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

Abstract In this paper, we introduce a necessary condition for the existence of characteristic zero liftings of certain smooth, proper varieties in positive characteristic, using étale homotopy theory and Wall’s finiteness obstruction. For a variety with finite étale fundamental group $$\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> , we define a notion of mod- l finite dominatedness based on the $$\mathbb {F}_l$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>l</mml:mi> </mml:msub> </mml:math> -chain complex of the universal cover of its l -profinite étale 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 $$\mathbb {F}_l [\pi ]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>l</mml:mi> </mml:msub> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>π</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -modules. To prove this result, we extend Wall’s discussions of finiteness obstructions to l -profinite complete spaces with finite fundamental group.