Faltings' annihilator theorem

In abstract algebra (specifically commutative ring theory), Faltings' annihilator theorem states: given a finitely generated module M over a Noetherian commutative ring A and ideals I, J, the following are equivalent:

• for any ,
• there is an ideal in A such that and annihilates the local cohomologies ,

provided either A has a dualizing complex or is a quotient of a regular ring.

The theorem was first proved by Faltings in (Faltings 1981).