Exactness property of Breuil-Kisin functors and Bloch-Kato Selmer groups

Abstract

Let $K$ be a $p$-adic field and $T$ a lattice in a semistable representation of $\mathrm{Gal}(\overline{K}/K)$ with Hodge-Tate weights in $[0, r]$. Assuming $0\leq r<p-1$, we prove that for a semistable extension of $\mathbb{Z}_p$ by $T$, the corresponding sequence of strongly divisible modules is exact. Analogous statements are proved for Breuil-Kisin modules and for prismatic $F$-crystals for all $r\geq 0$. In the crystalline case, we deduce that the integral Bloch-Kato Selmer group $H^1_f(K, T)$ is computed by $\mathrm{Ext}^1$ in the category of crystalline strongly divisible modules. Using further exactness results, we define a tensor product of strongly divisible modules, which commutes with the functors to Galois representations. As an application, we show that for abelian varieties $A_1, A_2$ over $K$ with good reduction, the cup product map $\delta_1\cup\delta_2:A_1(K)\otimes A_2(K)\rightarrow H^2(K, T_p(A_1)\otimes T_p(A_2))$ induced by the Kummer sequences of $A_1, A_2$ factors through an $\mathrm{Ext}^2$ group of strongly divisible modules.