce a hom-set that is itself a difference object, but it obtains the "currying" isomorphism of hom-sets involving the tensor of two objects that is needed to develop a cohomological framework. Topos theory Dr Tomasic shows that such an enriched category is equivalent to the category of sets with an action of the monoid of natural numbers under addition, which is a Grothendieck topos, known as the classifying topos of the natural numbers. We therefore seek to develop our cohomological theories over an arbitrary base topos. By working over the classifying topos of a group or even a groupoid, one can then develop the corresponding group equivariant algebraic geometry. Therefore, in particular the topos of difference sets yields a respective difference algebraic geometry. Galois cohomology and Hochschild cohomology The specific topic I am researching is to develop Galois cohomology and Hochschild cohomology over an arbitrary base topos, with attention to development over the topos of difference sets. This research will be a natural continuation of my Master's thesis, which was an exposition of a graded Hochschild cohomological vanishing result in an upcoming paper by Dr Ambrus Pal. In particular, I will explore the Galois cohomology and Hochschild cohomology of difference algebras. This topic was agreed upon with Dr Tomasic, and it will form an intrinsic part of the programme. Dr Tomasic's project has already established the foundations of enriched/internal homological algebra, and further exploration of application to both Galois cohomology and Hochschild cohomology is a compelling continuation of this work.