Equivariant motivic sheaves

Question

Thanks to the work of Cisinski-Deglise: http://arxiv.org/abs/0912.2110, we now have a triangulated category of `motivic sheaves' available that admits the standard yoga of the six functors.

Is there any literature out there, or does anyone know how to use/adapt this to obtain a formalism of `equivariant motivic sheaves'. Slightly more precisely, I am interested in having a motivic version of the Bernstein-Lunts equivariant derived category $D_G(X)$ (say $G$ linear algebraic acting on $X$). See no. 51 at http://www.math.tau.ac.il/~bernstei/Publication_list/Publication_list.html for the Bernstein-Lunts construction in the topological setting.

Let me just briefly indicate why (I think) there are problems with just trying to run through the Bernstein-Lunts construction in the motivic setting. Very roughly what one wants is that $D_G(X)$ to be the same as $D(EG\times_G X)$ (a la Borel construction for equivariant cohomology). The space $EG\times_G X$ is a bit problematic (infinite dimensional, not a variety, etc.). Bernstein-Lunts solution is to recover "chunks" of $D(EG\times_G X)$ using sufficiently acyclic approximations to the universal bundle $EG \to BG$. The problem with adapting this naively to the motivic setting is that "chunks" means complexes of a certain length. I.e., you need some t-structure to start making sense of "chunks". And well, motivic t-structures haven't yet been constructed in any sort of generality that would make this issue moot. Does anyone know how to get around this?

Added later: since there seems to be some discussion in the comments about defining this equivariant motivic category (let me denote it $DM_G(X)$) via simplicial varieties, let me ask for two related properties that one would like (in order to justify calling it an equivariant derived category):

1) if the quotient $X/G$ exists, then $DM_G(X) \simeq DM(X/G)$;

2) a realization functor $DM_G(X) \to D_G(X)$, where the right hand side is the Bernstein-Lunts construction in the non-motivic setting.

(I am being deliberately sloppy about what I mean by quotient, and as to what base field I am working over, make whatever assumptions about these things as you see fit.)

Perhaps I should also point out that I am a bit skeptical about getting a suitable formalism using just simplicial varieties, since even in the ordinary complex algebraic setting I am not aware of nor have any idea how to get a functor yoga going using just simplicial varieties. I would assume that if one could do such a thing in the motivic setting, then one should also be able to do it (perhaps even in a simpler way) in the topological setting (which would be quite interesting in my opinion).

Answer 1

Concerning a potential definition of $DA^{et}(BG)$:

The category $DA^{et}(BG,\Delta^{op})$, i.e., the category of motives over the diagram of schemes $(BG,\Delta^{op})$ is probably too big to serve as a reasonable candidate of the category of motives over $BG$. A more reasonable category to consider would be the full subcategory of $DA^{et}(BG,\Delta^{op})$ consisting of ``cartesian diagrams of motives'', i.e., motives $M$ such that the natural map $d^*M_m \to M_n$ is an isomorphism for every $d:\Delta^m \to \Delta^n$.

To explain the notations it is better to look at the general situation: let $(X,I)$ be a diagram of motives and let $M\in DA^{et}(X,I)$ be a diagram of motives over $(X,I)$. For every $i\in I$, one has a motive $M_i \in DA^{et}(X(i))$ over $X(i)$ which is something like the restriction of $M$ to the sub diagram $X(i)$ of $X$ consisting of one scheme. For every arrow $r:j\to i$ in $I$, one has a map $r^* M_i \to M_j$ were $r^*$ is the pull-back along the map $X(j) \to X(i)$. One say that $M$ is cartesian if all these morphisms are isomorphisms. For instance, if $I$ has a final objet $o\in I$, then cartesian motives over $(X,I)$ are equivalent to ordinary motives over $X(o)$.

Another reference for motives over a diagram of schemes is my paper with S. Zucker:

``Relative Artin motives and the reductive Borel-Serre compactification of a locally symmetric variety''

and more precisely section 1.2 (sections 1.3, 1.4 and 1.5 contain also some useful information).

Answer 2

Concerning a potential definition of equivariant motives via approximations of classifying spaces - it works, and it works with the homotopy $t$-structure on motives. Basically, you have to follow the paths of Totaro' definition of Chow groups of classifying spaces resp. the more general equivariant intersection theory of Edidin and Graham.

The important thing (which is the basis for the abovementioned definitions) is that for each linear group $G$, you can find representations $V$ such that the complement of the open subset $V^{\operatorname{free}}$ where $G$ acts freely has arbitrarily high codimension $s$. The quotient $V^{\operatorname{free}}/G$ is then a finite-dimensional approximation of the classifying space, and it computes the right Chow groups in dimensions up to $s$.

Now you do the same thing with motives. The point that makes it work (and the reason why you would want to use the homotopy $t$-structure) is that the open part $V^{\operatorname{free}}$ is $(s-2)$-acyclic for the homotopy $t$-structure, assuming that its closed complement is of codimension $s$. This fact can be proved using Gersten resolution or some such method. With this technical statement in hand, you can develop a category of Borel-equivariant motives along the lines of the book of Bernstein and Lunts. Eventually, this definition coincides with the one suggested in Joseph Ayoub's answer, with an adaptation of the arguments in Bernstein-Lunts appendix B.