Let $ A $ be a ring.
Let $ I $ be a preordered set, filtering.
Let $ \Sigma $ a multiplicative subset of $ A $.
Suppose for any given $ i \in I $ a multiplicative subset $ S_i $ of $ A $ contained in $ \Sigma $.
Make the following assumptions:
$ \Sigma = \displaystyle \bigcup_i S_i $
For $ (i, j) \in I^2 $ with $ i \leq j $, elements $ S_i $ become invertible in $ S_ {j}^{-1} A $.
$ \mathcal{D} = ((S_{i}^{-1} A)_{i \in I} \ , \ (S_{i}^{-1} A \to S_{j}^{-1} A)_{i \leq j}) $ is a commutative diagram in the category of filter $ A $ - algebras.
Question :
Show that $ \displaystyle \lim_{\longrightarrow} \mathcal{D} \simeq \Sigma^{-1} A $.
To do that, we need to show that :
$ 1) $ exist morphisms $ \pi_i: S_{i}^{-1} A \to \Sigma^{-1} A $ such that $ \pi_{i + p} f_{i, i + p } = \pi_{i} $.
$ 2) $ for any group $ B $ equipped with morphisms $ \pi_i: S_{i}^{-1} A \to B $ such that $ p_{i + p} f_{i, i + p} = p_{i} $ there exists a unique morphism $ g: \Sigma^{-1} A \to B $ such that $ p_i = g \circ \pi_i $.
But, i d'ont know how to do it.
Thank you in advance for your help.