Compact objects in persistence modules and interval decomposition

Question

$\newcommand\Mod{\mathrm{Mod}}\DeclareMathOperator\Fun{Fun}$If $k$ is a field, a persistent $k$-module is a functor $\mathbb{R}\to \Mod_k$ where $\mathbb{R}$ is a poset under the natural ordering of $\mathbb{R}$. A persistent $k$-module $X$ is said to be of finite type when each $X_t$ is finite dimensional and there exist a finite sequence $t_1 < \cdots < t_n$ such that the maps $X_s \to X_t$ are isomorphisms whenever $t_i \leq s < t < t_{i+1}$ or $s\geq t_n$ or $t<t_1$. A persistent $k$-module $X$ of finite type admits an ‘interval decomposition’, ie. a decomposition as a direct sum $$ X\cong\bigoplus_{i} I(s_i, t_i) $$ of interval modules, where for $s<t$ the interval module $I(s,t)$ consists of the ground field $k$ in the interval $[s,t]$ and zero otherwise, the structure maps being either identities or zero.

I have two questions about these objects:

• It is clear that the persistent $k$-modules of finite type are compact objects in the category $\Fun(\mathbb{R}, \Mod_k)$ of persistent modules. Are they all the compact objects in this category?

• What is an example of a persistent $k$-module that does not admit an interval decomposition?

Any answer to these questions, even when the characteristic of $k$ is zero, would be of great interest to me.

Answer 1

Ok so I found answers to both questions, so thought I'd post them for future reference. Concerning the first question, there exists an isomorphism of categories $Fun(\mathbb{R}, Mod_k)\cong grMod_{k[x]}$, where $k[x]$ denotes the ring of polynomials with real exponents, generated as a $k$-module by the monomials $x^t$ with $t\in\mathbb{R}$. This ring admits an $\mathbb{R}$-grading $$ k[x]\cong \bigoplus_{t\in \mathbb{R}} kx^t $$ and $grMod_{k[x]}$ denotes the associated category of graded modules. Consequently, the compact objects in persistent $k$-modules are exactly the finitely presented graded $k[x]$-modules. For such a module $M=\bigoplus_{t\in \mathbb{R}} M_t$, each $M_t$ has to be a finite dimensional $k$-module. Moreover, if $Z_1, \cdots, Z_n$ are generators of the $k[x]$-module $M$, with degrees $z_1, \cdots, z_n \in \mathbb{R}$ a finite presentation of $M$ amounts to a finite set of relations $\{r_j\}$ each of the form $\sum_i \lambda_{ij} X^{t_{ij}} Z_i=0$ for which $t_{ij} + z_i=\tau_j$ is independent of $i$. Reordering if necessary, we can assume $\tau_1 < \cdots < \tau_m $. This means that the $k$-module maps $X^{t-s}:M_s \to M_t$ are isomorphisms whenever $\tau_i \leq s < t < \tau_{i+1}$ or $s\geq \tau_m$ or $t< \tau_1$. This seems to prove that every compact persistent $k$-module is of finite type.

Regarding the second question, there is a nice example due to Webb of a persistent $k$-module that does not admit an interval decomposition: for each $t\in \mathbb{R}$, denote $$ M_t=\prod_{n\in \mathbb N \:,\: n\geq - \text{min}(\lfloor t \rfloor, 0)} k $$ with the natural inclusion maps, and where $\lfloor - \rfloor$ is the floor function $\mathbb{R}\to \mathbb{Z}$. This means that $M_t$ is the space of sequences $(x_n)$ of elements of $k$ when $t\geq 0$, and the subspace of sequences such that $x_n=0$ for $n\leq -\lfloor t \rfloor$ when $t<0$. It is argued in S. Oudot's book https://www.ams.org/bookstore/pspdf/surv-209-prev.pdf that this persistent module doesn't have an interval decomposition.