I have two questions about the Hochschild cohomology of algebraic groups. The first one will reveal the depth of my ignorance, and, because of this depth, I might be asking a question more appropriate for MSE. If so, then please let me know, and I will have no objection to migrating.
I encountered the Hochschild cohomology of an affine algebraic group $G$ with coefficients in a $G$-module in Chapitre II, section 3, of [DG] Demazure and Gabriel - Groupes algébriques (MSN). However, when I Google "Hochschild cohomology", all the results seem to be discussing the cohomology of associative algebras. There is an obvious way to get an associative algebra $k[G]$ from an affine algebraic group $G$, but it is quite uninformative: $k[G]$ considers $G$ only as a scheme, and makes no use of its algebraic-group structure. So I assume the Hochschild cohomology of $G$, in the sense of [DG], is not much related to the Hochschild cohomology of $k[G]$, in what Google suggests is the "usual" sense. Corollaire 3.5 (in Chapitre III, Section 3) of [DG] says, if I understand it correctly, that I should instead be looking at a kind of cohomology associated to comodules of $k[G]$ viewed as a coalgebra. Is this related to the "usual" Hochschild cohomology? Or perhaps homology? (I have no idea how the various duals interact.)
Because of my ignorance regarding (1), I do not know how even to begin to search for my answer to this second question. Over a characteristic-$0$ field, if $G$ is reductive, then the Hochschild cohomology of $G$ with coefficients in any module vanishes (Proposition 3.7 of [DG]). What, if anything, is known about the Hochschild cohomology of a reductive group $G$ over a positive-characteristic field? I would prefer not to assume that the field be algebraically closed, but I can assume that it is separably closed if necessary.
