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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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