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Graded absolute integral closures and Picard groups

Primarily AI-generated textHuman understanding: all partsmath.AC — Commutative Algebramath.AG — Algebraic Geometry

Contributed by Akhil Mathew ↗

SubmitterAkhil Mathew

Version 1 / Sep 26, 2026 / CC BY 4.0

Abstract

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

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References

  1. B. Bhatt, Cohen--Macaulayness of absolute integral closures , arXiv:2008.08070v2 (2021). https://arxiv.org/abs/2008.08070v2 .arXiv
  2. B. Bhatt, A. Mathew, and V. Vologodsky, Sheared Witt vectors , arXiv:2607.01178v1 (2026), 3.1. https://arxiv.org/abs/2607.01178v1 .arXiv
  3. Z. Rosengarten, Tate duality in positive dimension over function fields , arXiv:1805.00522v9 (2023), Remark 2.2.16. https://arxiv.org/abs/1805.00522v9 .arXiv
  4. The Stacks Project Authors, The Stacks Project . https://stacks.math.columbia.edu .link
  5. R. G. Swan, On seminormality , J. Algebra 67 (1980), no. 1, 210--229. https://doi.org/10.1016/0021-8693(80)90318-X doi:10.1016/0021-8693(80)90318-X .DOI
  6. C. Traverso, Seminormality and Picard group , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 24 (1970), no. 4, 585--595. https://www.numdam.org/item/ASNSP_1970_3_24_4_585_0/ .link
  7. C. A. Weibel, The $K$-book: An introduction to algebraic $K$-theory , Graduate Studies in Mathematics, vol. 145, American Mathematical Society, Providence, RI, 2013. https://sites.math.rutgers.edu/ weibel/Kbook/Kbook.I.pdf .link

Version history

  1. v1Initial depositCurrentSep 26, 2026