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

Let $K$ be a $p$-adic field and $T$ a lattice in a semistable representation of $\mathrm{G}\mathrm{a}\mathrm{l}(\bar{K}\mathrm{/}K)$ with Hodge-Tate weights in $[0,r]$. Assuming $0\le r<p-1$, 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 $r\ge 0$. In the crystalline case, we deduce that the integral Bloch-Kato Selmer group $H_f^1(K,T)$ is computed by ${\mathrm{E}\mathrm{x}\mathrm{t}}^1$ 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 ${\delta }_1\cup {\delta }_2:A_1(K)\otimes A_2(K)\to H^2(K,T_p(A_1)\otimes T_p(A_2))$ induced by the Kummer sequences of $A_1,A_2$ factors through an ${\mathrm{E}\mathrm{x}\mathrm{t}}^2$ group of strongly divisible modules.