Ramanujam–Samuel theorem

In algebraic geometry, the Ramanujam–Samuel theorem gives conditions for a divisor of a local ring to be principal.

It was introduced independently by Samuel (1962) in answer to a question of Grothendieck and by C. P. Ramanujam in an appendix to a paper by Seshadri (1963), and was generalized by Grothendieck (1967, Theorem 21.14.1).

Statement

Grothendieck's version of the Ramanujam–Samuel theorem (Grothendieck 1967, theorem 21.14.1) is as follows. Suppose that A is a local Noetherian ring with maximal ideal m, whose completion is integral and integrally closed, and ρ is a local homomorphism from A to a local Noetherian ring B of larger dimension such that B is formally smooth over A and the residue field of B is finite over that of A. Then a cycle of codimension 1 in Spec(B) that is principal at the point mB is principal.

References

• Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS. 32: 5–361. doi:10.1007/bf02732123. MR 0238860.
• Samuel, Pierre (1962), "Sur une conjecture de Grothendieck", Les Comptes rendus de l'Académie des sciences, 255: 3101–3103, MR 0154887
• Seshadri, C. S. (1963), "Quotient space by an abelian variety", Mathematische Annalen, 152: 185–194, doi:10.1007/BF01470879, ISSN 0025-5831, MR 0164973