Let $X/K$ be a variety (scheme of finite type, geometricaly integral) over a finitely generated field $K$. If it is smooth and proper, we can formulate the Tate conjecture, and if $\text{char}(K) = 0$ also the Hodge conjecture.


Are there analogues in the general case (when cohomology is no longer pure)?


Of course there is a conjectural generalisation that says:



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> The $\ell$-adic (resp. Betti) realisation functors from the category of mixed motives over $K$ to the category of Galois representations (resp. mixed Hodge structures) is fully faithful.
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*(NB: "category of Galois representations" means that morphisms are equivariant for some open subgroup of $\mathrm{Gal}(\bar{K}/K)$.)*


The problem with this generalisation is that the statement refers to the category of mixed motives, and we don't know whether it exists (i.e., the conjecture is conjectural).



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> **Q:** Is there a generalisation that does not depend on the category of mixed motives (probably involving some sort of cycle maps)?
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On the one hand I expect this is possible, on the other hand maybe the same problem that pops up in defining mixed motives also appears when trying to formulate a generalisation of the Tate and Hodge conjectures. (I am still very much a novice when it comes to the mixed versions of motives and cohomology.)




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Note that there is a good generalisation of the Mumford–Tate conjecture. After all, we still have Mumford–Tate groups and images of Galois; we just don't expect them to be reductive (and in general they are not).


For some reason I could not find anything about this via a search on the general internet or specifically mathscinet.



Is there an analogue of the Tate (and Hodge) conjecture for varieties that are not proper smooth (i.e., the mixed case)?
(**Edited**) The short answer is yes. There are analogues of these conjectures, starting with work of Beilinson, Jannsen and perhaps others in the 1980's. Basically, they would say that cycle maps from motivic cohomology$^1$ $\otimes \mathbb{Q}$ (resp. $\mathbb{Q}\_\ell$) to $Hom\_{MHS}(\mathbb{Q}(-p), H^i(X))$ (resp. $Hom\_{G}(\mathbb{Q}\_{\ell}(-p), H^i(X))$ should be onto. However, without further qualifications, this is known to be false$^2$. 


Probably a good place to start to read about this is Jannsen's *Mixed motives and algebraic K-theory*. 


1. Motivic cohomology can be expressed as an $Ext$ group in Voevodsky's category of mixed motives (which exists!) or more concretely from my point of view in terms of Bloch's higher Chow groups. Since we are tensoring with $\mathbb{Q}$, this can also be defined using higher $K$-theory which is how it was done in the original papers.
2. It is expected to hold when $p=2i$ (usual Hodge/Tate) or $p=i$ ("Milnor" case), or over $\overline{\mathbb{Q}}$ for any $p,i$.


