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]
-
depth
M
p
+
ht
(
I
+
p
)
/
p
≥
n
{\displaystyle \operatorname {depth} M_{\mathfrak {p}}+\operatorname {ht} (I+{\mathfrak {p}})/{\mathfrak {p}}\geq n}
for any prime ideal p ∈ Spec ( A ) − V ( J ) {\displaystyle {\mathfrak {p}}\in \operatorname {Spec} (A)-V(J)}
,
- there is an ideal
b
{\displaystyle {\mathfrak {b}}}
in A such that b ⊃ J {\displaystyle {\mathfrak {b}}\supset J}
and b {\displaystyle {\mathfrak {b}}}
,
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
- Takesi Kawasaki, On Faltings' Annihilator Theorem, Proceedings of the American Mathematical Society, Vol. 136, No. 4 (Apr., 2008), pp. 1205–1211. NB: since
ht
(
(
I
+
p
)
/
p
)
=
inf
(
ht
(
r
/
p
)
∣
r
∈
V
(
p
)
∩
V
(
I
)
=
V
(
(
I
+
p
)
/
p
)
}
{\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.