Suppose that we have a mixed motif $M$ with only two (subsequent) weights, and have a subobject for each weight factor. How we can control whether there exists a submotif of $M$ with these two factors? It seems that here the existence of a mixed Hodge substructure with the corresponding weight factors in the singular cohomology of $M$ is not sufficient, since the corresponding mixed motivic $Ext^1$ does not inject into its Hodge 'realization'. Is this correct? What happens here if the base field is a number field? Are there any examples here that are well understood (certainly, the matter is very conjectural)?

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isan injection. Might the more general case be similar?) – Emerton Nov 17 '10 at 22:55