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:Takesi Kawasaki, On Faltings' Annihilator Theorem, Proceedings of the American Mathematical Society, Vol. 136, No. 4 (Apr., 2008), pp. 1205–1211. NB: since $\operatorname((I + \mathfrak)/\mathfrak) = \operatorname(\operatorname(\mathfrak/\mathfrak) \mid \mathfrak \in V(\mathfrak) \cap V(I) = V((I + \mathfrak)/\mathfrak) \}$, the statement here is the same as the one in the reference.
*$\operatorname M_ + \operatorname(I + \mathfrak)/\mathfrak \ge n$ for any $\mathfrak \in \operatorname(A) - V(J)$,
*there is an ideal $\mathfrak b$ in ''A'' such that $\mathfrak \supset J$ and $\mathfrak b$ annihilates the local cohomologies $\operatorname^i_I(M), 0 \le i \le n - 1$,
provided either ''A'' has a dualizing complex or is a quotient of a regular ring.

The theorem was first proved by Faltings in .

References

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Abstract algebra
Commutative algebra