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Syntomic complexes and \(p\)-adic nearby cycles. (English) Zbl 1395.14013

The goal of this paper is to give a comparison up to universal constants between the truncated sheaves of \(p\)-adic nearby cycles and the syntomic cohomology sheaves on semistable formal schemes, and as an application to prove a version of the semistable conjecture of Fontaine and Jannsen (proved in its original formulation by T. Tsuji [Invent. Math. 137, No. 2, 233–411 (1999; Zbl 0945.14008)]) valid for proper formal schemes with semistable reduction (which avoids the use of Poincaré duality).
Let us be precise. Let \(\mathcal O_K\) be a complete discrete valuation ring of mixed characteristic and with uniformizer \(\varpi\), fraction field \(K\), and perfect residue field \(k\). Let \(X\) be a fine and saturated log-scheme log-smooth over \(\mathcal O_K\), with semistable reduction over \(\mathcal O_K\). Let \(X_{\mathrm{tr}}\) denote the open subscheme of \(X_K\) where the log-structure is trivial. For \(r \geq 0\), let \(\mathscr S_n(r)_X\) denote the (log) syntomic sheaf modulo \(p^n\) on \(X_{k, \mathrm{et}}\). In J.-M. Fontaine and W. Messing [Contemp. Math. 67, 179–207 (1987; Zbl 0632.14016)] and K. Kato [Astérisque 223, 269–293 (1994; Zbl 0847.14009)], they have constructed a period morphism (\(i: X_k \hookrightarrow X\) and \(j: X_{\mathrm{tr}} \hookrightarrow X\)) \[ \alpha^{\mathrm{FM}}_{r,n}: \mathscr S_n(r)_X \to i^* Rj_* \mathbb Z / p^n (r)'_{X_{\mathrm{tr}}}, \quad r \geq 0, \] where \(\mathbb Z_p (r)': = \frac 1{p^{\lfloor r/(p-1)\rfloor}}\mathbb Z_p (r)\).
The first main result of this paper (Theorem 1.1) says that, assuming that \(K\) “contains enough roots of unity”, the kernel and cokernel of the map \[ \alpha^{\mathrm{FM}}_{r,n}: \mathscr H^i\big( \mathscr S_n(r)_X \big) \to i^* R^ij_* \mathbb Z / p^n (r)'_{X_{\mathrm{tr}}}, \quad 0 \leq i \leq r, \] is annihilated by \(p^{Nr+c_p}\) for some constant \(N\) and \(c_p\) independent of \(X, n, r\). The condition that “\(K\) contains enough roots of unity” can be achieved by adjoining finitely many roots of unity to \(K\); without this condition, the kernel and cokernel of \(\alpha^{\mathrm{FM}}_{r,n}\) is still killed by a power of \(p\) independent of \(X\) and \(n\). We will return later to comment on its proof. Theorem 1.1 was previously known if \(0 \leq i \leq p-1\) (in which case \(\alpha_{r,n}^{\mathrm{FM}}\) is an isomorphism) by many works of Kato, Kurihara, and Tsuji; (cf. T. Tsuji [J. Reine Angew. Math. 472, 69–138 (1996; Zbl 0838.14015)] for the case of étale local system). This generalization has a long list of potential applications to relating \(p\)-adic motivic cohomology and (log-)syntomic cohomology. One of the immediate corollary says that for a quasi-compact formal semistable scheme \(\mathscr X\) over \(\mathcal O_K\), the map \[ \alpha_{r,i}: H^{i-1}_{\mathrm{dR}}(\mathscr X_{K, \mathrm{tr}}) \to H^i_{\mathrm{et}} (\mathscr X_{K, \mathrm{tr}}, \mathbb Q_p(r)) \] is an isomorphism when \(1\leq i\leq r-1\) and is injective if \(i =r\) (and the cokernel of \(\alpha_{r,r}\) can be very large if \(\dim \mathscr X_{K} \geq 1\)).
The second main result of this paper is to apply the first result to prove the following semistable conjecture: for \(\mathscr X\) a proper semistable formal scheme over \(\mathcal O_K\), there exists a natural \(\mathbb B_{\mathrm{st}}\)-linear Galois equivariant period isomorphism \[ \alpha: H^i(\mathscr X_{\overline K, \mathrm{tr}}, \mathbb Q_p) \otimes_{\mathbb Q_p} \mathbb B_{\mathrm{st}} \simeq H^i_{\mathrm{HK}}(\mathscr X) \otimes_F \mathbb B_{\mathrm{st}} \] that preserves the Frobenius, monodromy operators, and the filtration (after base changed to \(\mathbb B_{\mathrm{dR}}\)). This was previous proved by T. Tsuji [Invent. Math. 137, No. 2, 233–411 (1999; Zbl 0945.14008)] when \(\mathscr X\) comes from an algebraic variety over \(K\). The novelty lies in replacing the use of Poincaré duality as in most classical proofs, by the theory of finite dimensional Banach spaces introduced by P. Colmez [J. Inst. Math. Jussieu 1, No. 3, 331–439 (2002; Zbl 1044.11102).] together with the finiteness of étale cohomology of proper rigid analytic spaces proved by P. Scholze [Forum Math. Pi 1, Article ID e1, 77 p. (2013; Zbl 1297.14023)].
Now, we come back to comment on the proof of Theorem 1.1, which is the core of this paper. This computation is local. Let \(R\) be the \(p\)-adic completion of an étale algebra over \[ R_\square = \mathcal O_K[ X_1^{\pm 1}, \dots, X_a^{\pm 1}, X_{a+1}, \dots, X_{d+1} ] \big/ (X_{d+1}X_{a+1} \cdots X_{a+b} - \varpi ), \] and equipped the log-structure induced by the “divisor at infinity” \(X_{d+1} X_{a+1} \cdots X_{a+b} =0\). By choosing a way to present \(\mathcal O_K\) as a quotient of \(W(k)[[X_0]]\), one may embed \(\mathrm{Spf} R\) into a formal log-scheme \(\mathrm{Spf} R_\varpi^+\) that is log-smooth over \(W(k)\), and let \(R_\varpi^{\mathrm{PD}}\) denote the \(p\)-adic PD-envelope of \(R\) in \(R^+_\varpi\). Then explicitly, the syntomic cohomology of \(R\) can be computed by the complex \[ \mathrm{Syn} (R,r): = \mathrm{Cone} \Big( F^r \Omega^\bullet_{R_\varpi^{\mathrm{PD}}} \rightarrow p^r - p^\bullet \varphi_{\mathrm{Kum}} \Omega^\bullet_{R_\varpi^{\mathrm{PD}}} \Big) [-1], \] where \(\varphi_{\mathrm{Kum}}\) is the Frobenius induced by \(X_i\mapsto X_i^p\) for \(0\leq i\leq d+1\).
Now the proof of Theorem 1.1 breaks down into two steps: first introducing a slightly different map (when \(K\) contains “enough roots of unity”) \[ \alpha^{\text{Laz}}_{r,n}: \tau_{\leq r} \mathrm{Syn}(R,r)_n \to \tau_{\leq r} \mathrm{R} \Gamma_{\mathrm{cont}} (G_R, \mathbb Z/p^n(r)) \to \tau_{\leq r} \mathrm{R} \Gamma_{\mathrm{cont}} \big((\mathrm{Sp} R[\tfrac 1p])_{\mathrm{tr},\mathrm{et}}, , \mathbb Z/p^n(r)\big) \] and prove that it is a quasi-isomorphism up to \(p^{Nr}\), for a universal constant \(N\). The superscript Laz was to suggest that this map, modulo some \((\varphi, \Gamma)\)-module theory reduction, is an integral Lazard isomorphism between Lie algebra cohomology and continuous group cohomology, and this is how the first step is proved. The second step is to show that \(\alpha_{r,n}^{\text{Laz}}\) is equal to \(\alpha_{r,n}^{\mathrm{FM}}\) up to \(p^{Nr+c_p}\). This step is formal.

MSC:

14F42 Motivic cohomology; motivic homotopy theory
14F20 Étale and other Grothendieck topologies and (co)homologies
11G25 Varieties over finite and local fields
14F30 \(p\)-adic cohomology, crystalline cohomology
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