Graded absolute integral closures and Picard groups
We deduce from Bhatt's graded Cohen--Macaulay and vanishing results that an integer-graded absolute integral closure of a weighted polynomial ring over $\Z_p$, modulo $p$, is free with basis degrees smaller than the sum of the weights. A direct $p$-adic expansion turns this degree bound into a uniform solution of the cusp relation modulo nilpotents. Consequently every ring $R$ admits a faithfully flat algebra with seminormal reduction, and every invertible $R[T]$-module becomes free after a faithfully flat extension of $R$, answering a question of Drinfeld.
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