Signed Barcodes for Multi-parameter Persistence via Rank Decompositions and Rank-Exact Resolutions

[10, 22] Let \(P\) be a totally ordered set. Then, any \(M\in {{\,\textrm{rep}\,}}P\) decomposes asfor a unique multiset of intervals \(\text {Bar}\,M\) called the persistence barcode of \(M\).

Consider the module \(M\) on the left-hand side of Fig. 2, which is a representation of the grid poset \(\llbracket 1,3\rrbracket \times \llbracket 1,3\rrbracket \subset {\mathbb {R}}^2\), and assume there is a decomposition \({{\,\textrm{Rk}\,}}M= \sum _{{I}\in \mathcal {I}} {{\,\textrm{Rk}\,}}\textbf{k}_{I}\). Since \(M((1,2)\rightarrow (1,3)) = \textrm{id}= M((1,3)\rightarrow (2,3))\), the square \(\llbracket 1,2\rrbracket \times \llbracket 2,3\rrbracket \) must be contained in one of the intervals, say \({I}\). Similarly, the square \(\llbracket 2,3\rrbracket \times \llbracket 1,2\rrbracket \) must be contained in some interval \({I}'\) involved in the decomposition. If \({I}\) and \({I}'\) are the same interval, then the morphism \((2,1)\rightarrow (2,3)\) has non-zero rank in the decomposition, whereas it has zero rank in \(M\), a contradiction. But if \({I}\) and \({I}'\) are not the same interval, then no other interval may appear in the decomposition because the dimension vector of \(M\) is then saturated by \({I}\) and \({I}'\), so the morphism \((1,2)\rightarrow (3,2)\) has zero rank in the decomposition, whereas it has non-zero rank in \(M\), again a contradiction.

Consider again the module \(M\) on the left-hand side of Fig. 2, and take its direct sum with the module \(N\) highlighted in red on the right-hand side. Then, as shown in the figure, \({{\,\textrm{Rk}\,}}(M\oplus N)\) does admit a decomposition as a sum of rank invariants of interval modules, however these intervals differ significantly from the indecomposable direct summands \(M\) and \(N\): in particular, the first interval in the decomposition creates the illusion that it should be spanned by some feature in \(M\oplus N\), whereas in fact no feature does so, as explained in Example 1.2.

Let \(M\in {{\,\textrm{Rep}\,}}P\).

Given an interval \({I}\subseteq P\), the generalized rank of \(M\) over \({I}\), denoted by \({{\,\textrm{Rk}\,}}_{I}M\), is defined by:where \(\varprojlim M|_{I}\rightarrow \varinjlim M|_{I}\) is the natural morphism from the limit to the co-limit of the diagram \(M|_{I}\) and is well-defined because \({I}\) is both convex and connected. Given a collection \(\mathcal {I} \) of intervals, the generalized rank invariant of \(M\) over \(\mathcal {I} \) is the map \({{\,\textrm{Rk}\,}}_\mathcal {I} M:\mathcal {I} \rightarrow {\mathbb {N}}\cup \{ \infty \}\) defined by \({{\,\textrm{Rk}\,}}_\mathcal {I} M({I}) = {{\,\textrm{Rk}\,}}_{I}M\).

Given a collection \(\mathcal {I} \) of intervals in \(P\), and a function \(r:\mathcal {I} \rightarrow \mathbb {Z}\), a (signed) rank decomposition of r over \(\mathcal {I} \) is given by the following identity:where \(\mathcal {R} \) and \(\mathcal {S} \) are multi-sets of elements taken from \(\mathcal {I} \) such that \( \textbf{k}_{\mathcal {R}} \) and \( \textbf{k}_{\mathcal {S}} \) lie in \( {{\,\textrm{rep}\,}}_{\mathcal {I}} P\), and where by definition \( \textbf{k}_\mathcal {R} = \bigoplus _{{R}\in \mathcal {R}} \textbf{k}_{R}\) and \( \textbf{k}_\mathcal {S} = \bigoplus _{{S}\in \mathcal {S}} \textbf{k}_{S}\) (note that elements \({R}\in \mathcal {R} \) and \({S}\in \mathcal {S} \) are considered with multiplicity in the direct sums). By extension, we call the pair \((\mathcal {R}, \mathcal {S})\) itself a rank decomposition of r over \(\mathcal {I} \). It is a minimal rank decomposition if \(\mathcal {R} \) and \(\mathcal {S} \) are disjoint as multi-sets.

(Corollary 2.7) The minimal rank decomposition \((\mathcal {R} ^*, \mathcal {S} ^*)\) of any map \(r:\mathcal {I} \rightarrow {\mathbb {Z}}\) is unique if it exists. Furthermore, if a rank decomposition \((\mathcal {R}, \mathcal {S})\) of r exists, then a minimal one exists and is obtained from it by removing common intervals, that is:

(Theorem 2.12) Let \( \mathcal {I} \) be a locally finite collection of intervals in \( P\). Then any function \( r :\mathcal {I} \rightarrow {\mathbb {Z}}\) with upward finite support can uniquely be written as a (possibly infinite, but pointwise finite) \({\mathbb {Z}}\)-linear combination of the functions \( {{\,\textrm{Rk}\,}}_\mathcal {I} \textbf{k}_{{I}} \) with \( {I}\in \mathcal {I} \).

(Corollary 3.18) Let \( M \in {{\,\textrm{rep}\,}}\mathbb {R}^d \) be finitely presented. Let \( \mathcal {I} \) either be the collection of half-open rectangles or the collection of all half-open intervals. Then there are unique disjoint multi-sets \( \mathcal {R} \) and \( \mathcal {S} \) of elements of \( \mathcal {I} \), such that

(Corollary 5.12) Let \(M, M'\) be pfd persistence modules indexed over \({\mathbb {R}}^d\). Then, for any rank decompositions \((\mathcal {R}, \mathcal {S})\) and \((\mathcal {R} ', \mathcal {S} ')\) of \({{\,\textrm{Rk}\,}}M\) and \({{\,\textrm{Rk}\,}}M'\) respectively over all rectangles in \({\mathbb {R}}^d\), we have:

(Theorem 4.13) Let \( P\) be a finite poset. Thenis an isomorphism. Moreover, the following three sets are bases for \( {\textrm{K}_{0}}(\mathcal {E}_{\textrm{Rk}}) \):

Let \(\mathcal {I} \) denote a collection of intervals in \(P\). If \( \mathcal {R} \) and \( \mathcal {R} ' \) are two multi-sets of elements in \( \widehat{\mathcal {I}} \), such that \( {{\,\textrm{Rk}\,}}_\mathcal {I} \textbf{k}_{\mathcal {R}} = {{\,\textrm{Rk}\,}}_\mathcal {I} \textbf{k}_{\mathcal {R} '} \) and this common rank invariant is finite for all intervals of \(\mathcal {I} \), then \( \mathcal {R} = \mathcal {R} ' \).

Since the rank of a direct sum is the sum of the ranks we may remove the common elements from \( \mathcal {R} \) and \( \mathcal {R} ' \), and thus assume that the two multi-sets are disjoint. If \(\mathcal {R} \cup \mathcal {R} '\) is empty, then we are done. Otherwise, by Proposition 2.1 and the definition of \( \widehat{\mathcal {I}} \), there is \( {I}\in \mathcal {I} \) such that \( {{\,\textrm{Rk}\,}}_{{I}} \textbf{k}_{\mathcal {R}}={{\,\textrm{Rk}\,}}_{{I}} \textbf{k}_{\mathcal {R} '}>0 \). Since this rank is finite, it follows, again using Proposition 2.1, that there is a non-zero finite number of intervals containing \( {I}\) in \( \mathcal {R} \cup \mathcal {R} ' \). In particular there is a (not necessarily unique) maximal one, which we will call \( {J}\). Without loss of generality we assume \( {J}\in \mathcal {R} \). By definition of \( \widehat{\mathcal {I}} \) we have \( {J}= \bigcup X\) for some directed \( X \subseteq \mathcal {I} \). Now, by assumption, for every \({I}\in X\) we havewhich equals 1 by Proposition 2.1. It also follows from Proposition 2.1 that, for each \({I}\in X\), there is some interval \( {I}' \in \mathcal {R} ' \) such that \( {I}\subseteq {I}' \). Fix some \( {I}_0 \in X \) and for each \(I\in X\) choose an \( {I}' \) containing \( {I}_0 \cup {I}\) – employing the directness condition.

Since \( {{\,\textrm{Rk}\,}}_{{I}_0} \textbf{k}_{\mathcal {R} '} \) is finite, Corollary 2.3 says that there are actually only finitely many choices for \( {I}' \). It follows, using the fact that \( X \) is directed, that there is an \( {I}' \in \mathcal {R} ' \) containing all \( {I}\in X \). Thus \( {J}\subseteq {I}' \). If this is a proper inclusion then it contradicts the maximality of \( {J}\), otherwise it contradicts the disjointness of \( \mathcal {R} \) and \( \mathcal {R} ' \). \(\square \)

Let \( P\) be any poset, and denote by \( \mathcal {I} \) the intervals that arise as convex hulls of finitely many elements of \( P\). Then, \( \widehat{\mathcal {I}} \) is the collection of all intervals. Thus, by Proposition 2.4, the ranks of finitely spanned intervals are a complete invariant on pfd interval-decomposable modules (supported on arbitrary intervals).

Let \(\mathcal {R}, \mathcal {S}, \mathcal {R} ^*, \mathcal {S} ^*\) be multi-sets of elements of \(\widehat{\mathcal {I}}\), whose corresponding representations lie in \( {{\,\textrm{rep}\,}}_{\mathcal {I}} P\), and such that \(\mathcal {R} ^*\cap \mathcal {S} ^* = \emptyset \). Ifthen \(\mathcal {R} \supseteq \mathcal {R} ^*\), \(\mathcal {S} \supseteq \mathcal {S} ^*\), and \(\mathcal {R} \setminus \mathcal {R} ^* = \mathcal {S} \setminus \mathcal {S} ^*\).

Rewriting the equation yieldsand by additivity of the rank invariantBy Proposition 2.4 it follows that \(\mathcal {R} \cup \mathcal {S} ^* = \mathcal {R} ^*\cup \mathcal {S} \). As \(\mathcal {R} ^*\cap \mathcal {S} ^* = \emptyset \), we conclude that \(\mathcal {R} \supseteq \mathcal {R} ^*\), \(\mathcal {S} \supseteq \mathcal {S} ^*\), and \(\mathcal {R} \setminus \mathcal {R} ^* = \mathcal {S} \setminus \mathcal {S} ^*\). \(\square \)

The minimal rank decomposition \((\mathcal {R} ^*, \mathcal {S} ^*)\) of any map \(r:\mathcal {I} \rightarrow {\mathbb {Z}}\) is unique if it exists. Furthermore, if a rank decomposition \((\mathcal {R}, \mathcal {S})\) of r exists, then a minimal one exists and is obtained from it by removing common intervals, that is:

Any two rank decompositions \((\mathcal {R}, \mathcal {S})\) and \((\mathcal {R} ', \mathcal {S} ')\) of \(r:\mathcal {I} \rightarrow {\mathbb {Z}}\) satisfy \(\mathcal {R} \cup \mathcal {S} ' = \mathcal {R} ' \cup \mathcal {S} \).

Let \((\mathcal {R} ^*, \mathcal {S} ^*)\) be the minimal rank decomposition of r. From Theorem 2.6, we have \(\mathcal {R} = \mathcal {R} ^* \cup \mathcal {T} \) and \(\mathcal {S} = \mathcal {S} ^* \cup \mathcal {T} \) for some finite multi-set \(\mathcal {T} \) of elements of \(\mathcal {I} \), and similarly, we have \(\mathcal {R} ' = \mathcal {R} ^* \cup \mathcal {T} '\) and \(\mathcal {S} ' = \mathcal {S} ^* \cup \mathcal {T} '\) for some multi-set \(\mathcal {T} '\). Then,\(\square \)

A poset \((P, \le )\) is locally finite if for all \(p,q\in P\), the segment \(\langle p,q\rangle = \{r\in P \mid p\le r\le q\}\) is a finite set.

Let \(\mathcal {R} \) be a multi-set of elements in \( \mathcal {I} \). If \( \textbf{k}_{\mathcal {R}} \in {{\,\textrm{rep}\,}}_{\mathcal {I}} P\), then the map \( {{\,\textrm{Rk}\,}}_{\mathcal {I}} \textbf{k}_{\mathcal {R}} \) has upward finite support.

For any fixed \( {I}\in \mathcal {I} \), we know from Corollary 2.3 that \({{\,\textrm{Rk}\,}}_{I} \textbf{k}_{R} \ne 0\) for a finite number intervals \(R\in \mathcal {R} \). Let us denote this collection of intervals by \(\mathcal {R} '\). Furthermore, if \(I\subset I'\), then \({{\,\textrm{Rk}\,}}_{I'} \textbf{k}_{R} \ne 0\) implies that \(R\in \mathcal {R} '\) by Proposition 2.1. In particular,and this sum is non-zero for a finite number of \(I'\subseteq I\) since each \({{\,\textrm{Rk}\,}}_{\mathcal {I}} \textbf{k}_{R}\) has upward finite support. \(\square \)

The poset \(\mathbb {Z}\) is locally finite but \(\mathbb {R}\) is not. Moreover, let \(R=\langle 0,1\rangle \) and let \(\mathcal {I} \) be the collection of intervals in \(\mathbb {R}\). Then, \( {{\,\textrm{Rk}\,}}_\mathcal {I} \textbf{k}_R\) does not have upward finite support: there is an infinite number of intervals I satisfying \(\langle 0.5, 0.5\rangle \subseteq I\subseteq \langle 0,1\rangle \).

Let \( \mathcal {I} \) be a locally finite collection of intervals in \( P\). Then any function \( r :\mathcal {I} \rightarrow {\mathbb {Z}}\) with upward finite support can uniquely be written as a (possibly infinite, but pointwise finite) \({\mathbb {Z}}\)-linear combination of the functions \( {{\,\textrm{Rk}\,}}_\mathcal {I} \textbf{k}_{{I}} \) with \( {I}\in \mathcal {I} \).

Existence: For any \( {I}\in \mathcal {I} \) we set \( S_{{I}} = \{ {J}\supseteq {I}\mid \exists {K}\supseteq {J}\text { with } r({K}) \ne 0 \} \).

Since \( r \) has upward finite support, its support restricted to the upset of \( {I}\) is finite, and so is \(S_{{I}}\) since \( \mathcal {I} \) is locally finite.

Now we define a collection of scalars \( \alpha _{{I}} \in {\mathbb {Z}}\) for \( {I}\in \mathcal {I} \), inductively on the size of \( S_{{I}} \): If \( S_{{I}} = \varnothing \) we set \( \alpha _{{I}} = 0 \). Otherwise we setNote that for \( {J}\in S_{{I}} \setminus \{ {I}\} \) we have \( S_{{J}} \subsetneq S_{{I}} \), so the terms on the right hand side are already defined.

Now, using the description of the map \( {{\,\textrm{Rk}\,}}_\mathcal {I} \textbf{k}_{{I}} \) in Proposition 2.1, one immediately verifies that \( r = \sum _{{I}\in \mathcal {I}} \alpha _{{I}} {{\,\textrm{Rk}\,}}_\mathcal {I} \textbf{k}_{{I}} \). (Note in particular that this infinite sum is pointwise finite — on a given interval \( {J}\) the only possibly non-zero terms are the ones in \( S_{{J}} \) —hence well-defined.)

Uniqueness: subtracting two different \({\mathbb {Z}}\)-linear combinations realizing \( r \) from each other, we get a single linear combination \( \sum _{{I}\in \mathcal {I}} \alpha _{{I}} {{\,\textrm{Rk}\,}}_\mathcal {I} \textbf{k}_{{I}} \) with non-zero coefficients which sums up to zero. Note that there is at least one maximal \( {I}\in \mathcal {I} \) such that \( \alpha _{{I}} \ne 0 \), for otherwise the sum would not be defined. It follows, again using Proposition 2.1, that \( (\sum _{{J}\in \mathcal {I}} \alpha _{{J}} {{\,\textrm{Rk}\,}}_\mathcal {I} \textbf{k}_{{J}})({I}) = \alpha _{{I}} \ne 0 \), contradicting our assumption. \(\square \)

Let \(\mathcal {I} \) be a locally finite collection of intervals in \(P\). Then, for any map \(r :\mathcal {I} \rightarrow {\mathbb {Z}}\) with upward finite support, there is a unique pair \(\mathcal {R}, \mathcal {S} \) of disjoint multi-sets of elements of \(\mathcal {I} \) such that \( \textbf{k}_{\mathcal {R}} \) and \( \textbf{k}_{\mathcal {S}} \) lie in \({{\,\textrm{rep}\,}}_\mathcal {I} P\) and satisfy the following identity:

By Theorem 2.12, there is a unique (possibly infinite, but pointwise finite) \({\mathbb {Z}}\)-linear combination of functions \(r=\sum _{{I}\in \mathcal {I}} \alpha _{{I}}\, {{\,\textrm{Rk}\,}}_\mathcal {I} \textbf{k}_{{I}}\). Let then \(\mathcal {R} = \{ {I}\in \mathcal {I} \mid \alpha _{{I}}>0\}\) with multiplicities \({I}\mapsto \alpha _{{I}}\), and \(\mathcal {S} = \{ {I}\in \mathcal {I} \mid \alpha _{{I}}<0\}\) with multiplicities \({I}\mapsto |\alpha _{{I}}|\). It follows from the pointwise-finiteness of the linear combination that \( \mathcal {R} \) and \( \mathcal {S} \) satisfy the condition in Corollary 2.3, so in particular \( \textbf{k}_{\mathcal {R}} \) and \( \textbf{k}_{\mathcal {S}} \) lie in \( {{\,\textrm{rep}\,}}_{\mathcal {I}} P\). \(\square \)

Let \( P\) be a finite poset. Then the collection of maps \( {{\,\textrm{Rk}\,}}\textbf{k}_{\langle a, b\rangle } \) with \( a \le b \) is a basis for the free abelian group \( {\mathbb {Z}}^{\{ {\le }_{P} \}} \).

Given a finite poset \(P\), for any map \(r: \{ {\le }_{P} \} \rightarrow {\mathbb {Z}}\) there is a unique pair \(\mathcal {R}, \mathcal {S} \) of disjoint finite multi-sets of closed segments such that

Let \( M \in {{\,\textrm{rep}\,}}Q \). Then for any \( a \le b \in P \) we have

This follows directly from the construction of \( {\text {res}}_f \): \( ({\text {res}}_f M)(a) = M(f(a)) \), \( ({\text {res}}_f M)(b) = M(f(b)) \), and the structure map \( ({\text {res}}_f M)(a) \rightarrow ({\text {res}}_f M)(b) \) is the structure map \( M(f(a)) \rightarrow M(f(b)) \). \(\square \)

Let \( f :P \rightarrow Q \) be a poset homomorphism. Let \( M \in {{\,\textrm{rep}\,}}Q \), and assume that there are multi-sets \( \mathcal {R} \) and \( \mathcal {S} \) of intervals in \( P \) such that \( {{\,\textrm{Rk}\,}}M = {{\,\textrm{Rk}\,}}\textbf{k}_{\mathcal {R}} - {{\,\textrm{Rk}\,}}\textbf{k}_{\mathcal {S}} \).

Thenwhere \( f^{-1}(\mathcal {R}) = \{ f^{-1}(R) \mid R \in \mathcal {R} \} \) and similar for \( \mathcal {S} \).

Note that even if \( R \) is an interval, \( f^{-1} \) might not be: It is convex, but might be disconnected or empty.

For \( a \le b \in P \) we have \( {{\,\textrm{Rk}\,}}_{\langle a,b\rangle } {\text {res}}_f M = {{\,\textrm{Rk}\,}}_{\langle f(a), f(b)\rangle } M \) by Lemma 3.1. On the other handwhere the first and last equalities hold by Proposition 2.1. It follows that the given equality of ranks implies the desired one. \(\square \)

Let \( P = \mathbb {R} \), \( Q = \mathbb {R}^d \), and \( s \) and \( t \) points in \( \mathbb {R}^d \) such that \( s_i < t_i \text { for all } i \). Consider the mapThis is a poset homomorphism, whose image in Q is an upward sloping line. Let \( M \in {{\,\textrm{rep}\,}}Q \), and assume there is a decomposition of the usual rank asfor multi-sets \( \mathcal {R} \) and \( \mathcal {S} \) of half open (resp. open, closed, arbitrary) rectangles. Then we also have a decomposition of the usual rank invariant of the restriction aswhere the preimage of each half-open (resp. open, closed, arbitrary) rectangle is just the intersection of this rectangle with the upward sloping line, therefore it is either empty or a half-open (resp. open, closed, arbitrary) interval.

Let \( M \in {{\,\textrm{rep}\,}}\mathbb {R}^d \), and assume there is a decomposition of the usual rank asfor multi-sets \( \mathcal {R} \) and \( \mathcal {S} \) of half open rectangles. Then \( (\mathcal {R} \cap P, \mathcal {S} \cap P) \) is a usual rank decomposition of the restriction of \( M \) to \( P \). Here \( \mathcal {R} \cap P = \{ R \cap P \mid R \in \mathcal {R} \} \) and similar for \( \mathcal {S} \). Note that the intersection of any half-open rectangle with the finite grid \( P \) is either a segment or empty, whence we get the honest rank decomposition.

With \( P \) a finite grid in \( \mathbb {R}^d \) as above, let \( M \in {{\,\textrm{rep}\,}}P \). Let \( ( \mathcal {R}, \mathcal {S}) \) be a rank decomposition of \( M \). Then the left Kan extension of \( M \) to \( \mathbb {R}^d \) has a decomposition of the usual rank invariant aswhere \( \widehat{\mathcal {R}} = \{ \widehat{R} \mid R \in \mathcal {R} \} \), and similar for \( \widehat{\mathcal {S}} \) are multi-sets of half-open rectangles.

We consider the extendedfor the following reason: This inclusion has a right adjoint, which we will call \( \lfloor \cdot \rfloor \), given by “rounding down to the nearest grid-point”. We will apply Proposition 3.2 to this right adjoint. Clearly we can consider \( M \) a representation of this extended grid (by putting \( 0 \) in the new points). Likewise we can consider the segments in \( \mathcal {R} \) and \( \mathcal {S} \) as segments in the extended grid, and they still define a rank decomposition of \( M \).

Note that the adjoint pair of poset homomorphisms induces an adjoint pair of restriction functors (in the opposite order). In particular restriction along \( \lfloor \cdot \rfloor \) gives left Kan extensions along the inclusion of posets. Thus \( \textrm{Lan}\,M = {\text {res}}_{\lfloor \cdot \rfloor } M \).

Now by Proposition 3.2 we have the rank decompositionWe note that \( \widehat{R} \) is constructed to match \( \lfloor \cdot \rfloor ^{-1}(R) \), so the right hand side coincides with \( {{\,\textrm{Rk}\,}}\textbf{k}_{\widehat{\mathcal {R}}} - {{\,\textrm{Rk}\,}}\textbf{k}_{\widehat{\mathcal {S}}} \).

At first this equality holds in \( (\{ - \infty \} \cup \mathbb {R})^d \), but we may restrict to \( \mathbb {R}^d \) (and in fact all terms vanish on all points involving “\(- \infty \)”).

Finally, it follows from the grid structure of \( P \) that the intervals of the form \( \widehat{R} \) are half-open rectangles. \(\square \)

Let \( M \in {{\,\textrm{rep}\,}}\mathbb {R}^d \) be finitely presented. Then there are unique disjoint multi-sets \( \mathcal {R} \) and \( \mathcal {S} \) of half-open rectangles, such that we have a decomposition of the usual rank invariant as

Since \( M \) is finitely presented it is the left Kan extension of a representation of a finite grid. (Consider the positions of all generators and relations of \( M \), and then form the smallest grid containing all these points.) For this finite grid, we have rank decompositions by Corollary 2.13. Now apply Proposition 3.6. \(\square \)

In the situation above, for any \( M \in {{\,\textrm{Rep}\,}}Q \) we have a natural morphismsuch that for any \( i \in {I}\) the triangle formed by the above morphism and the maps \( \varprojlim M|_{\overline{f({I})}} \rightarrow M(f(i)) \) and \( \varprojlim ({\text {res}}_f M)|_{I}\rightarrow ({\text {res}}_f M)(i) = M(f(i)) \) commutes.

If moreover for any \( j \in \overline{f({I})} \) the subposetis connected, then the natural morphism above is an isomorphism. \(\square \)

Note that \( \varprojlim M|_{\overline{f({I})}} \) comes with a cone of maps to all \( M(j) \) for \( j \in \overline{f({I})} \). Considering only the \( j \) of the form \( f(i) \) for some \( i \in {I}\), we obtain a cone of maps to all \( M(f(i)) = ({\text {res}}_f M)(i) \). This cone induces the desired map \(\varprojlim M|_{\overline{f({I})}} \rightarrow \varprojlim ({\text {res}}_f M)|_{I}\). The claimed commutativities hold by construction.

For the “moreover” point we construct a map in the opposite direction. Again we start with the cone of maps from \( \varprojlim ({\text {res}}_f M)|_{I}\) to the various \( ({\text {res}}_f M)(i) = M(f(i)) \). We obtain maps to \( M(j) \) for \( j \in \overline{f({I})} \) by choosing \( i \in {I}\) such that \( f(i) \le j \), and composing with the structure map \( M(f(i)) \rightarrow M(j) \). The potential issue with this construction is well-definedness, i.e., independence of the choice of \( i \). Note that for a given \( j \) our choice for possible \( i \)’s is precisely the collection described in the extra assumption. Therefore, it suffices to check that two compatible choices for \( i' \le i'' \) lead to the same map to \( M(j) \). This is immediate from the commutativity of both triangles in the following diagram.

With well-definedness established, we also have a cone: for \( j' \le j'' \) choose \( i \) such that \( f(i) \le j' \) for both of them. From the universality of limits, we get a map \(\varprojlim ({\text {res}}_f M)|_{I}\rightarrow \varprojlim M|_{\overline{f({I})}} \).

As above, it follows from the construction that the triangles formed by this morphism and the respective maps to the \( M(f(i)) \) commute, and, in particular, that the composition of the two maps between the limits behaves like identity after composition with a map to \( M(f(i)) \). We conclude that the composition \( \varprojlim ({\text {res}}_f M)|_{I}\rightarrow \varprojlim M|_{\overline{f({I})}} \rightarrow \varprojlim ({\text {res}}_f M)|_{I}\) is the identity. For the other composition the same claim follows when observing that for all \( j \in \overline{f({I})} \) there is an \( i \) with \( f(i) \le j \). \(\square \)

Let \( f :P \rightarrow Q \) be a poset homomorphism. Then for any interval \( {I}\) in \( P \) and \( M \in {{\,\textrm{Rep}\,}}Q \) we haveIf moreover for any \( j \in \overline{f({I})} \) the posets \( \{ i \in {I}\mid f(i) \le j \} \) and \( \{ i \in {I}\mid f(i) \ge j \} \) are connected then we have equality.

Pick any \( i \in {I}\). In the diagram

the left vertical map is an isomorphism, such that the left triangle commutes, by Lemma 3.8. By the dual of that lemma the right vertical map is an isomorphism and the right triangle commutes. Thus the natural maps \( \varprojlim M|_{\overline{f({I})}} \rightarrow \varinjlim M|_{\overline{f({I})}} \) and \( \varprojlim ({\text {res}}_f M)|_{I}\rightarrow \varinjlim ({\text {res}}_f M)|_{I}\) are connected by isomorphisms, and in particular have the same rank. \(\square \)

One might ask if an invariant as natural as rank should always commute with restrictions, without the technical extra assumptions above. However, consider the subposets of \( \mathbb {R}^2 \) given as \( P = \{ (1,0), (0,1), (2,2) \} \) and \( Q = P \cup \{ (1,1) \} \). Consider the (indecomposable) representation of \( Q \) depicted as

Let \( f :P \rightarrow Q \) be the natural inclusion, \( {I}= P \), and then \( \overline{f({I})} = Q \). In this situation we haveAs one should expect from the strict inequality, the assumption in the last sentence of Proposition 3.9 are not met. Indeed \( \{ i \in {I}\mid f(i) \le (1,1) \} = \{(1,0), (0,1) \} \) is not connected.

Let \(f:P\rightarrow Q\) be a poset homomorphism. If \( {I}= \langle a,b\rangle \) is a segment, then \( \{ i \in {I}\mid f(i) \le j \} \) is connected, since it contains \( a \) and any other element is compatible with \( a \). Similarly \( \{ i \in {I}\mid f(i) \ge j \} \) is connected. In particular, Proposition 3.9 generalizes Lemma 3.1.

Let \( f :P \rightarrow Q \) be a morphism of lattices. Then for any interval \( {I}\) in \( P \), and \( j \in \overline{f({I})} \), the posets \( \{ i \in {I}\mid f(i) \le j \} \) and \( \{ i \in {I}\mid f(i) \ge j \} \) are connected.

For \( i_1, i_2 \in \{ i \in {I}\mid f(i) \le j \} \) also their join lies in this set. The proof for the other poset is dual. \(\square \)

Let \( f :P \rightarrow Q \) be a morphism of lattices. Then for any interval \( {I}\) in \( P \) we have

Let \( f :P \rightarrow Q \) be a poset homomorphism.

We call two collections of intervals \( \mathcal {I} _P \) and \( \mathcal {I} _Q \) in \( P \) and \( Q \) compatible, if

Let \( f :P \rightarrow Q \) be a lattice homomorphism, and let \( \mathcal {I} _P \) and \( \mathcal {I} _Q \) be compatible collections of intervals in \( P \) and \( Q \). Let M be a representation of \( Q \) admitting a rank decomposition \((\mathcal {R}, \mathcal {S})\) over \( \mathcal {I} _Q \),

Then \((f^{-1}(\mathcal {R}), f^{-1}(\mathcal {S}))\) is a rank decomposition of \( {\text {res}}_f M \) over \( \mathcal {I} _P \), where by definition \(f^{-1}(\mathcal {R}) = \{ f^{-1}(R) \mid R\in \mathcal {R}, R \cap f(P) \ne \emptyset \}\) and similarly for \( \mathcal {S} \).

Consider again the poset homomorphism \(P\rightarrow Q\) with \(P={\mathbb {R}}\) and \(Q={\mathbb {R}}^d\) from Example 3.4. Note that this is a lattice homomorphism. Let \( \mathcal {I} _P = \{ [a, b) \} \) be the collection of half-open intervals in \( \mathbb {R} \), and \( \mathcal {I} _Q \) be the collection of half-open intervals in the sense above. These are compatible collections of intervals. Therefore, for any \( M \in {{\,\textrm{Rep}\,}}Q \) admitting a rank decompositions \((\mathcal {R}, \mathcal {S})\) with elements in \(\mathcal {I} _Q\), its restriction to the line parametrized by \( f \) has an induced rank decomposition with elements in \(\mathcal {I} _P\).

The statement remains true if we restrict \( \mathcal {I} _Q \) to being any smaller collection of half-open intervals which includes all “half-open rectangles” \( \{ [a_1, b_1) \times \cdots \times [a_d, b_d) \} \).

We also get analogous results when considering \( \mathcal {I} _P \) to be the closed (resp. open, all) intervals in \( \mathbb {R} \), and \( \mathcal {I} _Q \) to be a collection of closed (resp. open, all) intervals containing the closed (resp. open, all) rectangles in \( \mathbb {R}^d \).

For a collection of finite subsets \( F_i \subset \mathbb {R} \), consider the finite grid \( \prod _{i=1}^d F_i \subset \mathbb {R}^d \). As in Proposition 3.6, we want to consider left Kan extensions from the grid to \( \mathbb {R}^d \). As in that proof, we add formal \(-\infty \) coordinates, and we notice that left Kan extensions are realized as restrictions along \( \lfloor \cdot \rfloor \).

We now observe that \( \lfloor \cdot \rfloor \) is a lattice homomorphism: right adjoints automatically commute with meets, but for joins we use the fact that we are specifically considering grids.

Consider either of the following situations:All of these have corresponding versions in the respective posets including “\( - \infty \)”-s. Then they form compatible collections of intervals, along the map \( \lfloor \cdot \rfloor \).

It then follows that any rank decomposition of \( M \in {{\,\textrm{rep}\,}}\prod _{i=1}^d F_i \subset \mathbb {R}^d \) induces a rank decomposition of its left Kan extension.

Let \( M \in {{\,\textrm{rep}\,}}\mathbb {R}^d \) be finitely presented. Let \( \mathcal {I} \) either be the collection of half-open rectangles or the collection of all half-open intervals. Then there are unique disjoint multi-sets \( \mathcal {R} \) and \( \mathcal {S} \) of elements of \( \mathcal {I} \), such that

Since \( M \) is finitely presented it is the left Kan extension of a representation of a finite grid. (Consider the positions of all generators and relations of \( M \), and then form the smallest grid containing all these points.) For this finite grid, the statement holds by Corollary 2.13. As a final step, by the above example we know that rank decompositions are preserved by left Kan extensions. \(\square \)

Let \( M \in {{\,\textrm{rep}\,}}\mathbb {R}^d \) be finitely presented. Let \( \mathcal {R} \) and \( \mathcal {S} \) be multi-sets of half-open rectangles. Then the following are equivalent: That is, we obtain the same decomposition over the half-open rectangles, independent of whether we consider the usual rank invariant or the generalized rank invariant.

This corollary shows that considering decompositions of the usual rank into half-open rectangles, while a priori outside the formal framework of Definition 1.5, may be considered a rank decomposition as in Definition 1.5 with respect to the class of half-open rectangles.

A short exact sequence \( 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 \) in \( {{\,\textrm{rep}\,}}P\) will be called rank-exact if \( {{\,\textrm{Rk}\,}}B = {{\,\textrm{Rk}\,}}A + {{\,\textrm{Rk}\,}}C \). We denote the collection of rank-exact sequences by \( \mathcal {E}_{\textrm{Rk}}\).

[24, Propositions 1.4 and 1.7] Let \( \mathcal {X} \) be a class of objects in an abelian category. Consider the class of exact sequencesThen \( \mathcal {E}_{\mathcal {X}} \) defines an exact structure.

It follows immediately from the construction that all objects in \( \mathcal {X} \) are \( \mathcal {E}_{\mathcal {X}} \)-projective, and therefore that finite direct sums of objects in \( \mathcal {X} \) are also \( \mathcal {E}_{\mathcal {X}} \)-projective.

Let \( \langle i,j\rangle \) be a segment in \( P\).

For \(i\in P\), let \(P_i = \textbf{k}_{\langle i, \infty \langle }\). Note that \( (-)_i \cong {{\,\textrm{Hom}\,}}( P_i, - ) \) and similarly for \( j \). Now the first claim follows from the exact sequenceand the left exactness of the \( {{\,\textrm{Hom}\,}}\)-functor.

For the second claim, observe that \( {{\,\textrm{Rk}\,}}_{\langle i,j\rangle } \) is the dimension of the image of the map \( (-)_i \rightarrow (-)_j \). In particular the formula for the rank is obtained by taking dimensions of the terms in the short exact sequence\(\square \)

For \( i < j \) in \( P\), and a short exact sequencein \( {{\,\textrm{rep}\,}}P\), the following are equivalent.

By Lemma 4.4(2), we see that (1) is equivalent toBy left-exactness of \({{\,\textrm{Hom}\,}}( \textbf{k}_{\langle i,j\langle }, -)\), this equation holds if and only if \( {{\,\textrm{Hom}\,}}( \textbf{k}_{\langle i, j\langle }, \mathbb E ) \) is exact. This shows that (1) is equivalent to (2).

The equivalence of (1) and (3) is dual. \(\square \)

The collection \(\mathcal {E}_{\textrm{Rk}}\) of rank-exact sequences defines the structure of an exact category on \( {{\,\textrm{rep}\,}}P\). Moreover, finite direct sums of objects of the form \( \textbf{k}_{\langle i, j\langle } \), where \( i < j \in P \cup \{ \infty \} \), are projective with respect to this exact structure. Dually, finite direct sums of objects \( \textbf{k}_{\rangle i, j\rangle } \), with \( i < j \in P \cup \{ - \infty \} \), are \( \mathcal {E}_{\textrm{Rk}}\)-injective.

In view of Proposition 4.5, we have \( \mathcal {E}_{\textrm{Rk}}= \mathcal {E}_{\{ \textbf{k}_{\langle i, j\langle } \mid i < j \}} \) in the notation of Proposition 4.2. It now follows from that proposition that \( \mathcal {E}_{\textrm{Rk}}\) is an exact structure. The claim on \( \mathcal {E}_{\textrm{Rk}}\)-projectives follows by Remark 4.3,

and the corresponding statement for injectives is dual. \(\square \)

Let \( \mathcal {E} \) be an exact category. The Grothendieck group \( {\textrm{K}_{0}}(\mathcal {E}) \) is the free abelian group on the objects of \( \mathcal {E} \), subject to the relation that \( [B] = [A] + [C] \) whenever \( 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 \) is a distinguished short exact sequence in \( \mathcal {E}. \)

Alternatively, the Grothendieck group may be characterized as a universal map from objects of \( \mathcal {E} \) to an abelian group which is additive on distinguished short exact sequences.

Grothendieck groups are central to homological algebra, and so it is unsurprising that they have already appeared in TDA in various forms. E.g., Patel [36] uses the standard Grothendieck group (i.e., the one associated with all the short exact sequences in the category) to define generalized persistence diagrams for one-parameter persistence modules valued in abelian categories, and Asashiba et al. [5] use the split Grothendieck group (i.e., the one associated the split exact sequences) to obtain interval approximations of persistence modules indexed by a two-parameter grid. As a result, they present the generalized persistence diagram of a module \(M\) given in [26] as a rank-preserving ‘projection’ of \(M\) onto the split Grothendieck group. The resulting approximation is thus essentially a minimal rank decomposition of \(M\) with respect to the generalized rank invariant over all intervals. The split Grothendieck group is natural in this context as it allows for the formal addition of (equivalence classes of) modules. This paper is the first in TDA to consider Grothendieck groups for exact categories.

Let \( P\) be a poset. Then \( {{\,\textrm{Rk}\,}}\) defines a homomorphism of abelian groups \( {\textrm{K}_{0}}(\mathcal {E}_{\textrm{Rk}}) \rightarrow \mathbb {Z}^{\{ {\le }_{P} \}} \).

By construction, \( {{\,\textrm{Rk}\,}}\) is additive on rank-exact sequences, therefore it factors through the Grothendieck group. \(\square \)

Let \( P\) be a finite poset. Then \( \mathcal {E}_{\textrm{Rk}}\) has enough projectives, which are finite direct sums of lower hook modules, and enough injectives, which are finite direct sums of upper hook modules. Moreover, every object has a finite \( \mathcal {E}_{\textrm{Rk}}\)-projective and a finite \( \mathcal {E}_{\textrm{Rk}}\)-injective resolution.

Since \( P\) is finite, the category \( {{\,\textrm{rep}\,}}P\) is equivalent to the module category of a finite dimensional algebra. In particular we can apply the results of [7]. Proposition 1.10 of that paper gives the desired description of \( \mathcal {E}_{\textrm{Rk}}\)-projectives, and Theorem 1.12 shows that there are enough projectives.

To see that any object has a finite \( \mathcal {E}_{\textrm{Rk}}\)-projective resolution, note thatand \( {{\,\textrm{End}\,}}( \textbf{k}_{\langle i, j\langle }) = \textbf{k}\). It follows that the indices appearing in a minimal \( \mathcal {E}_{\textrm{Rk}}\)-projective resolution increase from term to term, and thus we have a finite resolution since \(P\) is finite.

The claims for \( \mathcal {E}_{\textrm{Rk}}\)-injectives and \( \mathcal {E}_{\textrm{Rk}}\)-injective resolutions are dual. \(\square \)

The elements \( [ \textbf{k}_{\langle i, j\langle } ] \) generate \( {\textrm{K}_{0}}(\mathcal {E}_{\textrm{Rk}}) \).

For any \( M\in {{\,\textrm{rep}\,}}P \), consider the finite \( \mathcal {E}_{\textrm{Rk}}\)-projective resolutionBy definition,In \( {\textrm{K}_{0}}(\mathcal {E}_{\textrm{Rk}}) \) this gives\(\square \)

Let \( P\) be a finite poset. Thenis an isomorphism. Moreover, the following three sets are bases for \( {\textrm{K}_{0}}(\mathcal {E}_{\textrm{Rk}}) \):

By Theorem 2.12, the map \( {{\,\textrm{Rk}\,}}:{\textrm{K}_{0}}(\mathcal {E}_{\textrm{Rk}}) \rightarrow \mathbb {Z}^{\{ {\le }_{P} \}} \) is surjective. By Corollary 4.12, \( {\textrm{K}_{0}}(\mathcal {E}_{\textrm{Rk}}) \) is generated by \( | \{ {\le }_{P} \} | \) elements. For the map to be surjective, the image of a choice of \( | \{ {\le }_{P} \} | \) generators must be linearly independent. It follows that \( {{\,\textrm{Rk}\,}}\) is an isomorphism, and moreover that these generators are linearly independent. In particular we have established the second basis for \( {\textrm{K}_{0}}(\mathcal {E}_{\textrm{Rk}}) \). The final one is dual. The first one follows from Theorem 2.12, together with the fact that \( {{\,\textrm{Rk}\,}}\) is an isomorphism.

We revisit the example from Fig. 3. Note that we have the following rank-exact sequences, where each representation is indecomposable:

In the situation described above, left Kan extensions preserve rank-exact sequences.

This follows immediately from the description of the left Kan extensions. \(\square \)

Let \( f:S \rightarrow P\) be any morphism of posets. Thenfor any \( s < t \in S \cup \{ \infty \} \).

First, we recall the following general fact: let F and G be functors between abelian categories and assume that F is left adjoint to G. If G is exact, then F(M) is projective (with respect to the standard exact structure) if M is projective. To see this, observe that \({{\,\textrm{Hom}\,}}(F(M),-) \cong {{\,\textrm{Hom}\,}}(M,G(-))\) from the adjoint property, and that the right hand side is a composition of two exact functors. We conclude that \({{\,\textrm{Hom}\,}}(F(M),-) \) is exact, too.

The left Kan extension is right exact since it is left adjoint to the restriction functor (precomposition by f) . Furthermore, the restriction functor is exact, and therefore the left Kan extension sends projectives to the corresponding projectives. Now the claim follows from the fact that we have a projective presentation\(\square \)

Let \( P\) be an upper semi-lattice. Then finitely presented representations, together with rank-exact sequences, form an exact category with enough projectives and injectives. The projectives and injectives in this exact category are precisely upper and lower hooks, respectively. Moreover, any object has a finite projective and a finite injective resolution. \(\square \)

Let \( M\) be a finitely presented with respect to the standard exact structure. Let \( Q\) be the join-closure of all the indices appearing in a projective presentation of \( M\). Then it follows from the formula given for left Kan extensions above that \( M= \textrm{Lan}\,M_0 \), for some representation \( M_0\) of \( Q\).

By Theorem 4.11, \( M_0 \) has a finite rank-projective resolution. By Lemmas 4.15 and 4.16, the left Kan extension of this rank-projective resolution will be a new rank-projective resolution. In particular all terms appearing in this projective resolution are sums of upper hook representations. It follows that any projective is a summand of a sum of upper hook representations, thus itself a sum of upper hook representations. \(\square \)

The elements \( [ \textbf{k}_{\langle i, j\langle } ] \) generate \( {\textrm{K}_{0}}(\mathcal {E}_{\textrm{Rk}}^\textrm{fp}) \).

Let \( P\) be an upper semi-lattice. Then, \( {\textrm{K}_{0}}(\mathcal {E}_{\textrm{Rk}}^\textrm{fp}) \) is free, and the set \( \{ [\textbf{k}_{\langle i, j\langle }] \mid i < j \in P \cup \{ \infty \} \} \) is a basis of \( {\textrm{K}_{0}}(\mathcal {E}_{\textrm{Rk}}^\textrm{fp}) \), and furthermore, the mapis injective.

We already know that the family \( \{ [\textbf{k}_{\langle i, j\langle }] \mid i < j \in P \cup \{ \infty \} \} \) generates \( {\textrm{K}_{0}}(\mathcal {E}_{\textrm{Rk}}^\textrm{fp}) \). Assume that these elements are not linearly independent. That means that there is a finite subset which is not linearly independent. However, by Theorem 4.13, for any finite set the ranks will be linearly independent. It follows that the generating family is in fact a basis, and that \( {{\,\textrm{Rk}\,}}\) is injective. \(\square \)

Given an arbitrary collection \(\mathcal {I} \) of intervals in a finite grid \(G= \prod _{i=1}^d \llbracket 1, n_i\rrbracket \subset {\mathbb {R}}^d\), the generalized rank invariant \({{\,\textrm{Rk}\,}}_\mathcal {I} M\) of any pfd persistence module \(M\) indexed over \(G\) admits a unique minimal rank decomposition \((\mathcal {R}, \mathcal {S})\) over \(\mathcal {I} \).

The usual rank invariant of any pfd persistence module \(M\) indexed over a finite grid \(G= \prod _{i=1}^d \llbracket 1, n_i\rrbracket \subset {\mathbb {R}}^d\) admits a unique minimal rank decomposition \((\mathcal {R}, \mathcal {S})\), where \(\mathcal {R} \) and \(\mathcal {S} \) are finite multi-sets of (closed) rectangles in \(G\).

In the general case where \(G= \prod _{i=1}^d \llbracket 1, n_i\rrbracket \subset {\mathbb {R}}^d\), by applying (5.1) to every pair of comparable indices \(s\le t\in G\) in sequence, one computes the minimal rank decomposition of the usual rank invariant in time \(O\left( 2^{2d}\, \# \{ {\le }_{G} \}\right) \), assuming constant-time access to the ranks \({{\,\textrm{Rk}\,}}M(s', t')\) and constant-time arithmetic operations⁴. This bound is in \(O\left( 2^{2d}\, \prod _{i=1}^d n_i^2\right) \), and when d is fixed, it is linear in the size of the domain of the usual rank invariant viewed as a map \(\{ {\le }_{G} \}\rightarrow {\mathbb {Z}}\). When the module \(M\) comes from a simplicial filtration over the grid \(G\) with \(n = \max _{i} n_i\) simplices in total, the usual rank invariant itself can be pre-computed and stored, e.g. by naively computing the ranks \({{\,\textrm{Rk}\,}}M(s,t)\) for each pair \(s\le t\in G\) independently, which takes \(O(n^{2d+\omega })\) time in total, where \(2\le \omega <2.373\) is the exponent for matrix multiplication [34]. Adding in the computation time for the minimal rank decomposition yields a bound in \(O(n^{2d+\omega } + (2n)^{2d})\). While naive, this approach already compares favorably, in terms of running time, to the computation of other (stronger) invariants such as for instance the direct-sum decomposition of \(M\)—for which the best known algorithm runs in \(O(n^{d(2\omega +1)})\) time [23]. Moreover, the running time of computing the minimal rank decomposition is dominated by the running time of pre-computing the usual rank invariant, for which there is room for improvement. In the special case where \(d=2\) for instance, assuming the filtration is 1-critical (i.e., each simplex has a unique minimal time of appearance in the filtration), there is an \(O(n^4)\)-time algorithm to compute the usual rank invariant [11], and computing its minimal rank decomposition also takes \(O(n^4)\) time. By comparison, the best known algorithm to compute the direct-sum decomposition of \(M\) in this setting takes \(O(n^{2\omega +1})\) time [23].

The usual rank invariant of every finitely presented persistence module \(M\) over \({\mathbb {R}}^d\) admits a unique minimal rank decomposition over the half-open rectangles in \({\mathbb {R}}^d\).

The usual rank invariant of every finitely presented persistence module \(M\) over \({\mathbb {R}}^d\) admits a unique minimal rank decomposition over the lower hooks in \({\mathbb {R}}^d\).

Assume \( {I}= \bigcup _{i=1}^n {I}_i \), with each \({I}_i\) being a down-set in the poset \( {I}\). Then, the sequencewith component mapsis rank exact. Note that the assumption that each \({I}_i\) is a down-set in \({I}\) guarantees the projections in the component maps to be well-defined.

We employ induction on \( n \). Let \( {I}' = \bigcup _{i=2}^n {I}_i \). The short exact sequenceis rank exact. Inductively, we have rank exact sequences for \( {I}' \) and for \( {I}_1 \cap {I}' = \bigcup _{i=2}^n {I}_1 \cap {I}_i \) as in the two rows of the following diagram.

We can add the vertical arrows and check that the entire diagram commutes (for appropriate sign conventions). In particular its total complex is rank exact again. Putting this total complex in the upper row, we obtain the following commutative diagram.

Choosing the vertical arrows to be inclusion of and projection to the corresponding summand we obtain a new commutative diagram. Again we form the total complex, and observe that up to homotopy the two copies of \( \textbf{k}_{{I}'} \) and the two copies of \( \textbf{k}_{{I}_1 \cap {I}'} \) cancel. Thus we get the rank exact sequence in the statement of the lemma. \(\square \)

For the poset \( \mathbb {R}^d \), the setis a basis of \( {\textrm{K}_{0}}(\mathcal {E}_{\textrm{Rk}}^\textrm{fp}) \).

Note thatand all the sets on the right-hand side are down-sets in \( \langle \textbf{a}, \textbf{b} \langle \). Thus, Lemma 5.6 applies. In particular, any \( [ \textbf{k}_{\langle \textbf{a}, \textbf{b} \langle } ] \) is a linear combination of vectors in our candidate basis, which therefore generates \( {\textrm{K}_{0}}(\mathcal {E}_{\textrm{Rk}}^\textrm{fp}) \) by Theorem 4.19.

The linear independence follows from Proposition 2.4—alternatively, linear independence can also be verified by carefully analyzing the base change from the basis of Theorem 4.19 to the candidate basis here. \(\square \)

Let \(M\) be a pfd persistence module over \({\mathbb {R}}^d\) such that the usual rank invariant \({{\,\textrm{Rk}\,}}M\) admits a rank decomposition \((\mathcal {R}, \mathcal {S})\) over half-open (resp. open, closed, all) rectangles. Then, for any upward sloping line \(\ell \) in \({\mathbb {R}}^d\), \((\mathcal {R}, \mathcal {S})\) restricts to a rank decomposition \(({\mathcal {R}}_{|\ell }, {\mathcal {S}}_{|\ell })\) of \({{\,\textrm{Rk}\,}}{M}_{|\ell }\) over half-open (resp. open, closed, all) intervals.

For every pfd persistence module \(M\) indexed over the real line, the usual rank invariant \({{\,\textrm{Rk}\,}}M\) admits a unique minimal rank decomposition \((\mathcal {R}, \mathcal {S})\) over all intervals of \({\mathbb {R}}\), given by \(\mathcal {R} =\text {Bar}\,M\), the persistence barcode of \(M\), and \(\mathcal {S} =\emptyset \).

Call \(\mathcal {I} \) the collection of all closed intervals of \({\mathbb {R}}\), and note that \(\widehat{\mathcal {I}}\) as defined in (2.1) is then the collection of all intervals of \({\mathbb {R}}\). The structure theorem for pfd one-parameter persistence modules [22], combined with the fact that the usual rank invariant is additive on finite direct sums, implies that \((\text {Bar}\,M, \emptyset )\) is a rank decomposition of \({{\,\textrm{Rk}\,}}M={{\,\textrm{Rk}\,}}_\mathcal {I} M\) over \(\widehat{\mathcal {I}}\). This decomposition is minimal and Theorem 2.6 implies its uniqueness. \(\square \)

Let \(M, M'\) be pfd persistence modules indexed over \({\mathbb {R}}^d\). Then, for any rank decompositions \((\mathcal {R}, \mathcal {S})\) and \((\mathcal {R} ', \mathcal {S} ')\) of \({{\,\textrm{Rk}\,}}M\) and \({{\,\textrm{Rk}\,}}M'\) respectively over all rectangles in \({\mathbb {R}}^d\), we have:

Take any upward sloping line \(\ell \) in \({\mathbb {R}}^d\).

By (5.2), we have:Meanwhile, by Corollary 5.8, \((\mathcal {R} _{|\ell }, \mathcal {S} _{|\ell })\) is a rank decomposition of \({{\,\textrm{Rk}\,}}M_{|\ell }\), and \((\mathcal {R} '_{|\ell }, \mathcal {S} '_{|\ell })\) is a rank decomposition of \({{\,\textrm{Rk}\,}}M'_{|\ell }\) over all intervals of \({\mathbb {R}}\). By Proposition 2.4 and the structure theorem for pfd one-parameter persistence modules [22], we then have \(M_{|\ell } \oplus \textbf{k}_{\mathcal {S} _{|\ell }} \simeq \textbf{k}_{\mathcal {R} _{|\ell }}\) and \(M'_{|\ell } \oplus \textbf{k}_{\mathcal {S} '_{|\ell }} \simeq \textbf{k}_{\mathcal {R} '_{|\ell }}\), from which we deduce:Combined with the previous equation, this gives:The result follows then by taking the supremum on the left-hand side over all possible choices of upward sloping lines \(\ell \). \(\square \)

Let \(M, M'\) be pfd persistence modules indexed over \({\mathbb {R}}^d\). Then, for any rank decompositions \((\mathcal {R}, \mathcal {S})\) and \((\mathcal {R} ', \mathcal {S} ')\) of \({{\,\textrm{Rk}\,}}M\) and \({{\,\textrm{Rk}\,}}M'\) respectively over all rectangles in \({\mathbb {R}}^d\), we have:where \((\mathcal {R} ^*, \mathcal {S} ^*)\) and \((\mathcal {R} '^*, \mathcal {S} '^*)\) are the minimal rank decompositions of \({{\,\textrm{Rk}\,}}M\) and \({{\,\textrm{Rk}\,}}M'\) respectively over all rectangles in \({\mathbb {R}}^d\)—which exist as soon as \((\mathcal {R}, \mathcal {S})\) and \((\mathcal {R} ', \mathcal {S} ')\) do, by Theorem 2.6.

Let \(\mathcal {T}:= \mathcal {R} \setminus \mathcal {R} ^* = \mathcal {S} \setminus \mathcal {S} ^*\), and \(\mathcal {T} ' := \mathcal {R} ' \setminus \mathcal {R} '^* = \mathcal {S} ' \setminus \mathcal {S} '^*\). Note that \(\mathcal {T}, \mathcal {T} '\) are well-defined by Theorem 2.6. Then, for any upward sloping line \(\ell \), we have:The result follows then after multiplying by \(\omega (\ell )\) and taking the supremum on both sides over all possible choices of upward sloping lines \(\ell \). \(\square \)

Let \(M, M'\) be pfd persistence modules indexed over \({\mathbb {R}}^d\). Then, for any rank decompositions \((\mathcal {R}, \mathcal {S})\) and \((\mathcal {R} ', \mathcal {S} ')\) of \({{\,\textrm{Rk}\,}}M\) and \({{\,\textrm{Rk}\,}}M'\) respectively over all rectangles in \({\mathbb {R}}^d\), we have:

In the one-parameter setting, the signed prominence diagram encodes the lengths of the bars in the unsigned barcode, i.e., the vertical distances of the points to the diagonal in the persistence diagram. It is a discrete measure on the real line.

Given a persistence module \(M\) indexed over \({\mathbb {R}}^d\), and a vector \(\varepsilon \in {\mathbb {R}}_{\ge 0}^d\), the \(\varepsilon \)-shift \(M[\varepsilon ]\) is the module defined pointwise by \(M[\varepsilon ](t) = M(t+\varepsilon )\) and \(M[\varepsilon ](s\le t) = M(s+\varepsilon \le t+\varepsilon )\). There is a canonical morphism of persistence modules \(M\rightarrow M[\varepsilon ]\), whose image is called the \(\varepsilon \)-smoothing of \(M\), denoted by \(M^\varepsilon \).

The \(\varepsilon \)-shift of an arbitrary rectangle module \(\textbf{k}_{{R}}\) is the rectangle module \(\textbf{k}_{{R}-\varepsilon }\), where by definition \({R}-\varepsilon = \{t-\varepsilon \mid t\in {R}\}\). The \(\varepsilon \)-smoothing of \(\textbf{k}_{{R}}\) is the rectangle module \(\textbf{k}_{{R}^\varepsilon }\), where by definition \({R}^\varepsilon \) is the rectangle \({R}\cap ({R}-\varepsilon )\), obtained from \({R}\) by shifting the upper-right corner of \({R}\) by -\(\varepsilon \). Note that \(\textbf{k}_{{R}^\varepsilon }\) is the trivial module whenever \({R}\cap ({R}-\varepsilon )= \emptyset \).

Suppose the usual rank invariant of \(M: {\mathbb {R}}^d\rightarrow \textrm{Vec}_{\textbf{k}}\) admits a rank decomposition \((\mathcal {R}, \mathcal {S})\) over arbitrary rectangles in \({\mathbb {R}}^d\). Then, for any \(\varepsilon \in {\mathbb {R}}_{\ge 0}^d\), the pair \((\mathcal {R} ^\varepsilon , \mathcal {S} ^\varepsilon )\) where \(\mathcal {R} ^\varepsilon = \{{R}^\varepsilon \mid {R}\in \mathcal {R} \}\) and \(\mathcal {S} ^\varepsilon = \{{S}^\varepsilon \mid {S}\in \mathcal {S} \}\) decomposes the usual rank invariant of \(M^\varepsilon \). When \((\mathcal {R}, \mathcal {S})\) is minimal, so is \((\mathcal {R} ^\varepsilon , \mathcal {S} ^\varepsilon )\) after removing the empty rectangles from \(\mathcal {R} ^\varepsilon \) and \(\mathcal {S} ^\varepsilon \).

For any indices \(s\le t\in {\mathbb {R}}^d\), the commutativity of the squareimplies that \({{\,\textrm{Rk}\,}}M^\varepsilon (s, t) = {{\,\textrm{Rk}\,}}M(s, t+\varepsilon )\). Then, the usual rank decomposition of \(M\) at indices \((s, t+\varepsilon )\) gives:When \((\mathcal {R}, \mathcal {S})\) is minimal, the minimality of \((\mathcal {R} ^\varepsilon , \mathcal {S} ^\varepsilon )\) after removing the empty rectangles comes from the fact that each \({R}^\varepsilon \) is obtained from \({R}\) by shifting its upper-right corner by -\(\varepsilon \), so \({R}^\varepsilon = {S}^\varepsilon \ne \emptyset \) implies \({R}= {S}\). \(\square \)

Consider the \( 2 \times 2 \)-grid, and the interval \( {I}\) indicated below.

For this grid all indecomposables are thin, and we will denote them by their dimension vectors. Consider the following commutative diagram with exact rows.

Here the middle vertical arrow is the projection to the first summand. Note that \( {{\,\textrm{Rk}\,}}_{{I}} \) is zero on all objects in this diagram, except the right upper corner, where it is one. In particular \( {{\,\textrm{Rk}\,}}_{I}\) is additive on the lower sequence, but not on the upper one. But the upper sequence is a pull-back of the lower one. This shows that the collection of short exact sequences such that \( {{\,\textrm{Rk}\,}}_{{I}} \) is additive does not constitute an exact structure.