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In "Descent, motives and K-theory", Gillet and Soule define a weight filtration on integral cohomology $H^{*}_{c}(X, \mathbb{Z})$ of a complex variety with compact support.

They write that one can obtain a filtration on non-compactly supported cohomology $H^{*}(X, \mathbb{Z})$ using the work of Guillen and Vicente Navarro Aznar; however, it's not clear to me which paper they have in mind. In more recent writing of Cirici and Guillen (https://arxiv.org/pdf/1403.6805.pdf), only a reference to compactly-supported cohomology is made.

Question: Do the methods of Guillen and Navarro Aznar yield a weight filtration on $H^{*}(X, \mathbb{Z})$? Where can the construction of integral weight filtration in the non-compactly supported case be found?

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    $\begingroup$ The final paragraph of the Cirici-Guillen paper starts with: "The results of this paper are based on a non-published manuscript by the second named author together with V. Navarro-Aznar." Presumably, this manuscript is what Gillet-Soule had in mind. $\endgroup$
    – Dan Petersen
    Apr 16, 2023 at 19:34
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    $\begingroup$ Although she also only talks about cohomology with compact support, you may be interested in the following paper of my student Josefien Kuijper: arxiv.org/abs/2204.08968 $\endgroup$
    – Dan Petersen
    Apr 16, 2023 at 19:37
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    $\begingroup$ One more remark: one of the most useful properties of the weight filtration in mixed Hodge theory is that maps of varieties induce strict morphisms with respect to the weight filtrations. I believe that this fails for the integral weight filtrations constructed by Gillet-Soule. As for putting a weight filtration on ordinary cohomology, I don't know what to expect. $\endgroup$
    – Dan Petersen
    Apr 16, 2023 at 19:40

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The results of the paper you mention (https://arxiv.org/pdf/1403.6805.pdf) are for analytic spaces, but the same proofs (actually simplified) give a weight filtration on the cohomology with coefficients in any commutative ring, of any complex algebraic varietiy. You can also have a look at https://arxiv.org/pdf/1811.08642.pdf where we put a weight filtration on singular cochains with arbitrary coefficients, compatible with the E-infinity structure. In particular, in cohomology we get what you are asking for.

As Dan points out in the comments, morphisms need not be strictly compatible with the weight filtration over the integers.

To answer your first question, I believe that the paper that Gillet and Soulé have in mind is "Un critère d'extension des foncteurs définis sur les schémas lisses". This is mainly what we use as well.

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