  We investigate the maximal finite length submodule of the Breuil-Kisin
prismatic cohomology of a smooth proper formal scheme over a p-adic ring of
integers. This submodule governs pathology phenomena in integral p-adic
cohomology theories. Geometric applications include a control, in low degrees
and mild ramifications, of (1) the discrepancy between two naturally associated
Albanese varieties in characteristic p, and (2) kernel of the specialization
map in p-adic \'etale cohomology. As an arithmetic application, we study the
boundary case of the theory due to Fontaine-Laffaille, Fontaine-Messing, and
Kato. Also included is an interesting example, generalized from a construction
in Bhatt-Morrow-Scholze's work, which (1) illustrates some of our theoretical
results being sharp, and (2) negates a question of Breuil.
