Question

Let G be an algerbraic group, and X be a G-variety (that I will assume to be smooth). Then one can consider a simlicial variety whose terms are $G^i\times X$. This simplicial variety yields a 'complex of varieties' $\dots\to G\times G\times X\to G\times X\to X$ (the arrows are formal alternating sums of the corresponding morphisms of varieties), which yields a Voevodsky's motivic complex. Is the motif obtained isomorphic to anything 'nice'; did someone already consider it (or something similar)?

Certainly, one could speculate (mimicking the definition of the cohomology of a module over a group $G$) that $\dots\to G\times G\times G\to G\times G\to G$ is a 'resolution' of $\mathbb{Z}=M(pt)$ in a certain category of $G$-motives, and the complex I mentioned is $RHom(\mathbb{Z},X)$ in this category; yet this doesn't seem to help. Another association is Vishik's complex $\dots\to X\times X\times X\to X\times X\times X\to pt$; this does not seem to be helpful either.

Answer 1

The simplicial variety you write is a presentation of the stack $X/G$ as a simplicial sheaf, and the motive you obtain is just be the motive of $X/G$, no? Of course you might use this as the definition of the motive of $X/G$, so maybe this is orthogonal to your question. It's a motive over $BG$, or equivalently corresponds (by what some call "Koszul duality") to the class of $X$ in the dg-category of $G$-motives (comodules over the coalgebra given by the cochains (cohomological motive?) of $G$, in the $\infty$-categorical sense, or modules over the algebra object given by chains on $G$).