Link Floer homology splits into snake complexes and local systems

We classify isomorphism and chain homotopy equivalence classes of finitely generated $\mathbb{Z} \oplus \mathbb{Z}$ graded free chain complexes over $\frac{\mathbb{F}[U,V]}{(UV)}$. As a corollary, we establish that every link Floer complex $CFL(Y, L)$ over the ring $\frac{\mathbb{F}[U,V]}{(UV)}$ splits uniquely as a direct sum of snake complexes and local systems. This generalizes and extends the results of Petkova and Dai, Hom, Stoffregen, and Truong. We give the first example of an essentially infinite knot Floer-like complex, i.e., a complex satisfying all formal properties of link Floer complexes of knots in $S^3$ and whose chain homotopy equivalence class does not admit a representative of the form $C \otimes_{\mathbb{F}} \mathbb{F}[U,V]$. Finally, we also describe the first example of a knot Floer-like complex that does not admit a simultaneously vertically and horizontally simplified basis.