Hi everybody,

I have a lack of references concerning projective limits and injective limits. Up to my faults in Bourbaki there are only proj and inj limits indexed by a partially ordered set (not a category), and this set is almost systematically directed (i.e. for all $i,j$ there exists $k$ such that $k\geq i$, and $k\geq j$). The problem of combining projective limits with inductive limits is not treated at all as far as I can see.

I need a reference for this. Can someone help me ? Thank you !

My specific problem is the following: Assume given a system of modules $(M_{i,j})_{i,j}$ and arrows among them, over the same ring $A$.

We consider both

1) the projective limit with respect to the index $j$ of the injective limits with respect to the indexes $i$. Call it

$\projlim_j \;\;(\injlim_i \;\; (M_{i,j}))$

2) the injective limit with respect to the index $i$ of the projective limits with respect to the indexes $j$. Call it

$\injlim_i \;\;(\projlim_j\;\; (M_{i,j}))$

-- (Up to errors) There exists a canonical map

$\displaystyle CAN : \injlim_i \;\;(\projlim_j\;\; (M_{i,j})) \;\to\; \projlim_j\;\; (\injlim_i\;\; (M_{i,j})) $

Under which assumptions this map is INJECTIVE ?

As an example if there is no arrows at all between the $M_{i,j}$, then one has a direct sum instead of the injective limit, and a product instead of a projective limit. The arrow CAN becomes

$\displaystyle CAN : \oplus_i \;\;\prod_j \;M_{i,j} \;\to\; \prod_j \;\;\oplus_i \;M_{i,j}$

where

$((a_{i,j})_i)_j \mapsto ((a_{i,j})_j)_i$

In this case the arrow CAN is always injective independently on the nature of the $M_{i,j}$ (and up to errors it is an isomorphism if and only if one of the sets of index "$i$" or "$j$" is finite). I suspect that in the general case the injectivity only depends on the nature of the arrows, but not of the nature of the objects. Does anyone have a useful comment or a reference?

Many thanks !