Inside the derived category of Nisnevich sheaves with transfers there is the category $DM^{eff} $ of Voevodsky's effective motivic complexes (actually, Voevodsky only considered bounded above complexes, but it seems that the unbounded version is al well understood now). Abstract nonsense easily yields the existence of the right adjoint $R$ to this embedding. Did anyone study it? It seems to be much less popular than the left adjoint given by Suslin complexes. What can one say about $R(\Omega^i) $ (i.e. about the image of the sheaf of algebraic differentials put in degree $0$)?
$\begingroup$
$\endgroup$
2
asked Jan 6, 2014 at 14:07
-
$\begingroup$ The Suslin complex also describes this adjoint in the unbounded case. Therefore, the fact that you work with bounded above complexes or not does not affect your question about differentials. $\endgroup$– D.-C. CisinskiCommented Jan 6, 2014 at 14:14
-
$\begingroup$ I need unbounded complexes since I want motivic complexes to form a compactly generated category; then a Brown representability argument (due to Neeman) yields the right adjoint. I am not really interested in the left adjoint and Suslin complexes. $\endgroup$– Mikhail BondarkoCommented Jan 6, 2014 at 14:24
Add a comment
|