Faltings' annihilator theorem
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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]
- [math]\displaystyle{ \operatorname{depth} M_{\mathfrak{p}} + \operatorname{ht}(I + \mathfrak{p})/\mathfrak{p} \ge n }[/math] for any [math]\displaystyle{ \mathfrak{p} \in \operatorname{Spec}(A) - V(J) }[/math],
- there is an ideal [math]\displaystyle{ \mathfrak b }[/math] in A such that [math]\displaystyle{ \mathfrak{b} \supset J }[/math] and [math]\displaystyle{ \mathfrak b }[/math] annihilates the local cohomologies [math]\displaystyle{ \operatorname{H}^i_I(M), 0 \le i \le n - 1 }[/math],
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 [math]\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}) \} }[/math], 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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