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Hello,

Let $R$ be a commutative (not necessarily Noetherian) ring. Let $I$ be a small filtrant (not necessarily totally ordered) category. Let $(M_i)_{i\in I}$ be a projective system of $R$-modules with surjective transitions maps.

Assume that the natural morphism $(\lim_{i\in I} M_i)^{(X)}\rightarrow \lim_{i\in I}(M_i^{(X)})$ is an isomorphism for any set $X$ (where the superscript $(X)$ denotes the direct sum indexed by $X$).

Question: is it true that the projective system $(M_i)_{i\in I}$ is constant (i.e. isomorphic to its projective limit as a pro-object of $R$-modules)? In other words, does there exist $\overline i\in I$ such that for any $j\geq \overline i$ we have $M_j\cong \lim_{i\in I} M_i$?

Thanks for your hints,

Luisa Fiorot

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  • $\begingroup$ By constant, do you mean that there should exist $i$ such that $\lim M_i = M_i$ ? $\endgroup$
    – François Brunault
    Commented Oct 3, 2011 at 14:31
  • $\begingroup$ I've inserted the interpretation you wrote as an answer. It would help if you registered an account, because that would prevent you from making new user-IDs. A new user-ID prevents you from editing your question. $\endgroup$
    – S. Carnahan ♦
    Commented Oct 4, 2011 at 6:03
  • $\begingroup$ Write $K_i$ for the kernel of the map $\lim M_i \to M_i$. The $K_i$ form a projective system with respect to inclusion. It seems you're asking the following : if $\lim^1 K_i^{(X)} = 0$ for any set $X$, then does $K_i$ satisfy the Mittag-Leffler condition ? $\endgroup$
    – François Brunault
    Commented Oct 4, 2011 at 11:44

1 Answer 1

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No. There are nontrivial filtrant projective systems $(M_i)$ of $R$-modules with surjective transition maps such that $\varprojlim M_i=0$. Both $(\varprojlim M_i)^{(X)}$ and $\varprojlim(M_i^{(X)})$ may be identified with submodules of $(\varprojlim M_i)^X=0$, so the canonical map $(\varprojlim M_i)^{(X)}\to\varprojlim(M_i^{(X)})$ is an isomorphism.

For how to construct such examples, see G. Bergman, Some Empty Inverse Limits or G. Higman, A.H. Stone, On inverse systems with trivial limits. J. Lond. Math. Soc. 29, 233-236 (1954)

There is also Bourbaki, Topologie generale, III §7 Ex. 2, but the argument given there seems to be incomplete.

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