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: depth ⁡ M p + ht ⁡ ( I + p ) / p ≥ n {\displaystyle \operatorname {depth} M_{\mathfrak {p}}+\operatorname {ht} (I+{\mathfrak {p}})/{\mathfrak {p}}\geq n} for any prime ideal p ∈ Spec ⁡ ( A ) − V ( J ) {\displaystyle {\mathfrak {p}}\in \operatorname {Spec} (A)-V(J)} , there is an ideal b {\displaystyle {\mathfrak {b}}} in A such that b ⊃ J {\displaystyle {\mathfrak {b}}\supset J} and b {\displaystyle {\mathfrak {b}}} annihilates the local cohomologies H I i ⁡ ( M ) , 0 ≤ i ≤ n − 1 {\displaystyle \operatorname {H} _{I}^{i}(M),0\leq i\leq n-1} , 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).

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