  We give three proofs that valuation rings are derived splinters: a geometric
proof using the absolute integral closure, a homological proof which reduces
the problem to checking that valuation rings are splinters (which is done in
the second author's PhD thesis and which we reprise here), and a proof by
approximation which reduces the problem to Bhatt's proof of the derived direct
summand conjecture. The approximation property also shows that smooth algebras
over valuation rings are splinters.
