aa r X i v : . [ m a t h . A T ] A ug HOMOLOGICAL ALGEBRA OF MODULES OVER POSETS
EZRA MILLER
Abstract.
Homological algebra of modules over posets is developed, as closely par-allel as possible to that of finitely generated modules over noetherian commutativerings, in the direction of finite presentations and resolutions. Centrally at issue is howto define finiteness to replace the noetherian hypothesis which fails. The tamenesscondition introduced for this purpose captures finiteness for variation in families ofvector spaces indexed by posets in a way that is characterized equivalently by dis-tinct topological, algebraic, combinatorial, and homological manifestations. Tame-ness serves both theoretical and computational purposes: it guarantees finite presen-tations and resolutions of various sorts, all related by a syzygy theorem, amenable toalgorithmic manipulation. Tameness and its homological theory are new even in thefinitely generated discrete setting of N n -gradings, where tame is materially weakerthan noetherian. In the context of persistent homology of filtered topological spaces,especially with multiple real parameters, the algebraic theory of tameness yields topo-logically interpretable data structures in terms of birth and death of homology classes. Contents
1. Introduction 2Overview 2Acknowledgements 41.1. Modules over posets 51.2. Topological tameness 51.3. Combinatorial tameness: finite encoding 71.4. Algebraic tameness: fringe presentation 81.5. Homological tameness: the syzygy theorem 111.6. Bar codes and further developments 122. Tame poset modules 132.1. Modules over posets 132.2. Constant subdivisions 142.3. Auxiliary hypotheses 163. Fringe presentation by upsets and downsets 173.1. Upsets and downsets 183.2. Fringe presentations 214. Encoding poset modules 244.1. Finite encoding 24
Date : 10 August 2020.2020
Mathematics Subject Classification.
Primary: 05E40, 13E99, 06B15, 13D02, 55N31, 06A07,32B20, 14P10, 52B99, 13A02, 13P20, 68W30, 13P25, 62R40, 06A11, 06F20, 06F05, 68T09; Secondary:13C99, 05E16, 32S60, 14F07, 62R01, 62H35, 92D15, 92C15, 13F99, 20M14, 14P15, 06B35, 22A25. Z n -modules 315.1. Definitions 315.2. Injective hulls and resolutions 325.3. Flat covers and resolutions 335.4. Flange presentations 345.5. Syzygy theorem for Z n -modules 356. Homological algebra of poset modules 366.1. Indicator resolutions 366.2. Syzygy theorem for modules over posets 386.3. Syzygy theorem for complexes of modules 40References 411. Introduction
Overview.
A module over a poset is a family of vector spaces indexed by the posetelements with a homomorphism for each poset relation. The setup is inherently com-mutative: the homomorphism for a poset relation p (cid:22) q is the composite of homo-morphisms for the relations p (cid:22) r and r (cid:22) q whenever r lies between p and q . Thispaper lays the foundation for an extensive theory of modules over arbitrary posets,with a view toward abstract mathematical theory, algorithmic challenges, and statisti-cal implications. The mathematics includes commutative and homological algebra asthey interact with topological, analytic, algebraic, or polyhedral geometric structureon the poset, if any is given. The algorithmic challenges involve effectively encodingand manipulating arbitrary poset modules. The statistical considerations stem fromapplied topology, where modules over posets arise from persistent homology.This installment covers initial homological aspects: the extent to which modules overposets behave like multigraded modules over polynomial rings when it comes to finitepresentations and resolutions. The long-term investigation tests the frontier of multi-graded algebra regarding how far one can get without a ring and with no hypotheseson the multigrading other than a partial order. The syzygy theorem for poset moduleshere vastly generalizes the one for finitely generated modules over polynomial rings,along the way introducing finite data structures to enable algorithmic computation.The poset of utmost interest is the real vector space R n , with its usual compo-nentwise partial order. A module over R n is equivalently an R n -graded module overthe polynomial ring whose exponents are allowed to be nonnegative real numbers in-stead of integers. In this setting, the noetherian hypothesis fails spectacularly, andessentially nothing is known about homological behavior of its category of modules. OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 3
The infrastructure developed here meets the lack of noetherian hypotheses head on, toopen the possibility of working directly with modules over R n and, with no additionaldifficulty, arbitrary posets.The focus, and the most subtle point, is the nature of a suitable finiteness conditionto replace the noetherian hypothesis. The tame condition introduced here is the natu-ral candidate because it captures equivalent topological, algebraic, combinatorial, andhomological manifestations of finiteness for variation of vector spaces parametrized bya poset. Tameness serves both theoretical and computational purposes: it guaranteesvarious finite presentations and resolutions all related by a syzygy theorem, and thedata structures thus produced are amenable to algorithmic manipulation. Tameness,its syzygy theorem, and its data structures are new and theoretically as well as com-putationally valuable even in the discrete setting over the poset Z n , which is ordinarycommutative algebra of polynomial rings, where tame is much weaker than noetherian.No restriction on the underlying poset is required. For example, the lack of localfiniteness of R n is immaterial. Moreover, in that particular setting, if the partial order-ings and the modules possess supplementary geometry, be it subanalytic, semialgebraic,or piecewise-linear, for instance, then the data structures and transitions between thetopological, algebraic, combinatorial, and homological perspectives take advantage ofand preserve the geometry.Beyond the abstract route to graded module theory over real-exponent polynomialrings and arbitrary posets, one impetus for these developments lies in data scienceapplications, where the poset consists of “parameters” indexing a family of topologi-cal subspaces of a fixed topological space. Taking homology of the subspaces in thistopological filtration yields a poset module, called the persistent homology of the fil-tration, referring to how homology classes are born, persist for a while, and then dieas the parameter moves up in the poset. In ordinary persistent homology, the posetis totally ordered—usually the real numbers R , the integers Z , or a subset { , . . . , m } .This case is well studied (see [EH10], for example), and the algebra is correspondinglysimple [Cra13]. Persistence with multiple totally ordered parameters, introduced byCarlsson and Zomorodian [CZ09], has been developed in various ways, often assumingthat the poset is N n . That discrete framework has been preferred in part because itarises frequently when filtering finite simplicial complexes, but also because settingsinvolving continuous parameters unavoidably produce modules that fail to be finitelypresented in several fundamental ways. Tameness, with its data structures and syzygytheorem, circumvent these limitations.Multigraded algebra can be expressed equivalently in terms of modules, or sheaves,or functors, or derived categories, and the literature exhibits all of these. The expo-sition throughout this paper is intentionally kept at the most elementary level, withposets instead of thin skeletal categories, for instance, and with modules instead ofsheaves or functors on posets. At the risk of masking the depth of the content in theseenriched contexts, this choice of elementary language makes the exposition accessible EZRA MILLER to a wide audience, including statisticians applying persistent homology in addition totopologists, combinatorialists, algebraists, geometers, and programmers.The power of the foundations here is demonstrated by [Mil20b], for example, whichproves conjectures made by Kashiwara and Schapira concerning the relationship be-tween subanalytic and piecewise-linear stratifications of vector spaces and constructibil-ity of sheaves on real vector spaces in the derived category with microsupport restrictedto a cone; see [KS17, Conjecture 3.17] and [KS19, Conjecture 3.20]. The theory hereas well as in [Mil20a, Mil20c] was developed simultaneously and independently fromthat of Kashiwara and Schapira [KS18], cf. [Mil17]. The conical-microsupport theory isroughly equivalent to the subanalytic special case of poset module theory for partiallyordered real vector spaces, and similarly for the later PL theory [KS19]; this is essen-tially the content of [Mil20b]. A detailed comparison of the two viewpoints, includingkey differences, is left to [Mil20b], where the derived sheaf background is reviewed. Acknowledgements.
First, a special acknowledgement goes to Ashleigh Thomas,who has been a long-term collaborator on this project. She was listed as an author onearlier drafts of [Mil17] (of which this is roughly the first quarter), but her contributionslie more properly beyond these preliminaries (see [MT20], for example), so she declinedin the end to be named as an author on this installment. Early in the development ofthe ideas here, Thomas put her finger on the continuous rather than discrete natureof multiparameter persistence modules for fly wings. She computed the first examplesexplicitly, namely those in Example 1.2, and produced the biparameter persistencediagrams there as well as some of the figures in Example 3.21.Justin Curry pointed out connections from the combinatorial viewpoint taken here,in terms of modules over posets, to higher notions in algebra and category theory,particularly those involving constructible sheaves, which are in the same vein as Curry’sproposed uses of them in persistence [Cur14]; see Remarks 2.4, 3.2, 4.26, and 6.11.The author is indebted to David Houle, whose contribution to this project wasseminal and remains ongoing; in particular, he and his lab produced the fruit fly wingimages [Hou03]. Paul Bendich and Joshua Cruz took part in the genesis of this project,including early discussions concerning ways to tweak persistent (intersection [BH11])homology for investigations of fly wings. Ville Puuska discovered several errors in anearly version of Section 4, resulting in substantial correction and alteration; see Exam-ples 2.7 and 4.16. Banff International Research Station provided an opportunity forvaluable feedback and suggestions at the workshop there on Topological Data Analysis(August, 2017) as parts of this research were being completed; many participants, espe-cially the organizers, Uli Bauer and Anthea Monod, as well as Michael Lesnick, sharedimportant perspectives and insight. Thomas Kahle requested that Proposition 5.7 bean equivalence instead of merely the one implication it had stated. Hal Schenck gave Bibliographic note: this conjecture appears in v3 (the version cited here) and earlier versions ofthe cited arXiv preprint. It does not appear in the published version [KS18], which is v6 on the arXiv.
OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 5 helpful comments on an earlier version of the Introduction. Passages in Examples 1.1and 1.2 are based on or taken verbatim from [Mil15]. Portions of this work were fundedby NSF grant DMS-1702395.1.1.
Modules over posets.
There are many essentially equivalent ways to think ofa poset module. The definition in the first line of this Introduction is among the moreelementary formulations; see Definition 2.1 for additional precision. Others include a • representation of a poset [NR72]; • functor from a poset to the category of vector spaces (e.g., see [Cur19]); • vector-space valued sheaf on a poset (e.g., see [Cur14, § § • representation of a quiver with (commutative) relations (e.g., see [Oud15, § A.6]); • representation of the incidence algebra of a poset [DRS72]; or • module over a directed acyclic graph [CL18].The premise here is that commutative algebra provides an elemental framework outof which flows corresponding structure in these other contexts, in which the reader isencouraged to interpret all of the results. [Mil20b] provides an example of how that canlook, in that case from sheaf perspectives. Expressing the foundations via commutativealgebra is natural for its infrastructure surrounding resolutions. And as the objectsare merely graded vector spaces with linear maps among them—there are no rings toact—it is also the most elementary language available.Some of the formulations of poset modules are only valid when the poset is assumedto be locally finite (see [DRS72], for instance), or when the object being acted uponsatisfies a finitary hypothesis [KN09] in which the algebraic information is nonzeroon only finitely many points in any interval. This is not a failing of any particularformulation, but rather a signal that the theory has a different focus. Combinatorialformulations are built for enumeration. Representation theories are built for decom-position into and classification of irreducibles. While commutative algebra appreciatesa direct sum decomposition when one is available, such as over a noetherian ring ofdimension 0, its initial impulse is to relate arbitrary modules to simpler ones by lessrestrictive decomposition, such as primary decomposition, or by resolution, such as byprojective or injective modules. That is the tack taken here.1.2. Topological tameness.
The tame condition (Definitions 2.6 and 2.11) on a mod-ule M stipulates that the poset admit a partition into finitely many domains of con-stancy for M . This finiteness generalizes topological tameness for persistent homologyin a single parameter (see [CdS + § Example 1.1.
Let Q = R − × R + with the coordinatewise partial order, so ( r, s ) ∈ Q for any nonnegative real numbers − r and s . Let X = R be the plane containing anembedded planar graph. Define X rs ⊆ X to be the set of points at distance at least − r EZRA MILLER from every vertex and within s of some edge. Thus X rs is obtained by removing theunion of the balls of radius r around the vertices from the union of s -neighborhoods ofthe edges. In the following portion of an embedded graph, − r is approximately twice s : The biparameter persistent homology module M rs = H ( X rs ) summarizes the geometryof the embedded planar graph.Relevant properties of these modules are best highlighted in a simplified setting. Example 1.2.
Using the setup from Example 1.1, the zeroth persistent homology forthe toy-model embedded graph at left in Figure 1 is the R -module M shown at center. PSfrag replacements r → ↑ s PSfrag replacements kk k Figure 1. R -module and finite encodingEach point of R is colored according to the dimension of its associated vector spacein M , namely 3, 2, or 1 proceeding up (increasing s ) and to the right (increasing r ).The structure homomorphisms M rs → M r ′ s ′ are all surjective.This R -module fails to be finitely presented for three fundamental reasons. First,the three generators sit infinitely far back along the r -axis. (Fiddling with the sign on r does not help: the natural maps on homology proceed from infinitely large radius to 0regardless of how the picture is drawn.) Second, the relations that specify the transitionfrom vector spaces of dimension 3 to those of dimension 2 or 1 lie along a real algebraiccurve, as do those specifying the transition from dimension 2 to dimension 1. Thesecurves have uncountably many points. Third, even if the relations are discretized—restrict M to a lattice Z superimposed on R , say—the relations march off to infinityroughly diagonally away from the origin. (See Example 1.3 for the right-hand image.)Nonetheless, the R -module here is tame, with four constant regions: over thebottom-left region (yellow) the vector space is k ; over the middle (olive) region thevector space is k ; over the upper-right (blue) region the vector space is k ; and over theremainder of R the vector space is 0. The homomorphisms between these vector spacesdo not depend on which points in the regions are selected to represent them. For in-stance, k → k always identifies the two basis vectors corresponding to the connectedcomponents that are the left and right halves of the horizontally infinite red strip. OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 7
In principle, tameness can be reworked to serve directly as a data structure for algo-rithmic computation, especially in the presence of an auxiliary hypothesis to regulatethe geometry of the constant regions—when they are semialgebraic or piecewise linear(Definition 2.15.1 or 2.15.2), for example. The algorithms would generalize those forpolyhedral “sectors” in the discrete case [HM05] (or see [MS05, Chapter 13]).1.3.
Combinatorial tameness: finite encoding.
Whereas the topological notion oftameness requires little more than an arbitrary subdivision of the poset into regions ofconstancy (Definition 2.6), the combinatorial incarnation imposes additional structureon the constant regions, namely that they should be partially ordered in a natural way.More precisely, it stipulates that the module M should be pulled back from a P -modulealong a poset morphism Q → P in which P is a finite poset and the P -module hasfinite dimension as a vector space over the field k (Definition 4.1). Example 1.3.
The right-hand image in Example 1.2 is a finite encoding of M by athree-element poset P and the P -module H = k ⊕ k ⊕ k with each arrow in the imagecorresponding to a full-rank map between summands of H . Technically, this is only anencoding of M as a module over Q = R − × R + . The poset morphism Q → P takes allof the yellow rank 3 points to the bottom element of P , the olive rank 2 points to themiddle element of P , and the blue rank 1 points to the top element of P . (To makethis work over all of R , the region with vector space dimension 0 would have to besubdivided, for instance by introducing an antidiagonal extending downward from theorigin, thus yielding a morphism from R to a five-element poset.) This encoding issemialgebraic (Definition 2.15): its fibers are real semialgebraic sets.In general, constant regions need not be situated in a manner that makes them thefibers of a poset morphism (Example 4.4). Nonetheless, over arbitrary posets, modulesthat are tame by virtue of admitting a finite constant subdivision (Definition 2.11)always admit finite encodings (Theorem 4.22), although given constant regions aretypically subdivided by the constructed encoding poset morphism. This implication iswhat demands precision in the definition of tame via constant subdivision; it makessubtle use of the no-monodromy condition in Definition 2.6. In the case where the posetis a real vector space, if the constant regions have additional geometry (Definition 2.15),then a similarly geometric finite encoding is possible (Theorem 4.22.3). Remark 1.4.
Filtrations of finite simplicial complexes by products of intervals yieldpersistent homology modules that are not naturally modules over a polynomial ring in n (or any finite number of) variables. This is for the same reason that single-parameterpersistent homology is not naturally a module over a polynomial ring in one variable:though there might only be finitely many topological transitions, they can (and oftendo) occur at incommensurable real numbers. That said, filtering a finite simplicialcomplex automatically induces a finite encoding. Indeed, the parameter space mapsto the poset of simplicial subcomplexes of the original simplicial complex by sending aparameter to the simplicial subcomplex it indexes. EZRA MILLER
Remark 1.5.
The framework of poset modules arising from filtrations of topologicalspaces is more or less an instance of MacPherson’s exit path category [Tre09, § § Algebraic tameness: fringe presentation.
To compute with poset modulesalgebraically, in theoretical as well as algorithmic senses, requires presentations. Whenthe poset is Z n , so the modules are multigraded over the usual polynomial ring in n vari-ables, free presentations are available. But over arbitrary posets, there are no freemodules, and even when there are, requiring finite presentation is unreasonably re-strictive, cf. Example 1.2. Furthermore, there is nothing special about generators (intopological language, “births”) as opposed to cogenerators (“deaths”). These issuesare all resolved by (i) using arbitrary upsets instead the right-angled principal upsetsthat give rise to free modules and (ii) symmetrically involving downsets. The resultingnotion of fringe presentation (Definition 3.16) is a homomorphisms from a direct sumof interval modules for upsets to a direct sum of interval modules for downsets.Fringe presentation is expressed by a monomial matrix (Definition 3.17): an arrayof scalars with rows labeled by upsets and columns labeled by downsets. Example 1.6.
Over the poset R , the monomial matrix ϕ represents a fringe presentation of M = k as long as ϕ ∈ k is nonzero. That is, the monomial matrix specifies a homomorphism k (cid:2) (cid:3) → k (cid:2) (cid:3) with image M , which has M a = k over the yellow parameters a and 0elsewhere. The blue upset specifies the generators (births) at the lower boundary of M ; OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 9 unchecked, these persist all the way up and to the right. But the red downset specifiesthe cogenerators (deaths) along the upper boundary of M . This example illustrateshow fringe presentations are topologically interpretable in terms of birth and death ofhomology classes, when the modules in question are persistent homology.When birth upsets and death downsets are semialgebraic, or piecewise linear, orotherwise manageable algorithmically, monomial matrices render fringe presentationseffective data structures for multiparameter persistent homology ( multipersistence ).It is evidence for the naturality of the definitions that the algebraic condition ofadmitting a finite fringe presentation is equivalent to the topological and combinatorialnotions of tameness; this equivalence is part of the syzygy theorem (Theorem 6.12).Although the data structure of fringe presentation is aimed at R n -modules, it is newand lends insight already for finitely generated N n -modules (even when n = 2), wheremonomial matrices have their origins [Mil00, Section 3]. The context there is moreor less that of finitely determined modules; see Definition 5.14 in particular, which isreally just the special case of fringe presentation in which the upsets are localizationsof N n and the downsets are duals—that is, negatives—of those.It may be helpful to understand the relaxation from free presentation to fringe pre-sentation step by step over Z n or R n . First, tame modules can have generators thatsit infinitely far back along various axes, as in Example 1.2. This issue is solved byallowing flat modules instead of free ones. Over Z n , for instance, this means that(multigraded translates of) localizations of the polynomial ring by inverting variablesshould be used instead of only (translates of) the ring itself; see Remark 5.11.The next relaxation concerns cogenerators (deaths) as opposed to generators (births).Switching these means injective hulls and copresentations instead of flat covers andpresentations. Commutative algebra has considered multigraded injectives for decades[GW78] (see [MS05, Chapter 11] for an exposition), even algorithmically [Mil02, HM05].The goal, however, is to place flat presentations and injective copresentations onequal footing, so as to incorporate births and deaths simultaneously. These Matlisdual concepts (see Remark 5.11) are combined by composing a flat cover F ։ M withan injective hull M ֒ → E to get a homomorphism F → E whose image is M . Thishomomorphism is a flange presentation of M (Definition 5.12), which splices a flatresolution to an injective one in the same way that Tate resolutions (see [Coa03], forexample) transition from a free resolution to an injective one over a Gorenstein localring of dimension 0. Flange presentation is the most direct generalization to multipleparameters of the presentation corresponding to a bar code or persistence diagram. Thekey realization is that with multiple parameters, while births still correspond to gen-erators, deaths correspond to cogenerators rather than to relations among generators.The final relaxation, from summands that are flat or injective to arbitrary upsetor downset modules, provides finite data structures for tame modules even when theyhave infinte numbers of generators or cogenerators. Example 1.7.
The module M in Example 1.6 is tame but has uncountably manygenerators, uncountably many cogenerators, and an even worse set of relations. Thefringe presentation in Example 1.6 gathers the lower boundary points into a single upsetmodule and all upper boundary points into a single downset module (Definition 3.1).In contrast, a free R n -module of rank 1 has its nonzero components on a principalupset, which has exactly one lower corner. Thus fringe presentation sacrifices flatnessand injectivity for finiteness and flexibility to serve over arbitrary posets. Remark 1.8.
Any R n -module M can be approximated by a Z n -module, the resultof restricting M to, say, the rescaled lattice ε Z n . Suppose, for the sake of argument,that M is bounded, in the sense of being zero at parameters outside of a bounded subsetof R n ; think of Example 1.2, ignoring those parts of the module there that lie outsideof the depicted square. Ever better approximations, by smaller ε →
0, yield sets ofPSfrag replacements lattice points ever more closely hugging an algebraic curve. Neglecting the difficulty ofcomputing where those lattice points lie, how is a computer to store or manipulate sucha set? Listing the points individually is an option, and perhaps efficient for particularlycoarse approximations, but in n parameters the dimension of this storage problemis n −
1. As the approximations improve, the most efficient way to record such sets ofpoints is surely to describe them as the allowable ones on one side of the given algebraiccurve. And once the computer has the curve in memory, no approximation is required:just use the (points on the) curve itself. In this way, even in cases of multipersistencewhere the entire topological filtration setup can be approximated by finite simplicialcomplexes, understanding the continuous nature of the un-approximated setup can beat the same time more transparent and more efficient.
Remark 1.9. Z n -graded commutative algebra is decades old [GW78], but the per-spective arising from their equivalence with multipersistence is relatively new [CZ09].Initial steps have included descriptions of the set of isomorphism classes [CZ09], presen-tations [CSV17] and algorithms for computing [CSZ09, CSV12] or visualizing [LW15]them, as well as interactions with homological algebra of modules, such as persistenceinvariants [Knu08] and certain notions of multiparameter noise [SCL + OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 11 present only implicitly. And that is good, as nothing topologically new about persis-tence of homology classes is taught by the well known syzygies of monomial ideals (see[EMO20], for example), which in this setting are merely an interference pattern fromthe merging of separate birth points of the same class.1.5.
Homological tameness: the syzygy theorem.
Just as upsets and downsetscan be used to present poset modules, they can be used to resolve them. As in polyno-mial algebra, this line of thinking culminates in a syzygy theorem (Theorem 6.12 formodules; Theorem 6.17 for complexes) to the effect that, remarkably, the topological,algebraic, combinatorial, and homological notions of tameness available respectively via • constant subdivision (Definition 2.11), • fringe presentation (Definition 3.16), • poset encoding (Definition 4.1), and • indicator resolution (Definition 6.1)are equivalent. The moral is that the tame condition over arbitrary posets appearsto be the right notion to stand in lieu of the noetherian hypothesis over Z n : thetame condition is robust, has multiple characterizations from different mathematicalperspectives, and enables algorithmic computation in principle. The syzygy theoremis the main take-away from the paper. It engulfs the statements of its stepping stones,most notably Theorems 4.19 and 4.22, whose proofs isolate crucial ideas.The syzygy theorem directly reflects the more usual syzygy theorem for finitely de-termined Z n -modules (Theorem 5.19), with upset and downset resolutions being thearbitrary-poset analogues of free and injective resolutions, respectively, and fringe pre-sentation being the arbitrary-poset analogue of flange presentation. Indeed, the proofof the syzygy theorem works by reducing to the finitely determined case (Section 5)over Z n . The main point is that given a finite encoding of a module over an arbi-trary poset Q , the encoding poset can be embedded in Z n . The proof is completed bypushing the data forward to Z n , applying the more usual syzygy theorem to finitelydetermined modules there, and pulling back to Q .It bears mentioning that even if one is interested in ring-theoretic situations wherethe poset is Z n or R n , one can and should do homological algebra of tame modulesover a finite encoding poset rather than (only) over the original parameter space. Remark 1.10.
Topological tameness via constant subdivision is a priori weaker (thatis, more inclusive) than combinatorial tameness via finite encoding, and algebraic tame-ness via fringe presentation is a priori weaker than homological tameness via upsetor downset resolution. Thus the syzygy theorem leverages relatively weak topologi-cal structure into powerful homological structure. In particular, it provides concrete,computable, combinatorially describable representatives for objects in the derived cat-egory. The proof [Mil20b] of two conjectures due to Kashiwara and Schapira ([KS17,Conjecture 3.17] and [KS19, Conjecture 3.20]) relies on the fact that, although thetameness characterizations require no additional structure on the underlying poset, any additional structure that is present—subanalytic, semialgebraic, or piecewise-linear—ispreserved by the transitions among tameness characterizations in the syzygy theorem.1.6.
Bar codes and further developments.
Tame modules over the totally orderedset of integers or real numbers are, up to isomorphism, the same as “bar codes”: finitemultisets of intervals. The most general form of this bijection between algebraic objectsand essentially combinatorial objects over totally ordered sets is due to Crawley-Boevey[Cra13]. At its root this bijection is a manifestation of the tame representation theory ofthe type A quiver; that is the context in which bar codes were invented by Abeasis andDel Fra, who called them “diagrams of boxes” [AD80, ADK81]. Subsequent terminol-ogy for objects equivalent to these diagrams of boxes include bar codes themselves (see[Ghr08]) and planar depictions discovered effectively simultaneously in topological dataanalysis, where they are called persistence diagrams [ELZ02] (see [CEH07] for attribu-tion) and combinatorial algebraic geometry, where they are called lace arrays [KMS06].No combinatorial analogue of the bar code can classify modules over an arbitraryposet because there are too many indecomposable modules, even over seemingly wellbehaved posets like Z n [CZ09]: the indecomposables come in families of positive di-mension. Over arbitrary posets, every tame module does still admit a decompositionof the Krull–Schmidt sort, namely as a direct sum of indecomposables [BC19], butagain, there are too many indecomposables for this to be useful in general. Insteadof decomposing modules as direct sums of elemental pieces, which be arbitrarily com-plicated [BE20], the commutative algebra view advocates expressing poset modulesin terms of intervals, especially indicator modules for upsets and downsets, by wayof less rigid constructions like fringe presentation (Section 3), primary decomposition[Mil20a, MT20], or resolution (Section 6). This relaxes the direct sum in a K -theoreticway, allowing arbitrary complexes instead of split short exact sequences.Various aspects of bar codes are reflected in the equivalent concepts of tameness. Thefinitely many regions of constancy are seen in topological tameness by constant subdi-vision. The matching between left and right endpoints is seen in algebraic tameness byfringe presentation, where the left endpoints form lower borders of birth upsets and theright endpoints form upper borders of death downsets. The expressions of modules interms of bars is seen, in its relaxed form, in homological tameness, where modules be-come “virtual” sums, in the sense of being formal alternating combinations rather thandirect sums of intervals. Primary decomposition [Mil20a] isolates elements that would,in a bar code, lie in bars unbounded in fixed sets of directions (see also [HOST19]).Bar codes rely on some concept of minimality: left endpoints must correspond to minimal generators, and right endpoints to minimal cogenerators. These are not avail-able over arbitrary posets and are subtle to define and handle properly even for partiallyordered real vector spaces [Mil20c]. When minimality is available, instead of a bijection(perfect matching) from a multiset of births to a multiset of deaths, the best one cansettle for is a linear map from a filtration of the birth multiset to a filtration of thedeath multiset [Mil20d]. OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 13 Tame poset modules
Modules over posets.Definition 2.1.
Let Q be a partially ordered set ( poset ) and (cid:22) its partial order. A module over Q (or a Q -module ) is • a Q -graded vector space M = L q ∈ Q M q with • a homomorphism M q → M q ′ whenever q (cid:22) q ′ in Q such that • M q → M q ′′ equals the composite M q → M q ′ → M q ′′ whenever q (cid:22) q ′ (cid:22) q ′′ .A homomorphism M → N of Q -modules is a degree-preserving linear map, or equiva-lently a collection of vector space homomorphisms M q → N q , that commute with thestructure homomorphisms M q → M q ′ and N q → N q ′ .The last bulleted item is commutativity . In important instances (e.g., Example 2.3),it reflects that inclusions of topological subspaces induce functorial maps on homology. Example 2.2.
A module over the poset R n whose partial order is componentwisecomparison is the same thing as an R n -graded module over the monoid algebra k [ R n + ],where R + = { r ∈ R | r ≥ } is the additive monoid of nonnegative real numbers.(This is immediate from the definitions, see [Les15, § Z n -modules, which are Z n -graded modules over polynomial rings k [ N n ]:elements of k [ R n + ] are polynomials with real exponents. Example 2.3.
Let X be a topological space and Q a poset.1. A filtration of X indexed by Q is a choice of subspace X q ⊆ X for each q ∈ Q such that X q ⊆ X q ′ whenever q (cid:22) q ′ .2. The i th persistent homology of the filtered space X is the associated homologymodule, meaning the Q -module L q ∈ Q H i X q . Remark 2.4.
There are a number of abstract, equivalent ways to phrase Example 2.3.For example, a filtration is a functor from Q to the category S of subspaces of X or an S -valued sheaf on Q with its Alexandrov topology , whose base is the set of principal upsets(dual order ideals with unique minimal element). For background on and applicationsof many of these perspectives, see Curry’s dissertation [Cur14], particularly § Example 2.5. A real multifiltration of X is a filtration indexed by R n , with its partialorder by coordinatewise comparison. Example 1.1 is a real multifiltration of X = R with n = 2. The persistent homology of a real n -filtered space X is a multipersistencemodule , which is an R n -module. Constant subdivisions.Definition 2.6.
Fix a Q -module M . A constant subdivision of Q subordinate to M is apartition of Q into constant regions such that for each constant region I there is a singlevector space M I with an isomorphism M I → M i for all i ∈ I that has no monodromy :if J is some (perhaps different) constant region, then all comparable pairs i (cid:22) j with i ∈ I and j ∈ J induce the same composite homomorphism M I → M i → M j → M J . Example 2.7.
Consider the poset module (kindly provided by Ville Puuska [Puu18])PSfrag replacements
111 2 kk kk in which the structure morphisms M a → M b are all identity maps on k , except for therightmost one. This example demonstrates that module structures need not be recover-able from their isotypic subdivision , in which elements of Q lie in the same region whentheir vector spaces are isomorphic via a poset relation. In cases like this, refining the iso-typic subdivision appropriately yields a constant subdivision. Here, the two minimumelements must lie in different constant regions and the two maximum elements must liein different constant regions. Any partition accomplishing these separations—that is,any refinement of a partition that has a region consisting of precisely one maximum andone minimum—is a constant subdivision. Of course, a finite poset always admits a con-stant subdivision with finitely many regions, since the partition into singletons works. Example 2.8.
Constant subdivisions need not refine the isotypic subdivision in Ex-ample 2.7, one reason being that a single constant region can contain two or moreincomparable isotypic regions. For a concrete instance with a single constant regioncomprised of uncountably many incomparable isotypic regions, let M be the R -modulethat has M a = 0 for all a ∈ R except for those on the antidiagonal line spanned by (cid:2) − (cid:3) ∈ R , where M a = k . There is only one such R -module because all of the degreesof nonzero graded pieces of M are incomparable, so all of the structure homomorphisms M a → M b with a = b are zero. Every point on the line is a singleton isotypic region.This conclusion reverses entirely when the line is thickened to a strip of positive width,where the single isotypic region comprising the support yields a constant subdivision.The direction of the line in Example 2.8 is important: an antidiagonal line, whosepoints form an antichain in R , behaves radically differently than diagonal lines. Example 2.9.
Let M be an R -module with M a = k whenever a lies in the closeddiagonal strip between the lines of slope 1 passing through any pair of points. Thestructure homomorphisms M a → M b could all be zero, for instance, or some of them OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 15 could be nonzero. But the length | a − b | of any nonzero such homomorphism mustin any case be bounded above by the Manhattan (i.e., ℓ ∞ ) distance between the twopoints, since every longer structure homomorphism factors through a sequence thatexits and re-enters the strip.PSfrag replacements In particular, the structure homomorphism between any pair of points on the upperboundary line of the strip is zero because it factors through a homomorphism thatpoints upward first; therefore such pairs of points lie in distinct regions of any constantsubdivision. The same conclusion holds for pairs of points on the lower boundary lineof the strip. When the strip has width 0, so the upper and lower boundary coincide,the module is supported along a diagonal line whose uncountably many points mustall lie in distinct constant regions.The reference to “no monodromy” in Definition 2.6 agrees with the usual notion.
Lemma 2.10.
Fix a constant region I subordinate to a poset module M . The compositeisomorphism M I → M i → · · · → M i ′ → M I is independent of the path from i to i ′ through I , if one exists. In particular, when i = i ′ the composite is the identity on M I .Proof. The second claim follows from the first. When the path has length 0, the claimis that M I → M i → M I is the identity on M I , which follows by definition. For longerpaths the result is proved by induction on path length. (cid:3) Constant subdivision is the subtle part of the central finiteness concept of the paper.
Definition 2.11.
Fix a poset Q and a Q -module M .1. A constant subdivision of Q is finite if it has finitely many constant regions.2. The Q -module M is Q -finite if its components M q have finite dimension over k .3. The Q -module M is tame if it is Q -finite and Q admits a finite constant subdi-vision subordinate to M . Remark 2.12.
1. In ordinary totally ordered persistent homology, tameness means simply that thebar code (see Section 1.6) has finitely many bars, or equivalently, the persistence diagram has finitely many off-diagonal dots: finiteness of the set of constantregions precludes infinitely many non-overlapping bars (the bar code can’t be“too long”), while the vector space having finite dimension precludes a parametervalue over which lie infinitely many bars (the bar code can’t be “too wide”).2. The tameness condition here includes but is much less rigid than the compacttameness condition in [SCL + Z n in Q n .3. Some literature calls Definition 2.11.2 pointwise finite dimensional (PFD) . Theterminology here agrees with that in [Mil00], on which Section 5 here is based. Lemma 2.13.
Any refinement of a constant subdivision subordinate to a Q -module M is a constant subdivision subordinate to M .Proof. Choosing the same vector space M I for every region of the refinement containedin the constant region I , the lemma is immediate from Definition 2.6. (cid:3) Auxiliary hypotheses.
Effectively computing with real multifiltered spaces (Example 2.5) requires keepingtrack of the shapes of various regions, such as constant regions. (In later sections, otherregions along these lines include upsets, downsets, and fibers of poset morphisms.) Thefact that applications of persistent homology often arise from metric considerations,which are semialgebraic in nature, or are approximated by piecewise linear structuressuggests the following auxiliary hypotheses for algorithmic developments. The subana-lytic hypothesis is singled out for theoretical purposes surrounding conjectures of Kashi-wara and Schapira ([KS17, Conjecture 3.17], [KS19, Conjecture 3.20]; cf. [Mil20b]).
Definition 2.14.
An abelian group Q is partially ordered if it is generated by a sub-monoid Q + , called the positive cone , that has trivial unit group. The partial order is: q (cid:22) q ′ ⇔ q ′ − q ∈ Q + . A partially ordered group is1. real if the underlying abelian group is a real vector space of finite dimension;2. discrete if the underlying abelian group is free of finite rank. Definition 2.15.
Fix a subposet Q of a real partially ordered group. A partition of Q into subsets is1. semialgebraic if the subsets are real semialgebraic varieties;2. piecewise linear (PL) if the subsets are finite unions of convex polyhedra, wherea convex polyhedron is an intersection of finitely many closed or open half-spaces;3. subanalytic if the subsets are subanalytic varieties;4. of class X if the subsets lie in a family X of subsets of Q closed under complement,finite intersection, negation, and Minkowski sum with the positive cone Q + .A module over Q is semialgebraic , or PL , subanalytic , or of class X if Q + is and themodule is tamed by a subordinate finite constant subdivision of the corresponding type. OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 17
Remark 2.16.
Subposets of real partially ordered groups are allowed in Definition 2.15to be able to speak of, for example, piecewise linear sets in rational vector spaces, orsemialgebraic subsets of Z n , such as the set of lattice points in a right circular cone(e.g. [Mil20a, Example 5.9]). When Q is properly contained in the ambient real vectorspace, subsets of Q are semialgebraic, PL, or subanalytic when they are intersectionswith Q of the corresponding type of subset of the ambient real vector space. Proposition 2.17.
Fix a partially ordered real vector space Q .1. The classes of semialgebraic, PL, and subanalytic subsets of Q are each closedunder complements, finite intersections, and negation.2. The Minkowski sum S + Q + of a semialgebraic set S with the positive cone issemialgebraic if Q + is semialgebraic.3. The Minkowski sum S + Q + of a PL set with the positive cone is semialgebraicif Q + is polyhedral.4. The Minkowski sum S + Q + of a bounded subanalytic set S with the positive coneis subanalytic if Q + is subanalytic.Proof. See [Shi97] (for example) to treat the semialgebraic and subanalytic cases ofitem 1. The PL case reduces easily to checking that the complement of a single poly-hedron is PL, which in turn follows because a real vector space is the union of the(relatively open) faces in any finite hyperplane arrangement, so removing a single oneof these faces leaves a PL set remaining.For item 2, use that the image of a semialgebraic set under linear projection is asemialgebraic set, and then express S + Q + as the image of S × Q + under the projection Q × Q → Q that acts by ( q , q ′ ) q + q ′ . The same argument works for item 3. Thesame argument also works for item 4 but requires that the restriction of the projectionto the closure of S × Q + be a proper map, which always occurs when S is bounded. (cid:3) Fringe presentation by upsets and downsets
To define the concept of fringe presentation precisely requires some elementary back-ground on posets. That includes upsets and downsets and the modules constructedfrom them (Definition 3.1). Less obviously, notions of connectedness (Definition 3.5)play a key role, especially in computing vector spaces of homomorphisms between up-set and downset modules (Proposition 3.10). Situations where these Hom sets havedimension 1 (Corollary 3.11) are particularly key, leading to the notion of connectedhomomorphisms of interval modules (Definition 3.14). In general, the basic poset ma-terial in Section 3.1 should be useful as a reference more widely than for the applicationto fringe presentation here. Section 3.2 goes on to introduce fringe presentation (Defi-nition 3.16) and monomial matrix (Definition 3.17), along with some simple examples.
Upsets and downsets.Definition 3.1.
The vector space k [ Q ] = L q ∈ Q k that assigns k to every point of theposet Q is a Q -module with identity maps on k . More generally,1. an upset (also called a dual order ideal ) U ⊆ Q , meaning a subset closed undergoing upward in Q (so U + R n + = U , when Q = R n ) determines an indicatorsubmodule or upset module k [ U ] ⊆ k [ Q ]; and2. dually, a downset (also called an order ideal ) D ⊆ Q , meaning a subset closedunder going downward in Q (so D − R n + = D , when Q = R n ) determines an indicator quotient module or downset module k [ Q ] ։ k [ D ].When Q is a subposet of a real partially ordered group (Definition 2.14), an indicatormodule of either sort is semialgebraic, PL, subanalytic, or of class X if the correspondingupset or downset is of the same type (Definition 2.15). Remark 3.2.
Indicator submodules k [ U ] and quotient modules k [ D ] are Q -modules,not merely U -modules or D -modules, by setting the graded components indexed byelements outside of the ideals to 0. It is only by viewing indicator modules as Q -modules that they are forced to be submodules or quotients, respectively. For relationsbetween these notions and those in Remark 2.4, again see Curry’s thesis [Cur14]. Forexample, upsets form the open sets in the topology from Remark 2.4. Example 3.3.
Ising crystals at zero temperature, with polygonal boundary conditionsand fixed mesh size, are semialgebraic upsets in R n . That much is by definition: fixinga mesh size means that the crystals in question are (staircase surfaces of finitely gener-ated) monomial ideals in n variables. Remarkably, such crystals remain semialgebraicin the limit of zero mesh size; see [Oko16] for an exposition and references. Example 3.4.
Monomial ideals in polynomial rings with real exponents, which cor-respond to upsets in R n + , are considered in [ASW15], including aspects of primality,irreducible decomposition, and Krull dimension. Upsets in R n are also considered in[MMc15], where the combinatorics of their lower boundaries, and topology of relatedsimplicial complexes, are investigated in cases with locally finite generating sets. Definition 3.5.
A poset Q is1. connected if every pair of elements q, q ′ ∈ Q is joined by a path in Q : a sequence q = q (cid:22) q ′ (cid:23) q (cid:22) q ′ (cid:23) · · · (cid:23) q k (cid:22) q ′ k = q ′ in Q ;2. upper-connected if every pair of elements in Q has an upper bound in Q ;3. lower-connected if every pair of elements in Q has a lower bound in Q ; and4. strongly connected if Q is upper-connected and lower-connected. Example 3.6. R n is strongly connected. The same is true of any partially orderedabelian group. (See [Mil20a] for additional basic theory of those posets.) OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 19
Example 3.7.
A poset Q is upper-connected if (but not only if, cf. Example 3.6) ithas a maximum element—one that is preceded by every element of Q . Similarly, Q islower-connected if it has a minimum element—one that precedes every element of Q . Remark 3.8.
The relation q ∼ q ′ defined by the existence of a path joining q to q ′ asin Definition 3.5.1 is an equivalence relation. Definition 3.9.
Fix a poset Q . For any subset S ⊆ Q , write π S for the set ofconnected components of S : the maximal connected subsets of S , or equivalently theclasses under the relation from Remark 3.8. Proposition 3.10.
Fix a poset Q .1. For an upset U and a downset D , Hom Q ( k [ U ] , k [ D ]) = k π ( U ∩ D ) , a product of copies of k , one for each connected component of U ∩ D .2. For upsets U and U ′ , Hom Q ( k [ U ′ ] , k [ U ]) = k { S ∈ π U ′ | S ⊆ U } , a product of copies of k , one for each connected component of U ′ contained in U .3. For downsets D and D ′ , Hom Q ( k [ D ] , k [ D ′ ]) = k { S ∈ π D ′ | S ⊆ D } , a product of copies of k , one for each connected component of D ′ contained in D .Proof. For the first claim, the action ϕ q of ϕ : k [ U ] → k [ D ] on the copy of k in anydegree q ∈ U r D is 0 because k [ D ] q = 0, so assume q ∈ U ∩ D . Then ϕ q = ϕ q ′ : k → k if q (cid:22) q ′ ∈ U ∩ D because k [ U ] q → k [ U ] q ′ and k [ D ] q → k [ D ] q ′ are identity mapson k . Similarly, ϕ q = ϕ q ′ if q (cid:23) q ′ ∈ U ∩ D . Induction on the length of the path inDefinition 3.5.1 shows that ϕ q = ϕ q ′ if q and q ′ lie in the same connected componentof U ∩ D . Thus Hom Q ( k [ U ] , k [ D ]) ⊆ k π ( U ∩ D ) . On the other hand, specifying for eachcomponent S ∈ π ( U ∩ D ) a scalar α S ∈ k yields a homomorphism ϕ : k [ U ] → k [ D ], if ϕ q is defined to be multiplication by α S on the copies of k = k [ U ] q indexed by q ∈ S and 0 for q ∈ U r D ; that ϕ is indeed a Q -module homomorphism follows because k [ D ] q ′ = 0 (that is, q ′ D ) whenever q ′ (cid:23) q ∈ D but q ′ does not lie in the connectedcomponent of U ∩ D containing q . Said another way, pairs of elements of U ∩ D eitherlie in the same connected component of U ∩ D or they are incomparable.The proofs of the last two claims are similar (and dual to one another), particularlywhen it comes to showing that a homomorphism of indicator modules of the sametype—that is, source and target both upset or both downset—is constant on the rel-evant connected components. The only point not already covered is that if U ′ is aconnected upset and U ′ U then every homomorphism k [ U ′ ] → k [ U ] is 0 because q ′ ∈ U ′ r U implies k [ U ′ ] q ′ → k [ U ] q ′ . (cid:3) The cases of interest in this paper and its sequels [Mil20c, Mil20d], particulary realand discrete partially ordered groups (Definition 2.14) such as R n and Z n , have strongconnectivity properties, thereby simplifying the conclusions of Proposition 3.10. First,here is a convenient notation. Corollary 3.11.
Fix a poset Q with upsets U, U ′ and downsets D, D ′ .1. Hom Q ( k [ U ] , k [ D ]) = k if U ∩ D = ∅ and either U is lower-connected as asubposet of Q or D is upper-connected as a subposet of Q .2. If U and U ′ are upsets and Q is upper-connected, then Hom Q ( k [ U ′ ] , k [ U ]) = k if U ′ ⊆ U and otherwise.3. If D and D ′ are downsets and Q is lower-connected, then Hom Q ( k [ D ] , k [ D ′ ]) = k if D ⊇ D ′ and otherwise. (cid:3) Example 3.12.
Consider the poset N , the upset U = N r { } , and the downset D consisting of the origin and the two standard basis vectors. Then k [ U ] = m = h x, y i isthe graded maximal ideal of k [ N ] = k [ x, y ] and k [ D ] = k [ N ] / m . Now calculateHom N ( k [ U ] , k [ D ]) = Hom N ( m , k [ N ] / m ) = k , a vector space of dimension 2: one basis vector preserves the monomial x while killingthe monomial y , and the other basis vector preserves y while killing x . Example 3.13.
For an extreme case, consider the poset Q = R with U the closedhalf-plane above the antidiagonal line y = − x and D = − U , so that U ∩ D is totallydisconnected: π ( U ∩ D ) = U ∩ D . Then Hom Q ( k [ U ] , k [ D ]) = k R is a vector space ofbeyond continuum dimension, the copy of R in the exponent being the antidiagonal line.The proliferation of homomorphisms in Examples 3.12 and 3.13 is undesirable forboth computational and theoretical purposes; it motivates the following concept. Definition 3.14.
Let each of S and S ′ be a nonempty intersection of an upset in aposet Q with a downset in Q , so k [ S ] and k [ S ′ ] are subquotients of k [ Q ]. A homo-morphism ϕ : k [ S ] → k [ S ′ ] is connected if there is a scalar λ ∈ k such that ϕ acts asmultiplication by λ on the copy of k in degree q for all q ∈ S ∩ S ′ .The cases of interest in the rest of this paper concern three situations: both S and S ′ are upsets, or both are downsets, or S = U is an upset and S ′ = D is downset with U ∩ D = ∅ . However, the full generality of Definition 3.14 is required in the sequel tothis work [Mil20c]. Remark 3.15.
Corollary 3.11 says that homomorphisms among indicator modules areautomatically connected in the presence of appropriate upper- or lower-connectedness.
OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 21
Fringe presentations.Definition 3.16.
Fix any poset Q . A fringe presentation of a Q -module M is • a direct sum F of upset modules k [ U ], • a direct sum E of downset modules k [ D ], and • a homomorphism F → E of Q -modules with – image isomorphic to M and – components k [ U ] → k [ D ] that are connected (Definition 3.14).A fringe presentation1. is finite if the direct sums are finite;2. dominates a constant subdivision of M if the subdivision is subordinate to eachsummand k [ U ] of F and k [ D ] of E ; and3. is semialgebraic , PL , subanalytic , or of class X if Q is a subposet of a partially or-dered real vector space of finite dimension and the fringe presentation dominatesa constant subdivision of the corresponding type (Definition 2.15).Fringe presentations are effective data structures via the following notational trick.Topologically, it highlights that births occur along the lower boundaries of the upsetsand deaths occur along the upper boundaries of the downsets, with a linear map overthe ground field to relate them. Definition 3.17.
Fix a finite fringe presentation ϕ : L p k [ U p ] = F → E = L q k [ D q ].A monomial matrix for ϕ is an array of scalar entries ϕ pq whose columns are labeledby the birth upsets U p and whose rows are labeled by the death downsets D q : U ... U k D · · · D ℓ ϕ · · · ϕ ℓ ... . . . ... ϕ k · · · ϕ kℓ k [ U ] ⊕ · · · ⊕ k [ U k ] = F −−−−−−−−−−−−−−−→ E = k [ D ] ⊕ · · · ⊕ k [ D ℓ ] . Proposition 3.18.
With notation as in Definition 3.17, ϕ pq = 0 unless U p ∩ D q = ∅ .Conversely, if an array of scalars ϕ pq ∈ k with rows labeled by upsets and columns label-ed by downsets has ϕ pq = 0 unless U p ∩ D q = ∅ , then it represents a fringe presentation.Proof. Proposition 3.10.1 and Definition 3.14. (cid:3)
Example 3.19.
Fringe presentation in one parameter reflects the usual matching be-tween left endpoints and right endpoints of a module, once it has been decomposed as a direct sum of bars. A single bar, say an interval [ a, b ) that is closed on the left andopen on the right, has fringe presentationPSfrag replacements a b ↓ ֒ → ։ with imagein which a subset S ⊆ R is drawn instead of writing k [ S ]. With multiple bars, thebijection from left to right endpoints yields a monomial matrix whose scalar entriesform the identity, with rows labeled by positive rays having the specified left endpoints(the ray is the whole real line when the left endpoint is −∞ ) and columns labeled bynegative rays having the corresponding right endpoints (again, the whole line when theright endpoint is + ∞ ). In practical terms, the rows and columns can be labeled simplyby the endpoints themselves, with (say) a bar over a closed endpoint and a circle overan open one. Thus a standard bar code, in monomial matrix notation, has the form a ... a k ◦ b · · · ◦ b k . Example 3.20.
Although there are many opinions about what the multiparameteranalogue of a bar code should be, the analogue of a single bar is generally acceptedto be some kind of interval in the parameter poset—that is, k [ U ∩ D ], where U is anupset and D is a downset—sometimes with restrictions on the shape of the interval,depending on context. This case of a single bar explains the terminology “birth upset”and “death downset”. For instance, a fringe presentation of the yellow interval → with imagelocates the births along the lower boundary of the blue upset and the deaths along theupper boundary of the red downset. The scalar entries relate the births to the deaths.In this special case of one bar, the monomial matrix is 1 × OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 23
Example 3.21.
Consider an N -filtration of the following simplicial complex.Each simplex is present above the correspondingly colored rectangular curve in thefollowing diagram, which theoretically should extend infinitely far up and to the right.Each little square depicts the simplicial complex that is present at the parameter occu-pying its lower-left corner. Taking zeroth homology yields an N -module that replacesthe simplicial complex in each box with the vector space spanned by its connectedcomponents. A fringe presentation for this N -module is − − , where the grey square atop the third column represents the downset that is all of N .This fringe presentation means that, for example, the connected component that isthe blue endpoint of the simplicial complex is born along the union of the axes withthe origin removed but the point (cid:2) (cid:3) appended. The purple downset, correspondingto the left edge, records the death—along the upper purple boundary—of the homol-ogy class represented by the difference of the blue (left) and gold (middle) vertices.Computations and figures for this example were kindly provided by Ashleigh Thomas. Remark 3.22.
The term “fringe” is a portmanteau of “free” and “injective” (that is,“frinj”), the point being that it combines aspects of free and injective resolutions whilealso conveying that the data structure captures trailing topological features at boththe birth and death ends. 4.
Encoding poset modules
Sections 2 and 3 introduce two finiteness conditions: a topological one (tameness,Definition 2.11), which is the intuitive control of homological variation in a filtration ofa topological space, and an algebraic one (fringe presentation, Definition 3.16), whichprovides effective data structures. To interpolate between them, a third finiteness con-dition, this one combinatorial in nature (finite encoding, Definition 4.1), serves as a the-oretical tool whose functorial essence supports much of the development in this paper;the category of tame modules (Section 4.5) is best dealt with using this language, for in-stance. The main result of Section 4, namely Theorem 4.22, says that tame Q -modulescan be encoded in the manner of Definition 4.1. Theorems 4.19 and 4.22 are a substan-tial portion of the main result of the paper (Theorem 6.12), and their proofs contributekey arguments not repeated there although their statements are largely subsumed.4.1. Finite encoding.Definition 4.1.
Fix a poset Q . An encoding of a Q -module M by a poset P is a posetmorphism π : Q → P together with a P -module H such that M ∼ = π ∗ H = L q ∈ Q H π ( q ) ,the pullback of H along π , which is naturally a Q -module. The encoding is finite if1. the poset P is finite, and2. the vector space H p has finite dimension for all p ∈ P . Example 4.2.
Example 1.2 shows a constant isotypic subdivision of R which happensto form a poset and therefore produces an encoding. Example 4.3.
A finite encoding of the module in Example 3.21 is as follows.PSfrag replacements (cid:20) (cid:21)(cid:20) (cid:21) " (cid:21)(cid:20) (cid:21) (cid:2) (cid:3) (cid:2) (cid:3) kk k k k k k OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 25
Example 4.4.
There is no natural way to impose a poset structure on the set ofregions in a constant subdivision. Take, for example, Q = R and M = k ⊕ k [ R ],where k is the R -module whose only nonzero component is at the origin, where itis a vector space of dimension 1. This module M induces only two isotypic regions,namely the origin and its complement, and they constitute a constant subdivision.= ∪ Neither of the two regions has a stronger claim to precede the other, but at the sametime it would be difficult to justify forcing the regions to be incomparable.
Example 4.5.
Take Q = Z n and P = N n . The convex projection Z n → N n setsto 0 every negative coordinate. The pullback under convex projection is the ˇCech hull[Mil00, Definition 2.7]. More generally, suppose a (cid:22) b in Z n . The interval [ a , b ] ⊆ Z n is a box (rectangular parallelepiped) with lower corner at a and upper corner at b .The convex projection π : Z n → [ a , b ] takes every point in Z n to its closest point inthe box. A Z n -module is finitely determined if it is finitely encoded by π . Example 4.6.
The indicator module k [ Q ] is encoded by the morphism from Q to theone-point poset with vector space H = k . This generalizes to other indicator modules.1. Any upset module k [ U ] ⊆ k [ Q ] is encoded by a morphism from Q to the chain P of length 1, consisting of two points 0 <
1, that sends U to 1 and the complement U to 0. The P -module H that pulls back to k [ U ] has H = 0 and H = k .2. Dually, any downset module k [ D ] is also encoded by a morphism from Q to thechain P of length 1, but this one sends D to 0 and the complement D to 1, andthe P -module H that pulls back to k [ D ] has H = k and H = 0. Definition 4.7.
Fix a poset Q and a Q -module M .1. A poset morphism π : Q → P or an encoding of a Q -module (perhaps differentfrom M ) is subordinate to M if there is a P -module H such that M ∼ = π ∗ H .2. When Q is a subposet of a partially ordered real vector space, an encoding of M is semialgebraic , PL , subanalytic , or of class X if the partition of Q formed bythe fibers of π is of the corresponding type (Definition 2.15). Example 4.8.
The “antidiagonal” R -module M in Example 2.8 has a semialgebraicposet encoding by the chain with three elements, where the fiber over the middleelement is the antidiagonal line, and the fibers over the top and bottom elements arethe open half-spaces above and below the line, respectively. In contrast, using thediagonal line spanned by (cid:2) (cid:3) ∈ R instead of the antidiagonal line yields a modulewith no finite encoding; see Example 2.9. Lemma 4.9.
An indicator module is constant on every fiber of a poset morphism π : Q → P if and only if the module is the pullback along π of an indicator P -module. Proof.
The “if” direction is by definition. For the “only if” direction, observe that if U ⊆ Q is an upset that is a union of fibers of P , then the image π ( U ) ⊆ P is an upsetwhose preimage equals U . The same argument works for downsets. (cid:3) Example 4.10 (Pullbacks of flat and injective modules) . An indecomposable flat Z n -module k [ b + Z τ + N n ] is an upset module for the poset Z n . Pulling back to any posetunder a poset map to Z n therefore yields an upset module for the given poset. The dualstatement holds for any indecomposable injective module k [ b + Z τ − N n ]: its pullbackis a downset module.Pullbacks have particularly transparent monomial matrix interpretations. Proposition 4.11.
Fix a poset Q and an encoding of a Q -module M via a posetmorphism π : Q → P and P -module H . Any monomial matrix for a fringe presentationof H pulls back to a monomial matrix for a fringe presentation that dominates theencoding by replacing the row labels U , . . . , U k and column labels D , . . . , D ℓ with theirpreimages, namely π − ( U ) , . . . , π − ( U k ) and π − ( D ) , . . . , π − ( D ℓ ) . (cid:3) Uptight posets.
Constructing encodings from constant subdivisions uses general poset combinatorics.
Definition 4.12.
Fix a poset Q and a set Υ of upsets. For each poset element a ∈ Q ,let Υ a ⊆ Υ be the set of upsets from Υ that contain a . Two poset elements a , b ∈ Q lie in the same uptight region if Υ a = Υ b . Remark 4.13.
The partition of Q into uptight regions in Definition 4.12 is the commonrefinement of the partitions Q = U ·∪ ( Q r U ) for U ∈ Υ. Remark 4.14.
Every uptight region is the intersection of a single upset (not necessarilyone of the ones in Υ) with a single downset. Indeed, the intersection of any familyof upsets is an upset, the complement of an upset is a downset, and the intersectionof any family of downsets is a downset. Hence the uptight region containing a equals (cid:0)T U ∈ Υ a U (cid:1) ∩ (cid:0)T U Υ a U (cid:1) , the first intersection being an upset and the second a downset. Proposition 4.15.
In the situation of Definition 4.12, the relation on uptight regionsgiven by A (cid:22) B whenever a (cid:22) b for some a ∈ A and b ∈ B is reflexive and acyclic.Proof. The stipulated relation on the set of uptight regions is • reflexive because a (cid:22) a for any element a in any uptight region A ; and • acyclic because going up from a ∈ Q causes the set Υ a in Definition 4.12 to(weakly) increase, so a directed cycle can only occur with a constant sequenceof sets Υ a . (cid:3) OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 27
Example 4.16.
The relation in Proposition 4.15 makes the set of uptight regions intoa directed acyclic graph, but the relation need not be transitive. An example in theposet Q = N , kindly provided by Ville Puuska [Puu18], is as follows. Notationally, itis easier to work with monomial ideals in k [ x, y ] = k [ N ], which correspond to upsets(see Example 3.12). Let Υ = { U , . . . , U } consist of the upsets with indicator modules k [ U ] = h x , y i , k [ U ] = h x , y i , k [ U ] = h xy i , k [ U ] = h x y i . Identifying each monomial x a y b with the corresponding pair ( a, b ) ∈ N , it follows thatΥ x = { U } , and Υ x = Υ y = { U , U } , and Υ xy = { U , U , U } represent three distinctuptight regions; call them A , B , and C . They satisfy A ≺ B ≺ C because x ≺ x and y ≺ xy . However, A C because A = { x } while U forces C = xy k [ y ] to consist ofall lattice points in a vertical ray starting at xy . Definition 4.17.
In the situation of Definition 4.12, the uptight poset is the transitiveclosure P Υ of the directed acyclic graph of uptight regions in Proposition 4.15.4.3. Constant upsets.Definition 4.18.
Fix a constant subdivision of Q subordinate to M . A constant upset of Q is either1. an upset U I generated by a constant region I or2. the complement of a downset D I cogenerated by a constant region I . Theorem 4.19.
Let Υ be the set of constant upsets from a constant subdivision of Q subordinate to M . The partition of Q into uptight regions for Υ forms another constantsubdivision subordinate to M .Proof. Suppose that A is an uptight region that contains points from constant regions I and J . Any point in I ∩ A witnesses the containments A ⊆ D I and A ⊆ U I of A inside the constant upset and downset generated and cogenerated by I . Any point j ∈ J ∩ A is therefore sandwiched between elements i , i ′ ∈ I , so i (cid:22) j (cid:22) i ′ , because j ∈ U I (for i ) and j ∈ D I (for i ′ ). By symmetry, switching I and J , there exists j ′ ∈ J with i ′ (cid:22) j ′ . The sequence M I → M i → M j → M i ′ → M j ′ → M J , where the first and last isomorphisms come from Definition 2.6 and the homomorphismsin between are Q -module structure homomorphisms, induces isomorphisms M i → M i ′ and M j → M j ′ by definition of constant region. Elementary homological algebra impliesthat M i → M j is an isomorphism. The induced isomorphism M I → M J is independentof the choices of i , j , i ′ , and j ′ (in fact, merely considering independence of the choicesof i and j ′ would suffice) because constant subdivisions have no monodromy.The previous paragraph need not imply that I = J , but it does imply that all of thevector spaces M J for constant regions J that intersect A are—viewing the data of theoriginal constant subdivision as given—canonically isomorphic to M I , thereby allowing the choice of M A = M I . This, plus the lack of monodromy in constant subdivisions,ensures that M A → M a → M b → M B is independent of the choices of a ∈ A and b ∈ B with a (cid:22) b . Thus the uptight subdivision is also constant subordinate to M . (cid:3) Example 4.20.
Theorem 4.19 does not claim that I = U I ∩ D I , and in fact that claimis often not true, even if the isotypic subdivision (Example 2.7) is already constant.Consider Q = R and M = k ⊕ k [ R ], as in Example 4.4, and take I = R r { } .Then U I = D I = R , so U I ∩ D I contains the other isotypic region J = { } . Theuptight poset P M has precisely four elements:1. the origin { } = U J ∩ D J ;2. the complement U J r { } of the origin in U J ;3. the complement D J r { } of the origin in D J ; and4. the points R r ( U J ∪ D J ) lying only in I and in neither U J nor D J .Oddly, uptight region 4 has two connected components, the second and fourth quad-rants A and B , that are incomparable: any chain of relations from Definition 2.6 thatrealizes the equivalence a ∼ b for a ∈ A and b ∈ B must pass through the positivequadrant or the negative quadrant, each of which accidentally becomes comparable tothe other isotypic region J and hence lies in a different uptight region.4.4. Finite encoding from constant subdivisions.Definition 4.21. If Q is a subposet of a partially ordered real vector space, then a Q -module M has compact support if M has nonzero components M q only in a boundedset of degrees q ∈ Q . A constant subdivision subordinate to such a module is compact ifit has exactly one unbounded constant region (namely those q ∈ Q for which M q = 0). Theorem 4.22.
Fix a Q -finite module M over a poset Q .1. M admits a finite encoding if and only if there exists a finite constant subdivisionof Q subordinate to M . More precisely,2. the uptight poset of the set of constant upsets from any constant subdivision yieldsan uptight encoding of M that is finite if the constant subdivision is finite.3. If Q is a subposet of a partially ordered real vector space and the constant subdi-vision in the previous item is • semialgebraic, with Q + also semialgebraic; or • PL, with Q + also polyhedral; or • compact and subanalytic, with Q + also subanalytic; or • of class X ,then the relevant uptight encoding is semialgebraic, PL, subanalytic, or class X .Proof. One direction of item 1 is easy: a finite encoding induces a constant subdivisionalmost by definition. Indeed, if π : Q → P is a poset encoding of M by a P -module H ,then each fiber I of π is a constant region with M I = H π ( I ) . If i (cid:22) j with i ∈ I and j ∈ J , then the composite homomorphism M I → M i → M j → M J is merely thestructure morphism H π ( I ) → H π ( J ) of the P -module H . OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 29
The hard direction is producing a finite encoding from a constant subdivision. Forthat, it suffices to prove item 2. Let Υ be the set of constant upsets from a constantsubdivision of Q subordinate to M . Consider the quotient map Q → P Υ of sets thatsends each element of Q to the uptight region containing it. Proposition 4.15 andDefinition 4.17 imply that this map of sets is a morphism of posets. By Definition 2.6the vector spaces M A indexed by the uptight regions A ∈ P Υ constitute a P Υ -module H that is well defined by Theorem 4.19. The pullback of H to Q is isomorphic to M byconstruction. The claim about finiteness follows because the number of uptight regionsis bounded above by 2 r , where r is the number of constant regions in the originalconstant subdivision: every element of Q lies inside or outside of each constant upsetand inside or outside of each constant downset.For claim 3, every constant upset is a Minkowski sum I + Q + or the complement of I − Q + = − ( − I + Q + ) by Definition 4.18. These are semialgebraic, PL, subanalytic, orof class X , respectively, by Proposition 2.17 (or Definition 2.15 for class X ). Note thatin the compact subanalytic case, the unique unbounded constant region I afforded byDefinition 4.21 has I + Q + = I − Q + = Q , which is subanalytic. (cid:3) Example 4.23.
For the “antidiagonal” R -module M in Examples 2.8 and 4.8, everypoint on the line is a singleton isotypic region, but these uncountably many isotypicregions can be gathered together: the finite encoding there is the uptight poset for thetwo upsets that are the closed and open half-spaces bounded below by the antidiagonal. Example 4.24.
In any encoding of the “diagonal strip” R -module M in Example 4.4,the poset must be uncountable by Theorem 4.22.4.5. The category of tame modules.Example 4.25.
The kernel of a homomorphism of tame modules need not be tame.The upset U ⊆ R that is the closed half-space above the antidiagonal line L given by a + b = 1 has interior U ◦ , also an upset. The quotient module N = k [ U ] / k [ U ◦ ] is thetranslate by one unit (up or to the right) of the antidiagonal module in Examples 2.8,4.8, and 4.23. Both M = k [ U ] ⊕ k [ U ] and N are tame. The surjection ϕ : M ։ N that acts in every degree a = (cid:2) ab (cid:3) along L by sending the basis vectors of M a = k to b and − a in N a = k has kernel K = ker ϕ that is the submodule of M with • k in every degree from U ◦ , and • the line in k through (cid:2) (cid:3) and (cid:2) ab (cid:3) in every degree from L .0 → PSfrag replacements xy −→ PSfrag replacements xy −→ PSfrag replacements xy → That is, K a agrees with M a for degrees a outside of L , and K a is the line in M a of slope b/a through the origin when a lies on L . This kernel K is not tame. Indeed, if a and a ′ are distinct points on L , then the homomorphisms K a → K a ∨ a ′ and K a ′ → K a ∨ a ′ havedifferent images, so a and a ′ are forced to lie in different constant regions in everyconstant subdivision of R subordinate to K . (Note the relation between this exampleand Proposition 3.10.1 for Q = U ⊂ R and D = L ⊂ Q .) Remark 4.26.
Encoding of a Q -module M by a poset morphism is related to view-ing M as a sheaf on Q with its Alexandrov topology that is constructible in the senseof Lurie [Lur17, Definitions A.5.1 and A.5.2]. The difference is that poset encodingrequires constancy (in the sense of Definition 2.6) on fibers of the encoding morphism,whereas Alexandrov constructibility requires only local constancy in the sense of sheaftheory. This distinction is decisive for Example 4.25, where the kernel K is constructiblebut not finitely encoded.Because of Remark 4.26, allowing arbitrary homomorphisms between tame moduleswould step outside of the tame class. More formally, inside the category of Q -modules,the full subcategory generated by the tame modules contains modules that are nottame. To preserve tameness, it is thus necessary to restrict the allowable morphisms. Definition 4.27.
A homomorphism ϕ : M → N of Q -modules is tame if Q admits afinite constant subdivision subordinate to both M and N such that for each constantregion I the composite isomorphism M I → M i → N i → N I does not depend on i ∈ I .The map ϕ is semialgebraic, PL, subanalytic, or class X if this constant subdivision is. Lemma 4.28.
The kernel and cokernel of any tame homomorphism of Q -modules aretame morphisms of tame modules. The same is true when tameness is replaced bysemialgebraic, PL, subanalytic, or class X .Proof. Any constant subdivision as in Definition 4.27 is subordinate to both the kerneland cokernel of M → N , with the vector spaces assocated to any constant region I being ker( M I → N I ) and coker( M I → N I ). (cid:3) Definition 4.29.
The category of tame modules is the subcategory of Q -modules whoseobjects are the tame modules and whose morphisms are the tame homomorphisms. Remark 4.30.
To be precise with language, a morphism of tame modules is requiredto be tame, whereas a homomorphism of tame modules is not. That is, morphismsin the category of tame modules are called morphisms, whereas morphisms in thecategory of Q -modules are called homomorphisms. To avoid confusion, the set of tamemorphisms from a tame module M to another tame module N is denoted Mor( M, N )instead of Hom(
M, N ). Proposition 4.31.
Over any poset Q , the category of tame Q -modules is abelian.If Q is a subposet of a partially ordered real vector space of finite dimension, then thecategory of semialgebraic, PL, subanalytic, or class X modules is abelian. OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 31
Proof.
Over any poset, the category in question is a subcategory of the category of Q -modules, which is abelian. The subcategory is not full, but Mor( M, N ) is an abeliansubgroup of Hom(
M, N ); this is most easily seen via Theorem 4.22, for if ϕ : M → N and ϕ ′ : M → N ′ are finitely encoded by π : Q → P and π ′ : Q → P ′ , respectively, then ϕ + ϕ ′ is finitely encoded by π × π ′ : Q → P × P ′ . The same construction, but with thesource of π ′ being a new module M ′ instead of M , shows that the ordinary product anddirect sum of a pair of finitely encoded modules serves as a product and coproduct inthe tame category. Kernels and cokernels of morphisms in the tame category exist byLemma 4.28, which also implies that every monomorphism is a kernel (it is the kernelof its cokernel in the category of Q -modules) and every epimorphism is a cokernel (itis the cokernel of its kernel in the category of Q -modules).The semialgebraic, PL, and class X cases have the same proof, noting that π × π ′ hasfibers of the desired type if π and π ′ both do. The subanalytic case only follows fromthis argument when restricted to the category of modules whose nonzero graded pieceslie in a bounded subset of Q (the subset is allowed to depend on the module). However,the argument in the previous paragraph can be done directly with common refinementsof pairs of constant subdivisions, so reducing to Theorem 4.22 is not necessary. (cid:3) Finitely determined Z n -modules Unless otherwise stated, this section is presented over the discrete partially orderedgroup Q = Z n with Q + = N n . It begins by reviewing the structure of finitely de-termined Z n -modules (Section 5.1), including (minimal) injective and flat resolutions(Sections 5.2 and 5.3), before getting to flange presentations (Section 5.4) and thesyzygy theorem (Section 5.5). These latter two underlie the general syzygy theorem(Section 6.2), including existence of fringe presentations. They also serve as modelsfor the concepts of socle, cogenerator, and downset hull over real polyhedral groups,covered in the sequel [Mil20c], as well as dual notions of top, generator, and upset cover.The main references for Z n -modules used here are [Mil00, MS05]. The developmentof homological theory for injective and flat resolutions in the context of finitely de-termined modules is functorially equivalent to that for finitely generated modules, by[Mil00, Theorem 2.11], but it is convenient to have on hand statements in the finitelydetermined case. Flange presentation (Section 5.4) and the characterization of finitelydetermined modules in Proposition 5.7 and (hence) Theorem 5.19 are apparently new.5.1. Definitions.
The essence of the finiteness here is that all of the relevant information about therelevant modules should be recoverable from what happens in a bounded box in Z n . Definition 5.1. A Z n -finite module N is finitely determined if for each i = 1 , . . . , n the multiplication map · x i : N b → N b + e i is an isomorphism whenever b i lies outside ofsome bounded interval. For notation, k [ N n ] = k [ x ], where x = x , . . . , x n is a sequenceof variables, and e i is the standard basis vector whose only nonzero entry is 1 in slot i . Remark 5.2.
This notion of finitely determined is the same notion as in Example 4.5.A module is finitely determined if and only if, after perhaps translating its Z n -grading,it is a -determined for some a ∈ N n , as defined in [Mil00, Definition 2.1]. Remark 5.3.
For Z n -modules, the finitely determined condition is weaker—that is,more inclusive—than finitely generated, but it is much stronger than tame or (equiva-lently, by Theorem 4.22) finitely encoded. The reason is essentially Example 4.5, wherethe encoding has a very special nature. For a generic sort of example, the restrictionto Z n of any R n -finite R n -module with finitely many constant regions of sufficient widthis a tame Z n -module, and there is simply no reason why the constant regions should becommensurable with the coordinate directions in Z n . Already the toy-model fly wingmodules in Examples 1.2 and 1.3 yield infinitely generated but tame Z n -modules, andthis remains true when the discretization Z n of R n is rescaled by any factor. Example 5.4.
The local cohomology of an affine semigroup ring is tame but usu-ally not finitely determined; see [HM05] and [MS05, Chapter 13], particularly Theo-rem 13.20, Example 13.17, and Example 13.4 in the latter.5.2.
Injective hulls and resolutions.Remark 5.5.
Every Z n -finite module that is injective in the category of Z n -modulesis, by [MS05, Theorem 11.30], a direct sum of downset modules k [ D ] for coprincipal downsets D = a + τ − N n , said to be cogenerated by a along the face τ of N n . Notethat faces of N n correspond to subsets of [ n ] = { , . . . , n } via τ ↔ { i ∈ [ n ] | e i ∈ τ } .Minimal injective resolutions work for finitely determined modules just as they dofor finitely generated modules. The standard definitions are as follows. Definition 5.6.
Fix a Z n -module N .1. An injective hull of N is an injective homomorphism N → E in which E is aninjective Z n -module (see Remark 5.5). This injective hull is • finite if E has finitely many indecomposable summands and • minimal if the number of such summands is minimal.2. An injective resolution of N is a complex E • of injective Z n -modules whosedifferential E i → E i +1 for i ≥ H ( E • ) ∼ = N (so N ֒ → E and coker( E i − → E i ) ֒ → E i +1 are injective hulls for all i ≥ E • • has length ℓ if E i = 0 for i > ℓ and E ℓ = 0; • is finite if E • = L i E i has finitely many indecomposable summands; and • is minimal if N ֒ → E and coker( E i − → E i ) ֒ → E i +1 are minimal injectivehulls for all i ≥ Proposition 5.7.
The following are equivalent for a Z n -module N .1. N is finitely determined.2. N admits a finite injective resolution.3. N admits a finite minimal injective resolution. OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 33
Any finite minimal resolution is unique up to isomorphism and has length at most n .Proof. The proof is based on existence of finite minimal injective hulls and resolutionsfor finitely generated Z n -modules, along with uniqueness and length n given minimality,as proved by Goto and Watanabe [GW78].First assume N is finitely determined. Translating the Z n -grading affects nothingabout existence of a finite injective resolution. Therefore, using Remark 5.2, assumethat N is a -determined. Truncate by taking the N n -graded part of N to get a positively a -determined—and hence finitely generated—module N (cid:23) ; see [Mil00, Definition 2.1].Take any minimal injective resolution N (cid:23) → E • . Extend backward using the ˇCechhull [Mil00, Definition 2.7], which is exact [Mil00, Lemma 2.9], to get a finite minimalinjective resolution ˇ C ( N (cid:23) → E • ) = ( N → ˇ C E • ), noting that ˇ C fixes indecomposableinjective modules whose N n -graded parts are nonzero and is zero on all other indecom-posable injective modules [Mil00, Lemma 4.25]. This proves 1 ⇒ ⇒ ⇒
1, follows because everyindecomposable injective is finitely determined and the category of finitely determinedmodules is abelian. (The category of Z n -modules each of which is nonzero only in abounded set of degrees is abelian, and constructions such as kernels, cokernels, or directsums in the category of finitely determined modules are pulled back from there.) (cid:3) Flat covers and resolutions.
Minimal flat resolutions are not commonplace, but the notion is Matlis dual to thatof minimal injective resolution. In the context of finitely determined modules, flatresolutions work as well as injective resolutions. The definitions are as follows.
Definition 5.8.
Fix a Z n -module N .1. A flat cover of N is a surjective homomorphism F → N in which F is a flat Z n -module (see Remark 5.11). This flat cover is • finite if F has finitely many indecomposable summands and • minimal if the number of such summands is minimal.2. A flat resolution of N is a complex F • of flat Z n -modules whose differential F i +1 → F i for i ≥ H ( F • ) ∼ = N (so F ։ N and F i +1 ։ ker( F i → F i − ) are flat covers for all i ≥ F • • has length ℓ if F i = 0 for i > ℓ and F ℓ = 0; • is finite if F • = L i F i has finitely many indecomposable summands; and • is minimal if F ։ N and F i +1 ։ ker( F i → F i − ) are minimal flat coversfor all i ≥ Definition 5.9.
The
Matlis dual of a Z n -module M is the Z n -module M ∨ defined by( M ∨ ) a = Hom k ( M − a , k ) , so the homomorphism ( M ∨ ) a → ( M ∨ ) b is transpose to M − b → M − a . Lemma 5.10. ( M ∨ ) ∨ is canonically isomorphic to M for any Z n -finite module M . (cid:3) Remark 5.11.
By the adjunction between Hom and ⊗ , a module is flat if and only itsMatlis dual is injective (see [Mil00, § Z n -finite flat Z n -module is isomorphic to a direct sum of upsetmodules k [ U ] for upsets of the form U = b − τ + N n = b + N n + Z τ . These upsetmodules are the graded translates of localizations of k [ N n ] along faces.5.4. Flange presentations.Definition 5.12.
Fix a Z n -module N .1. A flange presentation of N is a Z n -module morphism ϕ : F → E , with imageisomorphic to N , where F is flat and E is injective in the category of Z n -modules.2. If F and E are expressed as direct sums of indecomposables, then ϕ is based .3. If F and E are finite direct sums of indecomposables, then ϕ is finite .4. If the number of indecomposable summands of F and E are simultaneouslyminimized then ϕ is minimal . Remark 5.13.
The term flange is a portmanteau of flat and injective (i.e., “flainj”)because a flange presentation is the composite of a flat cover and an injective hull.The same notational trick to make fringe presentations effective data structures(Definition 3.17) works on flange presentations.
Definition 5.14.
Fix a based finite flange presentation ϕ : L p F p = F → E = L q E q .A monomial matrix for ϕ is an array of scalar entries ϕ qp whose columns are labeledby the indecomposable flat summands F p and whose rows are labeled by the indecom-posable injective summands E q : F ... F k E · · · E ℓ ϕ · · · ϕ ℓ ... . . . ... ϕ k · · · ϕ kℓ F ⊕ · · · ⊕ F k = F −−−−−−−−−−−−−−−→ E = E ⊕ · · · ⊕ E ℓ . The entries of the matrix ϕ •• correspond to homomorphisms F p → E q . Lemma 5.15. If F = k [ a + Z τ ′ + N n ] is an indecomposable flat Z n -module and E = k [ b + Z τ − N n ] is an indecomposable injective Z n -module, then Hom Z n ( F, E ) = 0 unless ( a + Z τ ′ + N n ) ∩ ( b + Z τ − N n ) = ∅ , in which case Hom Z n ( F, E ) = k .Proof. Corollary 3.11.1. (cid:3)
Definition 5.16.
In the situation of Lemma 5.15, write F (cid:22) E if their degree setshave nonempty intersection: ( a + Z τ ′ + N n ) ∩ ( b + Z τ − N n ) = ∅ . OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 35
Proposition 5.17.
With notation as in Definition 5.14, ϕ pq = 0 unless F p (cid:22) E q .Conversely, if an array of scalars ϕ qp ∈ k with rows labeled by indecomposable flatmodules and columns labeled by indecomposable injectives has ϕ pq = 0 unless F q (cid:22) E q ,then it represents a flange presentation.Proof. Lemma 5.15 and Definition 5.16. (cid:3)
The unnatural hypothesis that a persistence module be finitely generated resultsin data types and structure theory that are asymmetric regarding births as opposedto deaths. In contrast, the notion of flange presentation is self-dual: their dualityinterchanges the roles of births ( F ) and deaths ( E ). Proposition 5.18. A Z n -module N has a finite flange presentation F → E if andonly if the Matlis dual E ∨ → F ∨ is a finite flange presentation of the Matlis dual N ∨ .Proof. Matlis duality is an exact, contravariant functor on Z n -modules that takes thesubcategory of finitely determined Z n -modules to itself (these properties are immediatefrom the definitions), interchanges flat and injective objects therein, and has the prop-erty that the natural map ( N ∨ ) ∨ → N is an isomorphism for finitely determined N (Lemma 5.10); see [Mil00, § (cid:3) Syzygy theorem for Z n -modules.Theorem 5.19. A Z n -module is finitely determined if and only if it admits one, andhence all, of the following:1. a finite flange presentation; or2. a finite flat presentation; or3. a finite injective copresentation; or4. a finite flat resolution; or5. a finite injective resolution; or6. a minimal one of any of the above.Any minimal one of these objects is unique up to noncanonical isomorphism, and theresolutions have length at most n .Proof. The hard work is done by Proposition 5.7. It implies that N is finitely deter-mined ⇔ N ∨ has a minimal injective resolution ⇔ N has a minimal flat resolution oflength at most n , since the Matlis dual of any finitely determined module N is finitelydetermined. Having both a minimal injective resolution and a minimal flat resolution isstronger than having any of items 1–3, minimal or otherwise, so it suffices to show that N is finitely determined if N has any of items 1–3. This follows, using that the categoryof finitely determined modules is abelian as in the proof of Proposition 5.7, from thefact that every indecomposable injective or flat Z n -module is finitely determined. (cid:3) Remark 5.20.
Conditions 1–6 in Theorem 5.19 remain equivalent for R n -modules,with the standard positive cone R n + , assuming that the finite flat and injective modulesin question are finite direct sums of localizations of R n along faces and their Matlisduals. (The equivalence, including minimality, is a consequence of the generator andcogenerator theory over real polyhedral groups [Mil20c].) The equivalent conditionsdo not characterize R n -modules that are pulled back under convex projection fromarbitrary modules over an interval in R n , though, because all sorts of infinite thingscan can happen inside of a box, such as having generators along a curve.6. Homological algebra of poset modules
Indicator resolutions.Definition 6.1.
Fix any poset Q and a Q -module M .1. An upset resolution of M is a complex F • of Q -modules, each a direct sum ofupset submodules of k [ Q ], whose differential F i → F i − decreases homologicaldegrees, has components k [ U ] → k [ U ′ ] that are connected (Definition 3.14), andhas only one nonzero homology H ( F • ) ∼ = M .2. A downset resolution of M is a complex E • of Q -modules, each a direct sumof downset quotient modules of k [ Q ], whose differential E i → E i +1 increasescohomological degrees, has components k [ D ′ ] → k [ D ] that are connected, andhas only one nonzero homology H ( E • ) ∼ = M .An upset or downset resolution is called an indicator resolution if the “up-” or “down-”nature is unspecified. The length of an indicator resolution is the largest (co)homolog-ical degree in which the complex is nonzero. An indicator resolution3. is finite if the number of indicator module summands is finite,4. dominates a constant subdivision or encoding of M if the subdivision or encodingis subordinate to each indicator summand, and5. is semialgebraic , PL , subanalytic , or of class X if Q is a subposet of a real partiallyordered group and the resolution dominates a constant subdivision or encodingof the corresponding type. Definition 6.2.
Monomial matrices for indicator resolutions are defined similarly tothose for fringe presentations in Definition 3.17, except that for the cohomological casethe row and column labels are source and target downsets, respectively, while in thehomological case the row and column labels are target and source upsets, respectively:... D ip ... · · · D i +1 q · · · ϕ pq E i −−−−−−−−−−−−−−−−→ E i +1 and ... U pi ... · · · U qi +1 · · · ϕ pq F i ←−−−−−−−−−−−−−−− F i +1 . OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 37 (Note the switch of source and target from cohomological to homological, so the mapgoes from right to left in the homological case, with decreasing homological indices.)As in Proposition 4.11, pullbacks have transparent monomial matrix interpretations.
Proposition 6.3.
Fix a poset Q and an encoding of a Q -module M by a poset mor-phism π : Q → P and P -module H . Monomial matrices for any indicator resolutionof H pull back to monomial matrices for an indicator resolution of M that dominatesthe encoding by replacing the row and column labels with their preimages under π . (cid:3) Definition 6.4.
Fix any poset Q and a Q -module M .1. An upset presentation of M is an expression of M as the cokernel of a homo-morphism F → F such that each F i is a direct sum of upset modules and everycomponent k [ U ′ ] → k [ U ] of the homomorphism is connected (Definition 3.14).2. A downset copresentation of M is an expression of M as the kernel of a homo-morphism E → E such that each E i is a direct sum of downset modules andevery component k [ D ] → k [ D ′ ] of the homomorphism is connected.These indicator presentations are finite , or dominate a constant subdivision or encodingof M , or are semialgebraic , PL , subanalytic , or of class X as in Definition 6.1. Example 6.5.
In one parameter, the bar [ a, b ) in Example 3.19, has upset presentationPSfrag replacements a b ↓ ֒ → ։ with cokernelisomorphic to the single bar. When there are multiple bars, the bijection from left toright endpoints yields a monomial matrix whose scalar entries again form an identitymatrix, with rows labeled by positive rays having the specified left endpoints (the rayis the whole real line when the left endpoint is −∞ ) and columns labeled by positiverays having the corresponding right endpoints—but with their open or closed naturereversed—as left endpoints (the ray is empty when the specified right endpoint is + ∞ ). Example 6.6. is the cokernel of ← ֓ Lemma 6.7.
The homomorphisms in indicator presentations and resolutions are tame,so their kernels and cokernels are tame. If the indicator modules in question are semi-algebraic, PL, subanalytic, or of class X then the morphisms are, as well. Proof.
Any connected homomorphism among indicator modules is tame—and satisfiesone of the auxiliary hypotheses, if the source and target do—by Definition 4.27, so theconclusion follows from Proposition 4.31. (cid:3)
Example 6.8.
The poset module in Example 2.7 has an upset presentation LR (cid:20) T B − − (cid:21) k [ L ] ⊕ k [ R ] ←−−−−−−−−− k [ T ]in which the monomial matrix has row and column labels • L , the upset generated by the leftmost element; • R , the upset generated by the rightmost element; • T , the upset consisting solely of the maximal element depicted on top; and • B , the upset consisting solely of the maximal element depicted on the bottom.Although the disjoint union of T and B is an upset, and there is a homomorphism ϕ : k [ T ∪ B ] → k [ L ] ⊕ k [ R ] whose cokernel is the desired poset module, there is no wayto arrange for the homomorphism ϕ to be connected. Remark 6.9.
It is tempting to think that a fringe presentation is nothing more thanthe concatenation of the augmentation map of an upset resolution (that is, the sur-jection at the end) with the augmentation map of a downset resolution (that is, theinjection at the beginning), but there is no guarantee that the components F i → E j of the homomorphism thus produced are connected (Definition 3.14). In contrast, aflange presentation (Definition 5.12) is in fact nothing more than the concatenation ofthe augmentation maps of a flat resolution and an injective resolution, since connectedhomomorphisms are forced by Lemma 5.15.6.2. Syzygy theorem for modules over posets.Proposition 6.10.
For any inclusion ι : P → Z of posets and P -module H there is a Z -module ι ∗ H , the pushforward to Z , whose restriction to ι ( P ) is H and is universallyrepelling: ι ∗ H has a canonical map to every Z -module whose restriction to ι ( P ) is H .Proof. At z ∈ Z the pushforward ι ∗ H places the colimit lim −→ H (cid:22) z of the diagram ofvector spaces indexed by the elements of P whose images precede z . The universal pro-perty of colimits implies that ι ∗ H is a Z -module with the desired universal property. (cid:3) Remark 6.11.
With perspectives as in Remark 2.4, the pushforward is a left Kan ex-tension [Cur14, Remark 4.2.9]. This instance is a special case of [Cur19, Example 4.4].
OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 39
Theorem 6.12 (Syzygy theorem) . A module M over a poset Q is tame if and only ifit admits one, and hence all, of the following:1. a finite constant subdivision of Q subordinate to M ; or2. a finite poset encoding subordinate to M ; or3. a finite fringe presentation; or4. a finite upset presentation; or5. a finite downset copresentation; or6. a finite upset resolution; or7. a finite downset resolution; or8. any of the above dominating any given finite encoding; or9. a finite encoding subordinate to any given one of items 1–7; or10. a finite constant subdivision subordinate to any given one of items 1–7.The statement remains true over any subposet of a real partially ordered group if “tame”and all occurrences of “finite” are replaced by “semialgebraic”, “PL”, or “class X ”.Moreover, any tame or semialgebraic, PL, or class X morphism M → M ′ lifts to asimilarly well behaved morphism of presentations or resolutions as in parts 3–7. All ofthese results except item 9 hold in the subanalytic case if M has compact support.Proof. Tame is equivalent to item 1 without auxiliary hypotheses by Definition 2.11and with auxiliary hypotheses by Definition 2.15. Tame is equivalent to item 2 byTheorem 4.22.1. With auxiliary hypotheses, 1 ⇒ M , construct a compact such subdivision by keeping thebounded constant regions as they are and taking the union of all unbounded constantregions to get a single unbounded one. The implication 2 ⇒ Q -module M has finite upset and downsetresolutions and (co)presentations, as well as a finite fringe presentation, all dominatingthe given encoding. (As noted in the first paragraph, the fibers of the encoding mor-phism are already a constant subdivision of the relevant type.) The domination takescare of the cases with auxiliary hypotheses by Definitions 3.16.3, 4.7.2, 6.1.5, and 6.4.Fix a Q -module M finitely encoded by a poset morphism π : Q → P and P -module H .The finite poset P has order dimension n for some positive integer n ; as such P has an embedding ι : P ֒ → Z n . The pushforward ι ∗ H (Proposition 6.10) is finitely determined(Definition 5.1; see also Example 4.5) as it is pulled back from any box containing ι ( P ).The desired presentation or resolution is pulled back to Q (via ι ◦ π : Q → Z n ) fromthe corresponding flange, flat, or injective presentation or resolution of ι ∗ H afforded byTheorem 5.19. These pullbacks are finite indicator resolutions of M dominating π byExample 4.10 and Lemma 4.9. The component homomorphisms are connected because,by Corollary 3.11 and Example 3.6 (see Definition 3.5), components of flange presen-tations, flat resolutions, and injective resolutions over Z n are automatically connected.The preceding argument proves the claim about a morphism M → M ′ , as well, since • only one poset morphism is required to encode the morphism M → M ′ ; • the push-pull constructions are functorial; and • morphisms of finitely determined modules can be lifted to the relevant presen-tations and resolutions, since the relevant covers, presentations, and resolutionsare free or injective in the category of finitely determined modules. (cid:3) Remark 6.13.
Comparing Theorems 6.12 and 5.19, what happened to minimality? Itis not clear in what generality minimality can be characterized. The sequel [Mil20c] tothis paper can be seen as a case study for posets arising from abelian groups that areeither finitely generated and free or real vector spaces of finite dimension. The answeris much more nuanced in the real case, obscuring how minimality might generalizebeyond these cases.
Remark 6.14.
In the situation of the proof of Theorem 6.12, composing two applica-tions of Proposition 4.11—one for the encoding π : Q → P and one for the embedding ι : P ֒ → Z n —yields a monomial matrix for a fringe presentation of M directly from amonomial matrix for a flange presentation. Remark 6.15.
Lesnick and Wright consider R n -modules [LW15, §
2] in finitely pre-sented cases. As they indicate, homological algebra of such R n -modules is no differentthan finitely generated Z n -modules. This can be seen by finite encoding: any finiteposet in R n is embeddable in Z n , because a product of finite chains is all that is needed.6.3. Syzygy theorem for complexes of modules.
Theorem 6.12 is stated for individual modules, but the proof works just as well forcomplexes, in a sense recorded here for reference in the proof of a version in the languageof derived categories of constructible sheaves [Mil20b, Theorem 4.5].
Definition 6.16.
Fix a complex M • of modules over a poset Q .1. M • is tame if its modules and morphisms are tame (Definitions 2.11 and 4.27).2. A constant subdivision or poset encoding is subordinate to M • if it is subordinateto all of the modules and morphisms therein, and in that case M • is said to dominate the subdivision or encoding. OMOLOGICAL ALGEBRA OF MODULES OVER POSETS 41
3. An upset resolution of M • is a complex of Q -modules in which each F i is a directsum of upset modules and the components k [ U ] → k [ U ′ ] are connected, with ahomomorphism F • → M • of complexes inducing an isomorphism on homology.4. A downset resolution of M • is a complex of Q -modules in which each E i is a directsum of downset modules and the components k [ D ] → k [ D ′ ] are connected, witha homomorphism M • → E • of complexes inducing an isomorphism on homology.These resolutions are finite , or dominate a constant subdivision or encoding, or are semialgebraic , PL , subanalytic , or of class X as in Definition 6.1. Theorem 6.17 (Syzygy theorem for complexes) . Theorem 6.12 holds verbatim for abounded complex M • in place of the module M as long as items 3, 4, and 5 are ignored.Proof. As already noted, the proof is the same. It bears mentioning that finite in-jective and flat resolutions of complexes exist in the category of finitely determined Z n -modules because finite injective resolutions do (Proposition 5.7): any of the stan-dard constructions that produce injective resolutions of complexes given that moduleshave injective resolutions works in this setting, and then Matlis duality (Definition 5.9)produces finite flat resolutions (see Remark 5.11). (cid:3) References [AD80] Silvana Abeasis and Alberto Del Fra,
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