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If we consider an oriented locally finite graphs $X$, and we suppose that $X$ is the projective limit of a familly of oriented locally finite graphs : $X=\lim\limits_{\overleftarrow{k\in\mathbb{N}}}X_{k}$, Is it true that $$\displaystyle H_{c}^{1}(X,\mathbb{C})=\lim\limits_{\overrightarrow{k\in \mathbb{N}}}H_{c}^{1}(X_{k},\mathbb{C})$$ where for an oriented graph $Y$, $H_{c}^{1}(Y,\mathbb{C})$ is the cohomology with compact support of $Y$. We know that if $Y$ is locally finite, the cohomolgy with compact support of $Y$ is the same of the singular cohomology of the geometric realization $|Y|$ of $Y$, e.q : $$\displaystyle H_{c}^{1}(Y,\mathbb{C})=H^{1}(|Y|,\mathbb{C})$$

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I am not really familiar with homology theories, but isn't $H^*$ a representable functor? – Marc Palm Feb 19 '12 at 7:00
Ordinary cohomology is representable, but that will not help. In general, for spaces or spectra, the above equality does not hold. Cohomology does not play well with limits. There is such a thing as continuous cohomology, which is the right notion for such limit systems. – Sean Tilson Feb 19 '12 at 7:37

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