Faltings' category of almost modules Hi,
Let $V$ be an integral domain with an ideal $m\subset V$ and put $K = S^{-1}V$ where
$S = {1}\cup m$ (a multiplicatively closed subset). Is it true that the category of almost
$(V,m)$-modules is equivalent to the category of $K$-modules via $-\otimes_V K$? By the category of almost $(V,m)$-modules I mean the category whose objects are $V$-modules and morphisms are defined by
$$Hom(M,N) = \varinjlim Hom(M',N/N')$$
where the direct limit is taken over $M'\subset M$ and $N'\subset N$ with $M/M'$ and $N'$
both $m$-torsion (a module $M$ is $m$-torsion if for every $x\in M$, there is some $u\in m$
such that $ux=0$). This is the Serre quotient of the category of $V$-modules by the
subcategory of $m$-torsion modules.
Thanks!
Graham: yes, thank you. Suppose that $m$ is the maximal ideal of a valuation ring (and remove zero!).
 A: At least in Faltings's setting, $m$ is the maximal ideal of a non-Noetherian valuation domain $V$.  If we let $S$ be the non-zero elements of $m$, then the localization of $V$ at $S$ will
be the field of fractions $K$ of $V$.    The category of $K$-modules (i.e. $K$-vector spaces)
is obtained as a Serre quotient of $V$-modules, but one quotients out modules which are
killed by some non-zero element of $V$, while the category of amost modules is obtained 
by quotienting out by a much smaller category, namely the modules which are killed by every element of $m$.  
In other words, I think you have misinterpreted the meaning of $m$-torsion, at least in so far it is used in the context of Faltings's "almost commutative algebra".
Added in response to the comment below:  Let me not use the word $m$-torsion anymore, since it can be interpreted in different ways, and seems to be causing confusion.
Let $\mathcal M$ be the category of $V$-modules.  Here are two Serre subcategories of $\mathcal M$:


*

*$\mathcal C$, the category of almost zero modules, i.e. modules for which $m M = 0$.
This is a Serre subcategory because $m^2 = m$ in the context of Faltings's theory.

*$\mathcal C'$, the category of torsion modules, i.e. modules such that $x M = 0$
for some non-zero $x \in m$.  This is a Serre category just because $V$ is a domain.
Clearly $\mathcal C$ is contained in $\mathcal C'$, but they are far from equal!
The quotient $\mathcal M/\mathcal C$ is the category of almost modules.  The quotient
$\mathcal M/\mathcal C'$ is equivalent to the category of $K$-vector spaces
(the equivalence being given by applying the functor $K\otimes\text{--}$, where $K$
is the fraction field of $V$).
In the statement of the question, you (mistakenly) define almost modules to be $\mathcal M/\mathcal C'$, and (correctly) conclude that this is equivalent to $K$-vector spaces.  Once you replace this mistaken definition with the correct definition of almost modules (i.e. $\mathcal M/\mathcal C$), you will see that the resulting category is not equivalent to $K$-vector spaces.
