Difference modules and difference cohomology

Abstract

We give some basics about homological algebra of difference representations. We consider both the difference discrete and the difference rational case. We define the corresponding cohomology theories and show the existence of spectral sequences relating these cohomology theories with the standard ones.

A topos-theoretic view of difference algebra

“…There is a number of independent papers that are related to parts of this manuscript, such as [2], [15] and especially the recent preprint [16]. This shows that difference algebra is decidedly entering the age of cohomology.…”

“…In order to avoid awkward notation for a general T, we will provide a proof in the concrete case of groups in 5. 15.…”

“…In [15], the authors study cohomology of difference algebraic groups by computing Ext functors in the category of difference comodules for a difference Hopf algebra. In our Section 21, we wanted to do difference algebraic group cohomology in the style of Demazure's article in [22].…”

“…In our Section 21, we wanted to do difference algebraic group cohomology in the style of Demazure's article in [22]. We do this in the context of enriched category theory over σ-Set, and derive a comparison theorem to the cohomology from [15]. We use internal homs to replace the assumption of invertibility of difference operators.…”

“…This context naturally leads to the notion of representations of affine difference algebraic groups as difference comodules of the associated difference Hopf algebra, allowing us to compare to the previous work of [51] and [15].…”