Interval Multiplicities of Persistence Modules

For any persistence module $M$ over a finite poset $\mathbf{P}$, and any interval $I$ of $\mathbf{P}$, we give a formula for the multiplicity $d_M(V_I)$ of the interval module $V_I$ in the indecomposable decomposition of $M$ in terms of the ranks of matrices consisting of structure linear maps of $M$. This generalizes the corresponding formula for 1-dimensional persistence modules. As applications, the formula enables us to compute the maximal interval-decomposable direct summand of $M$, to decide whether $M$ is interval-decomposable, and to detect properties determined by prescribed interval summands without decomposing $M$. We also give criteria, in terms of top and socle supports along minimal projective resolutions and injective coresolutions of $M$, restricting the intervals that can occur as direct summands of $M$ and thereby reduce the number of intervals to be computed in practice. Moreover, the formula tells us which morphisms of $\mathbf{P}$ are essential to compute $d_M(V_I)$. This leads to the notion of an order-preserving map $\zeta \colon Z \to \mathbf{P}$ essentially covering $I$, for which the multiplicity is preserved under the induced restriction functor $R \colon \operatorname{mod} \mathbf{P} \to \operatorname{mod} Z$. When $Z$ is of Dynkin type $\mathbb{A}$, also known as a zigzag poset, this allows the multiplicity to be computed more efficiently from the filtration level of topological spaces, without computing all structure linear maps of $M$. Finally, we give a formula for $d_M(V_I)$ in terms of a projective (or injective) (co)presentation of $M$. In the 2D-grid case, this is more practical since such resolutions can be computed from the filtration level of topological spaces.