proof of Artin-Rees theorem
Define the graded ring ,
where is an indeterminate by
Now, gives rise to a graded module, , over ,
namely
Observe that is a noetherian ring. For, if
generate in , then the
elements of are sums of degree monomials
in the
’s, i.e., if are independent indeterminates
the map
via is surjective, and as
is noetherian, so is .
Let generate over . Then, generate over . Therefore, is a noetherian module. Set
a submodule of .
Moreover, is a homogeneous submodule of and it is f.g.
as is noetherian.
Consequently, possesses a finite number of homogeneous
generators
: , where
. Let .
Given any and any , look at
. We have
where . Thus,
and
It follows that , so
Now, it is clear that the righthand side is contained in , as .
Title | proof of Artin-Rees theorem |
---|---|
Canonical name | ProofOfArtinReesTheorem |
Date of creation | 2013-03-22 14:28:43 |
Last modified on | 2013-03-22 14:28:43 |
Owner | mat_cross (707) |
Last modified by | mat_cross (707) |
Numerical id | 4 |
Author | mat_cross (707) |
Entry type | Proof |
Classification | msc 13C99 |