Rights statement: This is the author’s version of a work that was accepted for publication in Advances in Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Advances in Mathematics, 320, 2017 DOI: 10.1016/j.aim.2017.08.042
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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
}
TY - JOUR
T1 - The slice method for G-torsors
AU - Lötscher, Roland
AU - MacDonald, Mark Lewis
N1 - This is the author’s version of a work that was accepted for publication in Advances in Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Advances in Mathematics, 320, 2017 DOI: 10.1016/j.aim.2017.08.042
PY - 2017/11/7
Y1 - 2017/11/7
N2 - The notion of a (G,N)(G,N)-slice of a G-variety was introduced by P.I. Katsylo in the early 80's for an algebraically closed base field of characteristic 0. Slices (also known under the name of relative sections) have ever since provided a fundamental tool in invariant theory, allowing reduction of rational or regular invariants of an algebraic group G to invariants of a “simpler” group. We refine this notion for a G-scheme over an arbitrary field, and use it to get reduction of structure group results for G -torsors. Namely we show that any (G,N)(G,N)-slice of a versal G -scheme gives surjective maps H1(L,N)→H1(L,G)H1(L,N)→H1(L,G) in fppf-cohomology for infinite fields L containing F. We show that every stabilizer in general position H for a geometrically irreducible G-variety V gives rise to a (G,NG(H))(G,NG(H))-slice in our sense. The combination of these two results is applied in particular to obtain a striking new upper bound on the essential dimension of the simply connected split algebraic group of type E7E7.
AB - The notion of a (G,N)(G,N)-slice of a G-variety was introduced by P.I. Katsylo in the early 80's for an algebraically closed base field of characteristic 0. Slices (also known under the name of relative sections) have ever since provided a fundamental tool in invariant theory, allowing reduction of rational or regular invariants of an algebraic group G to invariants of a “simpler” group. We refine this notion for a G-scheme over an arbitrary field, and use it to get reduction of structure group results for G -torsors. Namely we show that any (G,N)(G,N)-slice of a versal G -scheme gives surjective maps H1(L,N)→H1(L,G)H1(L,N)→H1(L,G) in fppf-cohomology for infinite fields L containing F. We show that every stabilizer in general position H for a geometrically irreducible G-variety V gives rise to a (G,NG(H))(G,NG(H))-slice in our sense. The combination of these two results is applied in particular to obtain a striking new upper bound on the essential dimension of the simply connected split algebraic group of type E7E7.
KW - G-torsor
KW - Relative sections
KW - Essential dimension
KW - Stabilizer in general position
KW - E7
U2 - 10.1016/j.aim.2017.08.042
DO - 10.1016/j.aim.2017.08.042
M3 - Journal article
VL - 320
SP - 329
EP - 360
JO - Advances in Mathematics
JF - Advances in Mathematics
SN - 0001-8708
ER -