Galois Cohomology and Hochschild Cohomology over a Base Topos.

差环是一个有单位元的交换环，它具有一个自同态。自然地，我们定义与环结构和自同态交换的态射，并考虑差环的范畴;我们还定义差模，使得模结构与自同态交换，与类似的模交换，并考虑差模的范畴。在此之前，模型论的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物。然而，在上同调的情况下，这些方法都被证明是不成功的，而Tommy博士的项目的目的是从范畴论和拓扑论的观点继续哈基姆的专著。这有可能连接代数，相对计划和相对代数群的众多上同调理论。差上同调环模下的上同调代数依赖于一个简单的事实，即给定一个环R的两个R-mod对象，其hom-set本身也是R-mod中的一个对象。在这里，差的类比福尔斯不够的;一般来说，给定两个差集，它们的hom-set是一个裸集，也就是说，它不一定被赋予差自同态。同样，给定两个差R-模，它们的hom-set一般不是差R-模，因此在差R-模范畴中上同调的明显类比是不可能的。丰富范畴理论Tomorrow博士考虑的是丰富范畴内的差代数，通过定义任何两个差集的内部hom-object，通过其中一个对象与自然数的差集的直接和，配备了预期的移位自同态。然后，上面定义的差异对象的类别是这些丰富类别的基础类别。这不仅产生了一个本身就是差对象的hom-set，而且还获得了包含两个对象张量的hom-set的“currying”同构，这是开发上同调框架所需要的。拓扑理论托姆斯基博士表明，这样一个丰富的范畴是等价的范畴集的作用下，自然数的幺半群加法，这是一个格罗滕迪克拓扑，被称为分类拓扑的自然数。因此，我们寻求在任意基拓扑上发展我们的上同调理论。通过研究一个群甚至一个广群的分类拓扑，人们可以发展相应的群等变代数几何。因此，特别是拓扑的差集产生相应的差代数几何。伽罗瓦上同调和Hochschild上同调我正在研究的具体课题是在任意基拓扑上发展伽罗瓦上同调和Hochschild上同调，并注意在差集拓扑上的发展。这项研究将是一个自然的延续我的硕士论文，这是一个分级Hochschild上同调消失的结果在即将到来的论文由Ambrus博士的阐述。特别是，我将探讨差代数的伽罗瓦上同调和Hochschild上同调。这一主题是与汤姆博士商定的，它将成为该计划的一个固有部分。Tommy博士的项目已经建立了丰富/内部同调代数的基础，进一步探索Galois上同调和Hochschild上同调的应用是这项工作的一个引人注目的延续。

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.