Given a smooth manifold $M$ and a *relatively compact exhaustion* $M=\bigcup\_{n\in\mathbb N} M\_n$ with open and relatively compact $M\_n\subseteq M\_{n+1}$ (hence $M=\lim\limits\_{\to}M\_n$) do we always have $$H^k(M)=\lim\limits\_{\leftarrow} H^k(M\_n)?$$


This looks so natural that the answer should be known in which case I would like to get a reference.


EDIT (inspired by Weibel's book and an article of Milnor from 1962). For fixed $k$ the projective spectrum $\Omega^k(M\_n)$ of $k$-forms on $M\_n$ satisfies the (strict) Mittag-Leffler condition so that the derived functor ${\lim\limits\_{\leftarrow}}^1 \Omega^k(M\_n)$ vanishes. We thus get an exact sequence $0\to \Omega^\ast(M)\to \prod\_n \Omega^\ast(M\_n)\to \prod\_n \Omega^\ast(M\_n)\to 0$ of cochain complexes (the last map is the difference map $(\omega\_n)\_n\mapsto (\omega\_n-\rho\_{n+1}(\omega\_{n+1}))\_n$ with the restriction operator $\rho\_{n+1}$). The long cohomology sequence thus yields an exact sequence
$$\cdots \to H^k(M)\to \prod\_nH^k(M\_n) \to \prod\_nH^k(M\_n) \to H^{k+1}(M)\to \cdots.$$
From this one gets short exact sequences $$0\to {\lim\limits\_{\leftarrow}}^1 H^{k-1}(M\_n) \to H^k(M)\to \lim\limits\_{\leftarrow}H^k(M\_n)\to 0.$$ It seems to me that the spectrum $H^0(M\_n)$ of the spaces of locally constant functions again satisfies the strict Mittag-Leffler condition so that ${\lim\limits\_{\leftarrow}}^1H^0(M\_n)=0$. This means that $H^1(M)=\lim\limits\_{\leftarrow} H^1(M\_n)$ is always true.
Another consequence is $H^k(M)=\lim\limits\_{\leftarrow} H^k(M\_n)$ if all cohomologies $H^{k-1}(M\_n)$ are finite dimensional (because this implies a Mittag-Leffler condition).


However, in general the question remains open (and I would rather expect a counterexample).



Do de Rham cohomologies commute with direct limits?
$\def\RR{\mathbb{R}}\def\Hom{\mathrm{Hom}}$This seems too simple, so please tell me what I'm missing. We first prove the corresponding result in homology. Let $C\_i(X)$ be the group of singular $i$-chains in $X$. Then $C\_i(M) = \lim\_{j \to \infty} C\_i(M\_j)$ (because $i$-simplices are compact, so the image of any finite number of them must lie in some $M\_j$). Homology commutes with direct limits, so we deduce that $H\_i(M) = \lim\_{j \to \infty} H\_i(M\_j)$.


Now, the universal coefficient theorem tells us that $H^i(M, \RR) = \Hom(H\_i(M), \RR)$ and the same for $M\_j$. So the question is whether $\Hom( \ , \RR)$ turns direct limits into inverse limits, and the answer is yes. 


Both the universal coefficient theorem, and the identification of singular and de Rham cohomology, work for general manifolds.




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I also have a direct proof of Mittag-Leffler for $H^q(M\_0) \leftarrow H^q(M\_1) \leftarrow \cdots$. 


**Lemma** Let $M$ be a manifold (PL or better) and let $K$ be a compact subset of $M$. Then $H^q(M) \to H^q(K)$ has finite dimensional image.


**Proof** Take a PL triangulation of $M$. For each face $\sigma$ of the triangulation, let $U\_{\sigma}$ be the union of the relative interiors of those faces $\tau$ containing $\sigma$. So the $U\_{\sigma}$ form an open cover of $M$.


As $K$ is compact, it can be covered by finitely many $U\_{\sigma}$. Let $N$ be the union of the closures of those $U\_{\sigma}$. So $K \subset N$ and $H^q(M) \to H^q(K)$ factors through $H^q(N)$. But $N$ is a finite simplicial complex, so $H^q(N)$ is finite dimensional. $\square$


**Remark** Actually, the union of the $U\_{\sigma}$ also has finite cohomology, without taking the closure, but I didn't see a one line way of saying it.


Now, let $M\_0 \subset M\_1 \subset M\_2 \subset \cdots$ be an ascending chain of manifolds, with the closure $\overline{M\_i}$ (in $M\_{i+1}$) compact. Then $H^q(M\_{i+1}) \to H^q(M\_i)$ factors through $H^q(\overline{M}\_i)$, so it has finite dimensional image. For each $j \geq i+1$, the image of $H^q(M\_j)$ lies in that finite dimensional image. So the Mittag-Leffler condition holds and $\lim\_{\infty \leftarrow j}^1 H^q(M\_j)=0$.


