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: provided either A has a dualizing complex or is a quotient of a regular ring.
 * $$\operatorname{depth} M_{\mathfrak{p}} + \operatorname{ht}(I + \mathfrak{p})/\mathfrak{p} \ge n$$ for any $$\mathfrak{p} \in \operatorname{Spec}(A) - V(J)$$,
 * there is an ideal $$\mathfrak b$$ in A such that $$\mathfrak{b} \supset J$$ and $$\mathfrak b$$ annihilates the local cohomologies $$\operatorname{H}^i_I(M), 0 \le i \le n - 1$$,

The theorem was first proved by Faltings in.