Faltings' annihilator theorem explained

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:^[1]

\operatorname{depth}M_ak{p

} + \operatorname(I + \mathfrak)/\mathfrak \ge n for any

ak{p}\in\operatorname{Spec}(A)-V(J)

there is an ideal

akb

in A such that

ak{b}\supsetJ

and

akb

annihilates the local cohomologies

	i
\operatorname{H}
	I(M),

0\lei\len-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

Gerd . Faltings . Der Endlichkeitssatz in der lokalen Kohomologie . Mathematische Annalen . 1981 . 255 . 45–56.

Notes and References

Takesi Kawasaki, On Faltings' Annihilator Theorem, Proceedings of the American Mathematical Society, Vol. 136, No. 4 (Apr., 2008), pp. 1205–1211. NB: since
\operatorname{ht}((I+ak{p})/ak{p})=\operatorname{inf}(\operatorname{ht}(ak{r}/ak{p})\midak{r}\inV(ak{p})\capV(I)=V((I+ak{p})/ak{p})\}

, the statement here is the same as the one in the reference.