A necessary condition for liftings of positive characteristic varieties with finite fundamental groups

In this paper, we introduce a necessary condition for the existence of characteristic zero liftings of certain smooth, proper varieties in positive characteristic, using etale homotopy theory and Wall's finiteness obstruction. For a variety with finite etale fundamental group pi, we define a notion of mod-l finite dominatedness based on the F_l-chain complex of the universal cover of its l-profinite etale homotopy type. We prove that such a variety X can be lifted to characteristic zero only if the above chain complex of X is quasi-isomorphic to a bounded complex of finitely generated projective F_l[pi]-modules. To prove this result, we extend Wall's discussions of finiteness obstructions to l-profinite complete spaces with finite fundamental group.