10.87 Inverse systems
In many papers (and in this section) the term inverse system is used to indicate an inverse system over the partially ordered set (\mathbf{N}, \geq ). We briefly discuss such systems in this section. This material will be discussed more broadly in Homology, Section 12.31. Suppose we are given a ring R and a sequence of R-modules
M_1 \xleftarrow {\varphi _2} M_2 \xleftarrow {\varphi _3} M_3 \leftarrow \ldots
with maps as indicated. By composing successive maps we obtain maps \varphi _{ii'} : M_ i \to M_{i'} whenever i \geq i' such that moreover \varphi _{ii''} = \varphi _{i'i''} \circ \varphi _{i i'} whenever i \geq i' \geq i''. Conversely, given the system of maps \varphi _{ii'} we can set \varphi _ i = \varphi _{i(i-1)} and recover the maps displayed above. In this case
\mathop{\mathrm{lim}}\nolimits M_ i = \{ (x_ i) \in \prod M_ i \mid \varphi _ i(x_ i) = x_{i - 1}, \ i = 2, 3, \ldots \}
compare with Categories, Section 4.15. As explained in Homology, Section 12.31 this is actually a limit in the category of R-modules, as defined in Categories, Section 4.14.
Lemma 10.87.1. Let R be a ring. Let 0 \to K_ i \to L_ i \to M_ i \to 0 be short exact sequences of R-modules, i \geq 1 which fit into maps of short exact sequences
\xymatrix{ 0 \ar[r] & K_ i \ar[r] & L_ i \ar[r] & M_ i \ar[r] & 0 \\ 0 \ar[r] & K_{i + 1} \ar[r] \ar[u] & L_{i + 1} \ar[r] \ar[u] & M_{i + 1} \ar[r] \ar[u] & 0}
If for every i there exists a c = c(i) \geq i such that \mathop{\mathrm{Im}}(K_ c \to K_ i) = \mathop{\mathrm{Im}}(K_ j \to K_ i) for all j \geq c, then the sequence
0 \to \mathop{\mathrm{lim}}\nolimits K_ i \to \mathop{\mathrm{lim}}\nolimits L_ i \to \mathop{\mathrm{lim}}\nolimits M_ i \to 0
is exact.
Proof.
This is a special case of the more general Lemma 10.86.4.
\square
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