The authors extend the construction of \(A_{\mathrm{inf}}\)-cohomology of [B. Bhatt et al., Publ. Math., Inst. Hautes Étud. Sci. 128, 219–397 (2018; Zbl 1446.14011)] to the case of semistable formal schemes. Let \(K\) be a discretely valued field of characteristic \(0\) with perfect residue field \(k\) of characteristic \(p>0\), denote the completed algebraic closure \(\widehat{\overline{K}}\) by \(C\). For a proper semistable formal scheme \(\mathfrak{X}\) over \(\mathcal{O}_C\) the authors construct a perfect complex \(\mathrm{R}\Gamma_{A_{\mathrm{inf}}}(\mathfrak{X})\) of \(A_{\mathrm{inf}}\)-modules such that its specializations are related to classical cohomology theories by \[ \mathrm{R}\Gamma_{A_{\mathrm{inf}}}(\mathfrak{X})\otimes^{\mathbb{L}}_{A_{\mathrm{inf}}}W(C^{\flat})\simeq \mathrm{R}\Gamma_{\mathrm{et}}(\mathfrak{X}^{\mathrm{ad}}_C,\mathbb{Z}_p)\otimes^{\mathbb{L}}_{\mathbb{Z}_p}W(C^{\flat}) \tag{1} \] \[ \mathrm{R}\Gamma_{A_{\mathrm{inf}}}(\mathfrak{X})\otimes^{\mathbb{L}}_{A_{\mathrm{inf}},\theta}\mathcal{O}_C\simeq \mathrm{R}\Gamma_{\log\mathrm{dR}}(\mathfrak{X}/\mathcal{O}_C) \tag{2} \] \[ \mathrm{R}\Gamma_{A_{\mathrm{inf}}}(\mathfrak{X})\otimes^{\mathbb{L}}_{A_{\mathrm{inf}}}A_{\mathrm{cris}}\simeq \mathrm{R}\Gamma_{\log\mathrm{cris}}(\mathfrak{X}_{\mathcal{O}_C/p}/A_{\mathrm{cris}}) \tag{3} \]
When \(\mathfrak{X}\) appears as scalar extension of a semistable formal scheme \(\mathcal{X}\) over \(\mathcal{O}_K\) the object \(\mathrm{R}\Gamma_{A_{\mathrm{inf}}}(\mathfrak{X})\) gets equipped with the action of the Galois group \(\mathrm{Gal}(\overline{K}/K)\) and all of the comparison isomorphisms are compatible with Galois action. The authors deduce from this the semistable comparison isomorphism \[H^i_{\mathrm{et}}(\mathcal{X}_{\overline{K}},\mathbb{Z}_p)\otimes_{\mathbb{Z}_p}B_{\mathrm{st}}\simeq H^i_{\log\mathrm{cris}}(\mathcal{X}_k/W(k))\otimes_{W(k)}B_{\mathrm{st}}\] It was previously known in the case when \(\mathcal{X}\) is algebraizable by the works of Tsuji, Faltings, Bhatt and Beilinson. This comparison isomorphism appears as a formal consequence of (1) and (3) together with the observation which goes back to Faltings (end of Section 4 in [G. Faltings, in: Cohomologies \(p\)-adiques et applications arithmétiques (II). Paris: Société Mathématique de France. 185–270 (2002; Zbl 1027.14011)]) that there is a Galois-equivariant isomorphism (Proposition 9.2) \[ \mathrm{R}\Gamma_{\log\mathrm{cris}}(\mathcal{X}_{\mathcal{O}_{\overline{K}}/p}/A_{\mathrm{cris}})\otimes_{A_{\mathrm{cris}}}B_{\mathrm{st}}^+\simeq H^i_{\log\mathrm{cris}}(\mathcal{X}_k/W(k))\otimes_{W(k)}B_{\mathrm{st}}^+. \]
For any smooth proper variety \(X\) over \(K\) theory of Breuil-Kisin-Fargues modules provides a functorial \(\mathcal{O}_K\)-lattice inside the \(K\)-vector space \(H^i_{\mathrm{dR}}(X/K)\). The authors use the constructed \(A_{\mathrm{inf}}\)-cohomology theory to prove (Theorem 8.7) that this lattice coincides (under a certain torsion-freeness assumption on the de Rham cohomology) with the lattice \(H^i_{\log\mathrm{dR}}(\mathcal{X}/\mathcal{O}_K)\) provided by any semi-stable integral model \(\mathcal{X}\) of \(X\). In particular, any two models with torsion-free de Rham cohomology give rise to the same lattice in \(H^i_{\mathrm{dR}}(X/K)\).
The construction of \(\mathrm{R}\Gamma_{A_{\mathrm{inf}}}(\mathfrak{X})\) is essentially the same as the one given in the smooth case by [B. Bhatt et al., Publ. Math., Inst. Hautes Étud. Sci. 128, 219–397 (2018; Zbl 1446.14011)]. \(\mathrm{R}\Gamma_{A_{\mathrm{inf}}}(\mathfrak{X})\) is the derived global sections of the complex of sheaves on \(\mathfrak{X}_{\mathrm{et}}\) defined as \[A\Omega_{\mathfrak{X}}=L\mathrm{et}a_{\mu}R\nu_*\mathbb{A}_{\mathrm{inf}}(\widehat{\mathcal{O}}^+_{\mathfrak{X}_C^{\mathrm{ad}}})\] Here \(\nu:(\mathfrak{X}_C^{\mathrm{ad}})_{\mathrm{proet}}\to \mathfrak{X}_{\mathrm{et}}\) is the canonical morphism from the pro-étale site of the generic fiber and \(L\mathrm{et}a_{\mu}\) denotes the decalage functor with respect to the element \(\mu=[\varepsilon]-1\in A_{\mathrm{inf}}\). Authors are taking the pushforward to the étale site \(\mathfrak{X}_{\mathrm{et}}\) rather than the Zariski site \(\mathfrak{X}_{\mathrm{Zar}}\) as [B. Bhatt et al., Publ. Math., Inst. Hautes Étud. Sci. 128, 219–397 (2018; Zbl 1446.14011)] do because a semistable scheme in general admits standard coordinate charts only étale locally.
As compared to [B. Bhatt et al., Publ. Math., Inst. Hautes Étud. Sci. 128, 219–397 (2018; Zbl 1446.14011)], the authors have to overcome several additional difficulties. First, when constructing the de Rham comparison isomorphism (2) the key is the identification \[H^1(A\Omega_{\mathfrak{X}}\otimes^{\mathbb{L}}_{A_{\mathrm{inf}},\theta\circ\varphi^{-1}}\mathcal{O}_C)\simeq\Omega^{1,\log}_{\mathfrak{X}/\mathcal{O}_C}\{-1\}\] where \(\Omega^{1,\log}_{\mathfrak{X}/\mathcal{O}_C}\) is the sheaf of differential forms with logarithmic poles along the special fiber. As in [B. Bhatt et al., Publ. Math., Inst. Hautes Étud. Sci. 128, 219–397 (2018; Zbl 1446.14011)] a map from the sheaf of regular differential forms \(\Omega^1_{\mathfrak{X}/\mathcal{O}_C}\to H^1(A\Omega_{\mathfrak{X}}\otimes^{\mathbb{L}}_{A_{\mathrm{inf}},\theta\circ\varphi^{-1}}\mathcal{O}_C)\) can be constructed using the cotangent complex on the pro-étale site, and the authors then need to check (Theorem 4.11) that this map extends to logarithmic forms.
The other issue that comes up in the course of the proof of (3) is that the standard pro-étale perfectoid cover of the generic fiber of the semi-stable coordinate chart is not flat on the integral level. As an example, the map \(\mathcal{O}_C\langle x,y\rangle/(xy-p)\to \mathcal{O}_C\langle x^{1/p^{\mathrm{inf}ty}},y^{1/p^{\mathrm{inf}ty}}\rangle/(x^{1/p^n}y^{1/p^n}-p^{1/p^n}\mid n\in\mathbb{N})\) is not flat. In particular, the technique of quasi-syntomic descent is not available to the authors and they instead perform a careful explicit analysis of both sides of the comparison isomorphism over local coordinate charts.