 Kernel of localization away from one plus an ideal

ID: kernel-of-localization-away-from-one-plus-an-ideal

Let

R

I \subseteq R

an ideal,

S = 1 + I

, and

M

a finitely generated

R

-module. Then

ker (M ⟶ S^{- 1} M) = ⋂_{j \geq 1} I^{j} M .

(1 + a) m = 0

with

a \in I

, then

m = (- a)^{j} m \in I^{j} M

for every

j

. Conversely, apply the Artin-Rees lemma to

R m \subseteq M

. For sufficiently large

c

it gives

m \in I^{c + 1} M \cap R m = I (I^{c} M \cap R m) \subseteq I (R m),

m = am

for some

a \in I

and

(1 - a) m = 0

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