Vol. 6, No. 7, 2012

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ISSN: 1944-7833 (e-only)
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Annihilating the cohomology of group schemes

Bhargav Bhatt

Vol. 6 (2012), No. 7, 1561–1577
Abstract

Our goal in this note is to show that cohomology classes with coefficients in finite flat group schemes can be killed by finite covers of the base scheme, and similarly for abelian schemes with “finite covers” replaced by “proper covers.” We apply this result to commutative algebra to give by a conceptual proof of Hochster–Huneke’s theorem on the existence of big Cohen–Macaulay algebras in positive characteristic; all previous proofs of this result were equational or cocycle-theoretic in nature.

Keywords
group schemes, abelian varieties, étale cohomology, fppf cohomology, big Cohen–Macaulay algebras
Mathematical Subject Classification 2010
Primary: 14L15
Secondary: 13D45, 14K05, 14F20
Milestones
Received: 26 September 2011
Revised: 6 December 2011
Accepted: 3 January 2012
Published: 4 December 2012
Authors
Bhargav Bhatt
Department of Mathematics
University of Michigan
2074 East Hall
530 Church Street
Ann Arbor, MI 48109
United States
School of Mathematics
Institute for Advanced Study
Princeton, NJ 08540
United States
http://www-personal.umich.edu/~bhattb/