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Let MHS be the category of rational mixed Hodge structures. In particular, it contains extensions of Tate objects $\mathbb{Q}(n)$ for each integer $n$. Here $\mathbb{Q}(n)$ is the only one dimensional $\mathbb{Q}$-pure Hodge structure of weight $-2n$.

Can someone indicate me how to compute the group of extensions

$Ext^1_{MHS}(\mathbb{Q}(n), \mathbb{Q}(m))$?

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This extension group is $\mathbf C^\ast$ if $n < m$ and $0$ otherwise. See e.g. Carlson, "Extensions of mixed Hodge structures".

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I'm trying to read the paper but I find it hard. Can you give some help/intuition of why this result is true? – MHS93 May 26 '14 at 10:54

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