In mathematics, the Grothendieck–Ogg–Shafarevich formula describes the Euler characteristic of a complete curve with coefficients in an abelian variety or constructible sheaf, in terms of local data involving the Swan conductor. Andrew Ogg ( 1962) and Igor Shafarevich ( 1961) proved the formula for abelian varieties with tame ramification over curves, and Alexander Grothendieck ( 1977, Exp. X formula 7.2) extended the formula to constructible sheaves over a curve ( Raynaud 1965).
Suppose that F is a constructible sheaf over a genus g smooth projective curve C, of rank n outside a finite set X of points where it has stalk 0. Then
where Sw is the Swan conductor at a point.
- Grothendieck, Alexandre (1977), Séminaire de Géométrie Algébrique du Bois Marie – 1965–66 – Cohomologie l-adique et Fonctions L – (SGA 5), Lecture notes in mathematics (in French), vol. 589, Berlin; New York: Springer-Verlag, xii+484, doi: 10.1007/BFb0096802, ISBN 3540082484
- Ogg, Andrew P. (1962), "Cohomology of abelian varieties over function fields", Annals of Mathematics, Second Series, 76: 185–212, doi: 10.2307/1970272, ISSN 0003-486X, JSTOR 1970272, MR 0155824
- Raynaud, Michel (1965), "Caractéristique d'Euler–Poincaré d'un faisceau et cohomologie des variétés abéliennes", Séminaire Bourbaki, Vol. 9, Exp. No. 286, Paris: Société Mathématique de France, pp. 129–147, MR 1608794
- Shafarevich, Igor R. (1961), "Principal homogeneous spaces defined over a function field", Akademiya Nauk SSSR. Trudy Matematicheskogo Instituta imeni V. A. Steklova, 64: 316–346, ISSN 0371-9685, MR 0162806
In mathematics, the Grothendieck–Ogg–Shafarevich formula describes the Euler characteristic of a complete curve with coefficients in an abelian variety or constructible sheaf, in terms of local data involving the Swan conductor. Andrew Ogg ( 1962) and Igor Shafarevich ( 1961) proved the formula for abelian varieties with tame ramification over curves, and Alexander Grothendieck ( 1977, Exp. X formula 7.2) extended the formula to constructible sheaves over a curve ( Raynaud 1965).
Suppose that F is a constructible sheaf over a genus g smooth projective curve C, of rank n outside a finite set X of points where it has stalk 0. Then
where Sw is the Swan conductor at a point.
- Grothendieck, Alexandre (1977), Séminaire de Géométrie Algébrique du Bois Marie – 1965–66 – Cohomologie l-adique et Fonctions L – (SGA 5), Lecture notes in mathematics (in French), vol. 589, Berlin; New York: Springer-Verlag, xii+484, doi: 10.1007/BFb0096802, ISBN 3540082484
- Ogg, Andrew P. (1962), "Cohomology of abelian varieties over function fields", Annals of Mathematics, Second Series, 76: 185–212, doi: 10.2307/1970272, ISSN 0003-486X, JSTOR 1970272, MR 0155824
- Raynaud, Michel (1965), "Caractéristique d'Euler–Poincaré d'un faisceau et cohomologie des variétés abéliennes", Séminaire Bourbaki, Vol. 9, Exp. No. 286, Paris: Société Mathématique de France, pp. 129–147, MR 1608794
- Shafarevich, Igor R. (1961), "Principal homogeneous spaces defined over a function field", Akademiya Nauk SSSR. Trudy Matematicheskogo Instituta imeni V. A. Steklova, 64: 316–346, ISSN 0371-9685, MR 0162806