The Breuil–Kisin torsion monotonicity conjecture

Let $X/\mathcal{O}_K$ be the fixed scheme, let $\mathfrak{S}$ be the Breuil–Kisin ring, and let $E$ be its Eisenstein polynomial. For the Breuil–Kisin cohomology module

$$\mathfrak{M}^{i}=\mathrm{H}^i_{\mathfrak{S}}(X/\mathcal{O}_K),$$

write $\mathfrak{M}^{i,(n)}$ for the $n$-fold Frobenius pullback. Let $f(n)$ denote either

$$\ell_{\mathcal{O}_K}\left(\mathfrak{M}^{i}[u^\infty]^{(n+1)}[E]\right)$$

or

$$\ell_{\mathcal{O}_K}\left(\overline{\mathfrak{M}^{i}}^{(n+1)}[E]\right),$$

where $\mathfrak{M}^{i}[u^\infty]$ is its $u^\infty$-torsion submodule and $\overline{\mathfrak{M}^{i}}$ is the indicated torsion quotient. Breuil–Kisin torsion conjecture. The function $f$ satisfies

$$f(a)\leqslant e\cdot f(0),\qquad f(n)\leqslant f(n+1)\quad\text{for all }n.$$

This is a refinement of the main crystalline–de Rham torsion inequality, translating it into statements about structured torsion in Breuil–Kisin cohomology. The paper presents it as a conjectural framework; the general inequalities and monotonicity remain open.