We generalize, explain and simplify Langer's results concerning Frobenius direct images of line bundles on quadrics, describing explicitly the decompositions of higher Frobenius push-forwards of arithmetically Cohen-Macaulay bundles into indecomposables, with an additional emphasis on the case of characteristic two. These results are applied to check which Frobenius push-forwards of the structure sheaf are tilting.
We prove that every connected affine scheme of positive characteristic is a K (π, 1) space for the étale topology. The main ingredient is the special case of the affine space A k n over a field k. This is dealt with by induction on n, using a key "Bertini-type" statement regarding the wild ramification of -adic local systems on affine spaces, which might be of independent interest. Its proof uses in an essential way recent advances in higher ramification theory due to T. Saito. We also give rigid analytic and mixed characteristic versions of the main result.
Abstract. A technical ingredient in Faltings' original approach to p-adic comparison theorems involves the construction of K(π, 1)-neighborhoods for a smooth scheme X over a mixed characteristic dvr with a perfect residue field: every point x ∈ X has an open neighborhood U whose general fiber is a K(π, 1) scheme (a notion analogous to having a contractible universal cover). We show how to extend this result to the logarithmically smooth case, which might help to simplify some proofs in p-adic Hodge theory. The main ingredient of the proof is a variant of a trick of Nagata used in his proof of the Noether Normalization Lemma.
We present some simple examples of smooth projective varieties in positive characteristic, arising from linear algebra, which do not admit a lifting neither to characteristic zero, nor to the ring of second Witt vectors. Our first construction is the blow-up of the graph of the Frobenius morphism of a homogeneous space. The second example is a blow-up of P 3 in a 'purely characteristic-p' configuration of points and lines.
ABSTRACT. If X is a smooth toric variety over an algebraically closed field of positive characteristic and L is an invertible sheaf on X, it is known that F * L, the push-forward of L along the Frobenius morphism of X, splits into a direct sum of invertible sheaves. We show that this property characterizes smooth projective toric varieties.
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