Faltings' annihilator theorem

Last updated August 13, 2023

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]

$\operatorname {depth} M_{\mathfrak {p}}+\operatorname {ht} (I+{\mathfrak {p}})/{\mathfrak {p}}\geq n$ 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 $\operatorname {H} _{I}^{i}(M),0\leq i\leq n-1$ ,

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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References

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

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

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