Faltings' annihilator theorem

Template:Short description Template:More citations needed 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:

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

provided either A has a dualizing complex or is a quotient of a regular ring.

The theorem was first proved by Faltings in Template:Harv.

References

• Template:Cite journal

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