Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T19:30:25.748Z Has data issue: false hasContentIssue false
  • English
  • Français

Vanishing of the higher direct images of the structure sheaf

Part of: (Co)homology theory Birational geometry

Published online by Cambridge University Press:  30 June 2015

Andre Chatzistamatiou  and
Kay Rülling
Andre Chatzistamatiou
Affiliation:
Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany email a.chatzistamatiou@uni-due.de
Kay Rülling
Affiliation:
Freie Universität Berlin, Arnimallee 7, 14195 Berlin, Germany email kay.ruelling@fu-berlin.de
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.

We prove that the higher direct images of the structure sheaf under a birational and projective morphism between excellent and regular schemes vanish.

Keywords

birational mapshigher direct images

MSC classification

Primary: 14E05: Rational and birational maps 14F17: Vanishing theorems
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 11 , November 2015 , pp. 2131 - 2144
Copyright
© The Authors 2015 

References

Alonso Tarrío, L., Jeremías López, A. and Lipman, J., Bivariance, Grothendieck duality and Hochschild homology, II: the fundamental class of a flat scheme-map, Adv. Math. 257 (2014), 365–461.CrossRefGoogle Scholar
Angéniol, B. and El Zein, F., Appendice: ‘La classe fondamentale relative d’un cycle’, Bull. Soc. Math. France Mém. 58 (1978), 67–93.CrossRefGoogle Scholar
Berthelot, P., Bloch, S. and Esnault, H., On Witt vector cohomology for singular varieties, Compositio Math. 143 (2007), 363–392.CrossRefGoogle Scholar
Berthelot, P., Esnault, H. and Rülling, K., Rational points over finite fields for regular models of algebraic varieties of Hodge type ⩾1, Ann. of Math. (2) 176 (2012), 413–508.CrossRefGoogle Scholar
Chatzistamatiou, A. and Rülling, K., Higher direct images of the structure sheaf in positive characteristic, Algebra Number Theory 5 (2011), 693–775.CrossRefGoogle Scholar
Chatzistamatiou, A. and Rülling, K., Hodge–Witt cohomology and Witt-rational singularities, Doc. Math. 17 (2012), 663–781.CrossRefGoogle Scholar
Conrad, B., Grothendieck duality and base change, Lecture Notes in Mathematics, vol. 1750 (Springer, Berlin, 2000).CrossRefGoogle Scholar
El Zein, F., Complexe dualisant et applications à la classe fondamentale d’un cycle, Bull. Soc. Math. France Mém. 58 (1978), 1–93.Google Scholar
Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Publ. Math. Inst. Hautes Études Sci. 24 (1965), 1–231.Google Scholar
Fakhruddin, N. and Rajan, C. S., Congruences for rational points on varieties over finite fields, Math. Ann. 333 (2005), 797–809.CrossRefGoogle Scholar
Fulton, W., Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 2, second edition (Springer, Berlin, 1998).CrossRefGoogle Scholar
Hartshorne, R., Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, vol. 20 (Springer, Berlin, 1966).CrossRefGoogle Scholar
Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; 205–326.CrossRefGoogle Scholar
Lipman, J., Rational singularities, with applications to algebraic surfaces and unique factorization, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 195–279.CrossRefGoogle Scholar
Lipman, J., Cohen–Macaulayness in graded algebras, Math. Res. Lett. (2) 1 (1994), 149–157.CrossRefGoogle Scholar
Sancho de Salas, J. B., Blowing-up morphisms with Cohen–Macaulay associated graded rings, Géométrie algébrique et applications, I (La Rábida, 1984), Travaux en Cours 22 (1987), 201–209.Google Scholar
Berthelot, D. P., Grothendieck, A. and Illusie, L., Séminaire de Géométrie Algébrique du Bois Marie – 1966–67 – Théorie des intersections et théoréme de Riemann–Roch – (SGA6), Lecture Notes in Mathematics, vol. 225 (Springer, Berlin, 1971).Google Scholar