Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-08T14:56:51.817Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact functors on perverse coherent sheaves

Part of: Local theory (Co)homology theory Abelian categories

Published online by Cambridge University Press:  04 May 2015

Clemens Koppensteiner
Clemens Koppensteiner*
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Rd, Evanston, IL 60208, USA email clemens@math.northwestern.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Inspired by symplectic geometry and a microlocal characterizations of perverse (constructible) sheaves we consider an alternative definition of perverse coherent sheaves. We show that a coherent sheaf is perverse if and only if $R{\rm\Gamma}_{Z}{\mathcal{F}}$ is concentrated in degree $0$ for special subvarieties $Z$ of $X$. These subvarieties $Z$ are analogs of Lagrangians in the symplectic case.

Keywords

perverse coherent sheaveslocal cohomology

MSC classification

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions
Secondary: 18E30: Derived categories, triangulated categories 14B15: Local cohomology

Information

Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 9 , September 2015 , pp. 1688 - 1696
Copyright
© The Author 2015 

References

Arinkin, D. and Bezrukavnikov, R., Perverse coherent sheaves, Mosc. Math. J. 10 (2010), 3–29.CrossRefGoogle Scholar
Beauville, A., Symplectic singularities, Invent. Math. 139 (2000), 541–549.CrossRefGoogle Scholar
Bezrukavnikov, R., Perverse coherent sheaves (after Deligne), Preprint (2000), arXiv:math/0005152.Google Scholar
Brodmann, M. P. and Sharp, R. Y., Local cohomology: an algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics, vol. 60 (Cambridge University Press, Cambridge, 1998), doi:10.1017/CBO9780511629204.Google Scholar
Hartshorne, R., Residues and duality, Lecture Notes in Mathematics, vol. 20 (Springer, Berlin, 1966).Google Scholar
Kashiwara, M., t-structures on the derived categories of holonomic D-modules and coherent 𝓞-modules, Mosc. Math. J. 4 (2004), 847–868.CrossRefGoogle Scholar
Kashiwara, M. and Schapira, P., Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften, vol. 292 (Springer, Berlin, 1994).Google Scholar
Mirkovi, I. and Vilonen, K., Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), 95–143; doi:10.4007/annals.2007.166.95.CrossRefGoogle Scholar
Grothendieck, A. and Raynaud, M., Cohomologie locale des faisceaux co-hérents et théorémes de Lefschetz locaux et globaux (SGA2), Documents Mathématiques, vol. 4 (Société Mathématique de France, Paris, 2005); arXiv:math/0511279; MR 2171939.Google Scholar
Williamson, G., Vanishing of !-restriction of constructible sheaves, MathOverflow (2013), http://mathoverflow.net/a/129244.Google Scholar