Faltings' category of almost modules

Question

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!).

Answer 1

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 $\mathcal M$:

1. $\mathcal C$, 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.

2. $\mathcal C'$, 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.