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Abstract
We study certain questions in positive and mixed characteristic algebraic geometry related to the direct summand conjecture.
Our main result in positive characteristic is that the direct summand condition (see Condition 1.0.2) coincides with a derived enhancement (see Condition 1.0.1); one can view this result as asserting that the singularities satisfying the direct summand condition define a good positive characteristic analogue of the rational singularities of characteristic 0. Using this theorem, we are able to prove a number of results which, roughly speaking, assert that vanishing theorems familiar from complex geometry have positive characteristic analogues provided we ask for vanishing “up to passage to finite covers.” Moreover, our results are sharp in the sense that we give examples illustrating the necessity of our hypotheses.
We prove two theorems in mixed characteristic. The first is an analogue of the positive characteristic theorem alluded to above, except that “vanishing” is replaced by “divisibility by p.” Our proof of this theorem also provides a new proof of the pure positive characteristic result mentioned above. The second is that the direct summand conjecture holds in cases where the ramification is supported on a simple normal crossings divisor. To the best of our knowledge, this is the first family of examples where the direct summand conjecture is proven without putting any restrictions on the dimension; our proof uses methods from p-adic Hodge theory.
