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:^[1]

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

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).

References

1. Takesi Kawasaki, On Faltings' Annihilator Theorem, Proceedings of the American Mathematical Society, Vol. 136, No. 4 (Apr., 2008), pp. 1205–1211. NB: since ${\displaystyle \operatorname {ht} ((I+{\mathfrak {p}})/{\mathfrak {p}})=\operatorname {inf} (\operatorname {ht} ({\mathfrak {r}}/{\mathfrak {p}})\mid {\mathfrak {r}}\in V({\mathfrak {p}})\cap V(I)=V((I+{\mathfrak {p}})/{\mathfrak {p}})\}}$ , the statement here is the same as the one in the reference.

• Faltings, Gerd (1981). "Der Endlichkeitssatz in der lokalen Kohomologie". Mathematische Annalen. 255: 45–56.