For any persistence module $M$ over a finite poset $\mathbf{ P}$, and any interval $I$ of $\mathbf{ P}$, we give a formula of 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 the module $M$, which gives a generalization of the corresponding formula for 1-dimensional persistence modules. As an easy application, the formula enables us to compute the maximal interval-decomposable direct summand of $M$, which gives us a way to decide whether $M$ is interval-decomposable or not. In addition, when a set $V_{ I_1}, \dots, V_{ I_n}$ of interval direct summands of $M$ gives a specific property of $M$, we can study this property by directly computing the multiplicities $d_M(V_{ I_i})$ by our formula without decomposing $M$. Moreover, the formula tells us which morphisms of $\mathbf{ P}$ are essential to compute the multiplicity $d_M(V_I)$. This suggests us some order-preserving map $\zeta \colon Z \to \mathbf{ P}$ such that the induced restriction functor $R \colon \operatorname{ mod} \mathbf{ P} \to \operatorname{ mod} Z$ has the property that the multiplicity $d:= d_{ R(M)}(R(V_I))$ is equal to $d_M(V_I)$. In this case, we say that $\zeta$ essentially covers $I$. If $Z$ can be taken as a poset of Dynkin type $\mathbb{ A}$, also known as a zigzag poset, then the calculation of the multiplicity $d$ can be done more efficiently, starting from the filtration level of topological spaces. Thus this even makes it unnecessary to compute the structure linear maps of $M$. Finally, we also give a formula of $d_M(V_I)$ in terms of a projective (or injective) (co)presentation of $M$ rather than its structure linear maps. In the 2D-grid case, this formula is more practical because in that case resolutions of $M$ can be computed from the filtration level of topological spaces.
Categories: math.RT math.AT math.RA