Galois Cohomology and Hochschild Cohomology over a Base Topos.

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.