Associated graded module

qwirk · **Joined:** Fri, 3 Sep 2010 09:29:45 UTC **Posts:** 140

(A lot of what follows is from Matsumura - Commutative Ring Theory pp.174)

Let

be a ring

and A-module and

an ideal of

Set $\displaymath \text{gr}(A) = \bigoplus_{n \ge 0} I^n/I^{n+1}$

There are maps

$\gamma_n:(I^n/I^{n+1})\otimes_{A/I} M/IM \to I^nM/I^{n+1}M$ for $n \ge 0$

which gives a map

$\gamma:\text{gr}(A) \otimes_{A/I} M/IM \to \text{gr}(M)$

I am interested in showing that, in a particular case, this is an isomorphism.

One of the conditions Matsumura gives for this is that
M_0

is flat over A_0

and $I \otimes_A M = IM$
or (equivalently)
M_0

is flat over A_0

and $\text{Tor}_1^A(A_0,M)=0$

(Here A_0=A/I

and

)

Now in the case I am interested in

is a complete regular local Noetherian ring, so (letting I be the maximal ideal) A_0

is a field and so M_0

is flat over A_0

.

Is there anything in this particular case (complete local regular ring) that gives either $I \otimes_A M = IM$ or $\text{Tor}_1^A(A_0,M)=0$ ? (I'm guessing the answer is no, and that is probably not true in general)

S.O.S. Mathematics CyberBoard

Associated graded module

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