Vol. 3, No. 5, 2009

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A general homological Kleiman–Bertini theorem

Susan J. Sierra

Vol. 3 (2009), No. 5, 597–609
Abstract

Let $G$ be a smooth algebraic group acting on a variety $X$. Let $\mathsc{ℱ}$ and $\mathsc{ℰ}$ be coherent sheaves on $X$. We show that if all the higher $\mathsc{T}\phantom{\rule{0.3em}{0ex}}or$ sheaves of $\mathsc{ℱ}$ against $G$-orbits vanish, then for generic $g\in G$, the sheaf $\mathsc{T}\phantom{\rule{0.3em}{0ex}}{or}_{j}^{X}\left(g\mathsc{ℱ},\mathsc{ℰ}\right)$ vanishes for all $j\ge 1$. This generalizes a result of Miller and Speyer for transitive group actions and a result of Speiser, itself generalizing the classical Kleiman–Bertini theorem, on generic transversality, under a general group action, of smooth subvarieties over an algebraically closed field of characteristic 0.

Keywords
generic transversality, homological transversality, Kleiman's theorem, group action
Primary: 14L30
Secondary: 16S38