Essential graded algebra over polynomial rings with real exponents
aa r X i v : . [ m a t h . A C ] A ug ESSENTIAL GRADED ALGEBRA OVERPOLYNOMIAL RINGS WITH REAL EXPONENTS
EZRA MILLER
Abstract.
The geometric and algebraic theory of monomial ideals and multigradedmodules is initiated over real-exponent polynomial rings and, more generally, monoidalgebras for real polyhedral cones. The main results include the generalization ofNakayama’s lemma; complete theories of minimal and dense primary, secondary, andirreducible decomposition, including associated and attached faces; socles and tops;minimality and density for downset hulls, upset covers, and fringe presentations;Matlis duality; and geometric analysis of staircases. Modules that are semialgebraic orpiecewise-linear (PL) have the relevant property preserved by functorial constructionsas well as by minimal primary and secondary decompositions. And when the modulesin question are subquotients of the group itself, such as monomial ideals and quotientsmodulo them, minimal primary and secondary decompositions are canonical, as areirreducible decompositions up to the new real-exponent notion of density.
Contents
1. Introduction 2Overview 2Acknowledgements 31.1. Real exponent issues 41.2. Nakayama’s lemma and primary decomposition 51.3. Socles, cogenerators, and staircases 72. Algebra over partially ordered groups 102.1. Real and discrete polyhedral groups 102.2. Tame modules and morphisms 112.3. Localization and restriction 132.4. Support and primary decomposition 142.5. Matlis duality 153. Geometry of real staircases 173.1. Tangent cones of downsets 173.2. Upper boundary functors 194. Socles and cogenerators 214.1. Closed socles and closed cogenerator functors 224.2. Socles and cogenerator functors 23
Date : 9 August 2020.2020
Mathematics Subject Classification.
Primary: 05E40, 13C05, 13C70, 13A02, 06F05, 06F20,20M25, 13F55, 13J99, 06A11, 06B15, 55N31, 13P25, 62R40, 14P10, 52B99, 13F20, 13D05, 13D02,13E99, 20M14, 06B35, 22A25, 13F99, 13F70, 13P99; Secondary: 62R01, 68W30, 06A07.
Introduction
Overview.
Little is known about the algebra of rings of polynomials whose exponentsare allowed to be nonnegative real numbers instead of integers. The extreme failureof the noetherian condition—ideals can be uncountably generated—and the nontrivialtopology on the set of exponents present daunting technical difficulties. The smallamount of existing literature proceeds by restricting the study to monomial ideals thatare finitely generated, in an appropriate sense [ISW13, ASW15], or to multigradedmodules that are finitely presented [Les15]. Other work can be viewed as touching onthe continuous nature of the exponent set via nondiscrete-monoid algebras [ACHZ07].But the general behavior of modules over real-exponent polynomial rings remains wideopen, even in the special case of monomial ideals. The issue has risen to prominenceparticularly because modules over the real-exponent polynomial ring emerge in appliedtopology [CZ09] (see also [Mil15]), where the focus is on multigraded modules.This paper breaks ground on the earnest study of modules over real-exponent polyno-mial rings, including the usual setting where exponents lie in a right-angled nonnegative
SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 3 orthant but also real analogues of affine semigroup rings, with arbitrary pointed polyhe-dral cones of allowed exponents. This first step of the investigation concerns monomialideals and multigraded modules, which are general enough to exhibit the starkly dif-ferent behavior resulting from continuous exponents and deviation from noetherianitybut have enough combinatorial structure to allow complete treatment of basic theory,such as primary decomposition, Nakayama’s lemma, and minimal presentations.The algebraic development hinges on a number of foundations whose elementaryversions for finitely generated or noetherian modules fail when straightforwardly gen-eralized to real exponents but nonetheless admit fully functioning analogues whenappropriately enhanced. Most importantly, detecting • an injective homomorphism of modules by checking at all associated primes or • a surjective homomorphism by Nakayama’s lemma (Krull–Azumaya theorem)falter at the outset: modules need not contain copies of quotients by prime ideals, sothe notion of associated prime requires serious thought; and dually, modules—even assimple as monomial ideals—do not have minimal generators [ISW13], so considerationssurrounding Nakayama’s lemma require just as much attention.The solutions developed here to tackle these problems construct a topological frame-work for concepts of minimality, in the form of dense generator and cogenerator func-tors. For example, with definitions made properly, arbitrary real-exponent monomialideals have canonical monomial primary decompositions that are minimal in a strongsense, generalizing the situation for polynomial and other affine semigroup rings. Andthese decompositions are similarly derived from canonical irreducible decompositions.But the notion of “irredundant” for irreducible decomposition must be revised: compo-nents can be omitted as long as those that remain are dense in the sense developed here.Density engages with continuity of exponent sets in a way that gives hope of beingable to lift the lessons learned here for monomial ideals and multigraded modules toarbitrary ideals and modules. In the meantime, the results here generalize classicalstatements about monomial ideals to the much harder and uncharted context of realexponents. And they are made more important by virtue of the central role of realmultigraded algebra in the rapidly developing field of topological data analysis, wheremathemtical foundations for the real exponent case is sorely lacking from the theoryof persistent homology with multiple real parameters [CZ09]. Acknowledgements.
Justin Curry provided feedback after listening for hours aboutface posets infinitesimally near real persistence parameters; he enhanced the functorialviewpoint and provided references as well as insight on topics from sheaf theory to realalgebraic geometry. Ashleigh Thomas played a crucial role in the genesis of this alge-braic theory of real multipersistence, and continues to be a collaborator. Ville Puuskaread an inchoate version of this manuscript [Mil17, § EZRA MILLER
Real exponent issues.Example 1.1.
In the real-exponent polynomial ring k [ R n + ], a monomial ideal is an idealgenerated by monomials: I = h x a | a ∈ A i for some A ⊆ R n + , where x a = x a · · · x a n n .For example, the maximal graded ideal m = h x b , . . . , x b n n | b i > i = 1 , . . . , n i hasthe exponent setPSfrag replacements xy which is the nonnegative orthant with the origin missing. This ideal is not finitelygenerated, although it is countably generated by using any sequence of strictly positivevectors b = ( b , . . . , b n ) converging to . This ideal also has no minimal generatingset, because deleting any finite subset of a sequence converging to still results in asequence converging to . This phenomenon was observed in [ISW13]. Example 1.2.
The monomial ideal I = h x a | a + · · · + a n = 1 i ⊆ k [ R n + ] has exponentset depicted on the left:PSfrag replacements xy PSfrag replacements xy This ideal is uncountably generated, with the given generators forming the uniqueminimal monomial generating set. The quotient module M = k [ R n + ] /I , depicted onthe right, has open upper boundary. This module would appear to be primary tothe maximal ideal m , but as m d = m for all positive integers d , the module M isnot annihilated by any power of m . Worse, M has no elements annihilated by themaximal ideal: M has no simple submodules, so m is not an associated prime in theusual sense of M containing a copy of k = k [ R n + ] / m . In that usual sense, the setof such copies, namely the socle Hom( k , M ), would be an essential submodule of M ,which by definition intersects every nonzero submodule of M nontrivially. But in thispicture the Hom vanishes. Moreover, an essential submodule requires containing astrip of locally positive width near the upper boundary, but the intersection of allessential submodules is 0. SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 5
Question 1.3.
In Example 1.1, what should Nakayama’s lemma say?
Question 1.4.
In Example 1.2, what should the statement, “A homomorphism M → N is injective if and only if M p ֒ → N p is injective for all associated primes p of M ” say? These questions are precisely dual to each other when considered from a judicious angle.1.2.
Nakayama’s lemma and primary decomposition.
The first answers to Questions 1.3 and 1.4 are two major contributions of this paper, • Theorem 12.3 on detecting surjectivity by generator functors (tops),and the Matlis dual (Section 2.5) from which it is deduced, • Theorem 6.7 on detecting injectivity by cogenerator functors (socles).However, the final answers reflect the observations in Examples 1.1 and 1.2 that gen-erating sets and essential submodules in the setting of real exponents maintain thoseproperties when replaced with dense approximations. The precise formulation of thisdeeply non-discrete observation yields two additional major contributions, • the density enhancements in Theorem 12.15 and Theorem 7.27.Flowing from these foundational results, particularly from the socle injectivity cri-terion that is Theorem 6.7, are other staples of commutative algebra: • primary decompositions, minimal in a strong sense (Theorems 9.23 and 9.27); • canonical minimal primary decompositions of monomial ideals (Theorem 9.12); • irreducible decomposition for monomial ideals that are canonical and irredundantup to taking dense subsets (Theorem 9.7 and Corollary 9.8); • duals of all these for secondary decomposition and attached primes (Section 12). Example 1.5.
The upper boundary of the interval in R at the left end of the display k PSfrag replacements xy ֒ → k PSfrag replacements xy ⊕ k PSfrag replacements xy has the vertical axis as an asymptote, whereas the horizontal axis is exactly parallel tothe positive horizontal end of the upper boundary. The corresponding interval modulehas the indicated canonical minimal primary decomposition by Theorem 9.12.The minimality in Theorem 9.27 is a requirement that the socle of the module shouldmap isomorphically to the direct sum of the socles of the quotients modulo its primarycomponents (Definition 9.25). This isomorphism is stronger than usually proposedin noetherian commutative algebra. When applied to an injective hull or irreducible EZRA MILLER decomposition in noetherian situations, socle-minimality as in Definition 9.25 is equiv-alent to there being a minimal number of indecomposable summands. In contrast,minimal primary decompositions in noetherian commutative algebra do not requiresocle-minimality in any sense; they stipulate only minimal numbers of summands,with no conditions on socles. This has unfortunate consequences: even in noetheriansettings, different choices of primary components for an embedded (i.e., nonminimal)prime can strictly contain one another, for example. Requiring socle-minimality as inDefinition 9.25 recovers a modicum of uniqueness over arbitrary noetherian rings, assocles are functorial even if primary components themselves need not be. This tackis more commonly taken in combinatorial commutative algebra, typically involvingobjects such as monomial or binomial ideals. In particular, the “witnessed” forms ofminimality for mesoprimary decomposition [KM14, Definition 13.1 and Theorem 13.2]and irreducible decomposition of binomial ideals [KMO16] serve as models for the typeof minimality in primary decompositions considered here.In ordinary noetherian commutative algebra socle-minimal primary decompositionsare anyway automatically produced by the usual existence proof, which leverages thenoetherian hypothesis to create an irreducible decomposition. Indeed, a noetherian pri-mary decomposition is socle-minimal if and only if each primary component is obtainedby gathering some of the components in a minimal irreducible decomposition. Whenreal exponents enter, truly minimal irreducible decompositions are impossible by The-orem 9.7 and Corollary 9.8, which force us to settle for irredundancy up to taking densesubsets. Nonetheless, the primary component formed by gathering all irreducible com-ponents with a given associated face is well defined, regardless of which dense subset ofirreducible components was present. That is how uniqueness of the minimal primarydecomposition in Theorem 9.12 arises even from nonunique irreducible decomposition.Secondary decomposition is lesser known, even to algebraists, than its Matlis dual,primary decomposition, but secondary decomposition has been in the literature fordecades [Kir73, Mac73, Nor72] (see [Sha76, Section 1] for a brief summary of the mainconcepts). The unfamiliarity of secondary decomposition and its related functors isa primary reason why the bulk of the technical development over real exponents iscarried out in terms of cogenerators and socles instead of generators and tops.Minimal primary and dense irreducible decomposition owe their existence to defini-tions tailored to real exponents. These include especially • a definition of associated prime by socle nonvanishing (Definition 9.1) that yields • characterizations of coprimary modules as those with only one associated prime(Proposition 2.29 and Theorem 9.2).These, in turn, rely on the heart of the matter regarding density, which draws, at themost fundamental level, on the topological algebra surrounding real exponents: • the characterization of essential submodules by socle inclusion (Theorem 8.5). SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 7
Socles, cogenerators, and staircases.
The results discussed thus far all rest on the main socle injectivity criterion in Theo-rem 6.7. As such, the entire edifice is built on socles. Identifying the right definitionof socle in Section 4 to account for the departure from discrete exponents is the mostsubtle and difficult aspect of the theory. But the answer turns out to be pretty and,as luck would have it, finite.The problem to be overcome is seen in Example 1.2: the socle of the quotient mod-ule M there should lie along its upper boundary, but the upper boundary is missing: M is zero in the corresponding R n -graded degrees. The solution is to keep track ofthe directions in which limits must be taken to reach the missing boundary points.The finiteness of the answer comes down to the fact that it matters only which of thefinitely many faces of the exponent cone the limits are taken along. The main productof Section 4 is not a theorem, but nonetheless a major contribution, namely • the notion of socle in Definition 4.29.1 as well as • cogenerator and nadir in Definition 4.29.3.Readers from persistent homology should view these as functorializing the notion of“closed or open right endpoint of an interval” and generalizations to more parameters. Example 1.6.
Consider the module k [ y x ] at the left-hand end of Example 1.5. Anypoint along the curved portion of its upper boundary—that is, along the upper bound-ary of the middle illustration—represents a usual (“closed”) socle element of k [ y x ](Definition 4.1), because such an element is annihilated by moving up in any direction,including straight vertically or horizontally, so it yields an injection k ֒ → k [ y x ] in therelevant multigraded degree. In contrast, the horizontal ray in the upper boundaryrepresents a closed socle element of k [ y x ] along the x -axis (Definition 4.15), because it(i) extends infinitely far to the right, so it yields an injection k [ x -axis + ] ֒ → k [ y x ] but(ii) is annihilated upon moving upward in any direction, notably the vertical direction. Example 1.7.
In the right-hand illustration from Example 1.2, the module is 0 at anypoint along the antidiagonal upper boundary line segment. However, any such pointcan be approached within the interior of the triangle from below or from the left:PSfrag replacements xy This open boundary point represents an element in the socle (Definition 4.5). Limitscan be taken along any nonzero face of the cone R to reach this point, but the twominimal such faces, namely the positive x -axis and the positive y -axis, are the twonadirs of this cogenerator (Definition 4.29.3). This is appropriate to the English defintion of socle : the base of a column. It is an accident of history that in illustrations, socles lie along upper boundaries instead of alonglower boundaries at the bottoms of pictures, where tops quite unfortunately reside.
EZRA MILLER
Example 1.8.
As the upper boundary in Example 1.5 is closed, its has no opencogenerators: its socle is closed. So modify the interval there by omitting the horizontalray and leaving the rest of the points as they are. k PSfrag replacements xy localization k PSfrag replacements xy quotient-restriction k PSfrag replacements x y
The missing horizontal boundary ray represents an element in the socle along the x -axis.In more detail, the ray yields a summand of the upper boundary (Definition 3.15)that contains a copy of k [ x -axis + ]. Localizing along the x -axis yields the tranlsation-invariant version in the middle picture, which can be reached by vertical limits from theinterior of the lower half-plane. The quotient-restriction (Definition 2.22), thought of ei-ther as modding out by horizontal translation or by restricting to a vertical slice, yieldsa situation analogous to Example 1.7. The nadir is the face of R along the y -axis.The socle along a face τ of the exponent cone is a module over the quotient Q/ R τ of the ambient real vector space Q modulo its subspace generated by τ , via quotient-restriction (Definition 2.22). That aspect is not novel to real exponents; it is the ana-logue of the socle Hom R ( R/ p , M ) p at a prime p of positive dimension in a ring R beinga module over the local ring R p . What is novel, however, is the set of nadirs, as in theexamples. A nadir of positive dimension indicates an “open” cogenerator of M , whichis not an element of M but an element in its upper boundary δM (Definition 3.15).The upper boundary is a module not merely over a real-exponent polynomial ring, butalso over the face poset of the exponent cone. That is the crucial ingredient entailedby real exponents: every point of the grading group Q gets replaced by an infinitesimalcopy of the face poset of the exponent cone.This treatment of socles highlights the price to pay for real exponents. First, gen-eralizing standard constructions from noetherian commutative algebra demands care.Notably, for instance, localization fails to commute with Hom, materially complicatingproofs; see Remark 4.22, which explains how this failure to commute is not an arti-fact of the proofs but rather an intrinsic facet of the real-exponent theory. Second,socles of modules over real exponents are not submodules, but instead are functoriallymanufactured from auxiliary modules derived from M , namely upper boundary mod-ules δM . Honest submodules must be reconstructed from cogenerators (this is done inSection 8). Finally, it would have been nice to develop module theory over real expo-nents entirely within the language of monoid algebras, but the infinitesimal structureof real-exponent polynomial rings is unavoidably poset-theoretic in nature, the posetbeing the face lattice of the positive cone of exponents. Hence this paper is phrased interms of modules over posets [Mil20a, Mil20b], whose theory is reviewed in Section 2. SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 9
The geometry and combinatorics that rules the construction of socles, especially theentrance of face posets, is that of downsets and their boundaries, otherwise known as staircases [Mil02, § • the upper boundary functors (Definition 3.15), • what it means to divide an upper boundary element (Definition 3.19), and • computations of downset upper boundaries (Lemma 3.20 and Proposition 3.21)translate topological limits in partially ordered real vector spaces into algebraic colimitson modules. Taking ordinary socles of the upper boundary of a module M , to get amodule over the real exponent ring and the face poset of the exponent cone, completesthe detection of missing boundary points that the ordinary socle of M itself misses.Section 5 assures that functorial constructions surrounding socles preserve additionalsemialgebraic or piecewise linear structure when they are present in the input. To wit, • Theorem 5.2 says that left-exact functors with predictable actions on quotientsmodulo monomial ideals preserve additional geometric structure, and • Theorem 5.13 verifies the hypotheses to draw this conclusion for the cogeneratorfunctors, which take socles.This conclusion is reasonable, because socles take each downset to a well behaved subsetof its boundary (Lemma 3.20 and Proposition 3.21). Preserving additional geometricstructure is particularly crucial for algorithms: it is hopeless for a computer to manip-ulate an arbitrary real-exponent monomial ideal, for its staircase could be missing aCantor set or some more arbitrary, unfathomable antichain. Multigraded modules thatarise in practice—such as from persistent homology—come from finite, computable pro-cedures. Constraints from linear inequalities or comparisons among (squared) distancesbetween points or other simple geometric objects yield PL or semialgebraic modules.The entire theory for tame modules over real exponents has a simpler analogue overaffine semigroup rings. It is barely new, being based on more elementary foundations,but it is worth phrasing precisely and collecting the results for the record (Section 10).The dual discrete theory surrounding generator functors is interspersed with thecorresponding real-exponent theory of tops in Sections 11–12, which is Matlis dualityapplied to earlier sections; see especially • Theorem 11.31 on socle and top duality over real-exponent polynomials, and • Theorem 11.22 on closed socle and top duality over any partially ordered group.To locate results for affine semigroup rings, look for results over arbitrary partiallyordered groups, such as Theorem 11.22, or look for the keywords “discrete polyhedral group”; these indicate the same context as “affine semigroup” (Example 2.3) but referto modules over posets, which is the language adopted (of necessity) for real exponents.The paper closes with its final major results, in a discussion (Section 13) of • minimal presentations, including downset, upset, and fringe presentations; and • resolutions, including conjectures about minimal lengths of resolutions as wellas lengths of minimal resolutions along the lines of the Hilbert syzygy theorem.2. Algebra over partially ordered groups
The algebra surrounding localization, support, primary decomposition, and Matlisduality works over a broad class of partially ordered abelian groups with finitely manyfaces, appropriately defined [Mil20b], and sometimes in more genereality. The mainsettings of this paper restrict primarily to the continuous case of real vector spaces butalso secondarily the discrete case of finitely generated free abelian groups. Nonetheless,since some of the surrounding algebra works for all partially ordered abelian groups,this section reviews the basic setup, always indicating the allowed generality. Forreference, the definitions and claims in Sections 2.1, 2.2, and the start of 2.3 are takenfrom [Mil20a, § § § § § § § Real and discrete polyhedral groups.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 posets of interest in the paper are the following, primarily the real case in Exam-ple 2.3, although some results are naturally stated in the generality of Definition 2.2. Definition 2.2.
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 + . Example 2.3. A real polyhedral group Q is a real vector space of finite dimensionpartially ordered so that its positive cone Q + is an intersection of finitely many closedhalf-spaces. The notation R n and especially R n + is reserved for the case where the posi-tive cone is the nonnegative orthant, so the partial order is componentwise comparison. SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 11
Example 2.4. A discrete polyhedral group is a finitely generated free abelian grouppartially ordered so that its positive cone is a finitely generated submonoid. (Equiva-lently a discrete polyhedral group is the Grothendieck group of an affine semigroup withtrivial unit group.) The notation Z n is reserved for the special case where the positivecone is the nonnegative orthant N n , so the partial order is componentwise comparison. Remark 2.5.
Examples 2.3 and 2.4 are instances of the class of polyhedral partiallyordered groups , introduced in [Mil20b, Definition], which have finitely many faces:submonoids of the positive cone whose complements are ideals of the positive cone.
Example 2.6.
Fix a poset Q . The vector space k [ Q ] = L q ∈ Q k that assigns k to everypoint of 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 + Q + = U , when Q is a partially ordered group)determines an indicator submodule or upset module k [ U ] ⊆ k [ Q ];2. dually, a downset (also called an order ideal ) D ⊆ Q , meaning a subset closedunder going downward in Q (so D − Q + = D , when Q is a partially ordered group)determines an indicator quotient module or downset module k [ Q ] ։ k [ D ]; and3. an interval I ⊆ Q , meaning the intersection of an upset and a downset, deter-mines an interval module k [ I ] that is a subquotient of k [ Q ]: if I = U ∩ D then k [ I ] ֒ → k [ D ] and k [ U ] ։ k [ I ], since I is an upset in D and a downset in U . Remark 2.7.
For polyhedral groups the language of Q -modules is equivalent to that of Q -graded k [ Q + ]-modules. Indeed, a module over any partially ordered abelian group Q is the same thing as a Q -graded module over the monoid algebra k [ Q + ] of the positivecone [Mil20b, Lemma 2.6]. When Q = Z n and Q + = N n , the relevant monoid algebrais the polynomial ring k [ N n ] = k [ x ], where x = x , . . . , x n is a sequence of n variables.This is the classical case; see [MS05, § Q = R n and Q + = R n + ,the relevant monoid algebra is the real-exponent polynomial ring k [ R n + ], whose elementsare polynomials in x = x , . . . , x n with exponents that are nonnegative real numbers.2.2. Tame modules and morphisms.Definition 2.8. A constant subdivision of a poset Q subordinate to a Q -module 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 . Definition 2.9.
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 . Definition 2.10.
Fix a subposet Q of a partially ordered real vector space (e.g., a realpolyhedral 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.A module over Q is semialgebraic or PL if Q + is and the module is tamed by asubordinate finite constant subdivision of the corresponding type. Definition 2.11.
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 . Definition 2.12.
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 or PL if the constant subdivision of Q formed by the fibers of π [Mil20a, Theorem 4.22] is of the corresponding type (Definition 2.10). Definition 2.13.
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 .1. This constant subdivision is subordinate to the morphism ϕ .2. The morphism ϕ dominates a constant subdivision or poset encoding if thesubdivision or encoding is subordinate to ϕ .3. The morphism ϕ is semialgebraic or PL if it dominates a constant subdivisionof the corresponding type. Definition 2.14.
The category of tame modules is the subcategory of Q -modules whoseobjects are the tame modules and whose morphisms are the tame homomorphisms.The subcategories of semialgebraic modules and PL modules have the correspondinglyrestricted objects and tame morphisms.
Proposition 2.15 ([Mil20a, Proposition 4.31]) . Over any poset Q , the category oftame Q -modules is abelian. If Q is a subposet of a partially ordered real vector space,then the categories of semialgebraic and PL modules are abelian. Since socles involve essential submodules, which are divorced from generators, thetheory can often get by with less than tameness, which requires finite upset coversand downset hulls. In the pictures, only phenomena near upper boundaries matter forsocles, not anything near lower boundaries; see [Mil20b, Remark 5.6] for discussion.
SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 13
Definition 2.16. A downset hull of a module M over an arbitrary poset is an injection M ֒ → L j ∈ J E j with each E j being a downset module. The hull is finite if J is finite.The module M is downset-finite if it admits a finite downset hull.2.3. Localization and restriction.Definition 2.17.
Fix a face τ of a partially ordered group Q . The localization of a Q -module M along τ is the tensor product M τ = M ⊗ k [ Q + ] k [ Q + + Z τ ] , viewing M as a Q -graded k [ Q + ]-module. Example 2.18.
The localization D τ of a downset D is the downset with k [ D ] τ = k [ D τ ].The remainder of Section 2.3 is new in this generality, although Z n -graded versionsdate back to [Mil98, §
4] and [Mil00, § Definition 2.19.
For a partially ordered abelian group Q and a face τ of Q + , write Q/ Z τ for the quotient of Q modulo the subgroup generated by τ . If Q is a realpolyhedral group then write Q/ R τ = Q/ Z τ . Remark 2.20.
The image Q + / Z τ of Q + in Q/ Z τ is a submonoid that generates Q/ Z τ ,but Q + / Z τ can have units, so Q/ Z τ need not be partially ordered in a natural way.However, if Q is a real polyhedral group then the group of units (lineality space) of thecone Q + + R τ is just R τ itself, because Q + is pointed, so Q/ R τ is a real polyhedralgroup whose positive cone ( Q/ R τ ) + = Q + / R τ is the image of Q + . Similar reasoningapplies to the intersection of the real polyhedral situation with any subgroup of Q ; thisincludes the case of normal affine semigroups, where the subgroup of Q is discrete. Lemma 2.21.
The subgroup Z τ ⊆ Q of a partially ordered group Q acts freely on thelocalization M τ of any Q -module M along a face τ . Consequently, if I τ ⊆ k [ Q + ] is theaugmentation ideal h m − | m ∈ k [ τ ] is a monomial i , then the Q/ Z τ -graded module M/τ = M/I τ M over the monoid algebra k [ Q + / Z τ ] satisfies M τ ∼ = M a e a ( M/τ ) e a . Proof.
The monomials of k [ Q + ] corresponding to elements of τ are units on M τ actingas translations along τ . Since the augmentation ideal sets every monomial equal to 1,the quotient M → M/τ factors through M τ . (cid:3) Definition 2.22.
The k [ Q/ Z τ ]-module M/τ in Lemma 2.21 is the quotient-restriction of M along τ . Remark 2.23.
Over (any subgroup of) a real polyhedral group Q , the functor M τ M/τ has a “section”
M/τ M τ | τ ⊥ , where N | τ ⊥ = L a ∈ τ ⊥ N a is the restriction of N to any linear subspace τ ⊥ complementary to R τ . (When Q + = R n + , a complement iscanonically spanned by the face orthogonal to τ , or equivalently, the unique maximalface of R n + intersecting τ trivially.) The restriction is a module over the real polyhedralgroup τ ⊥ with positive cone ( Q + + R τ ) ∩ R τ ⊥ , which projects isomorphically to thepositive cone of Q/ R τ . Thus the quotient-restriction is both a quotient and a restrictionof M τ . While a section can exist over polyhedral partially ordered groups that are notreal, it need not. For example, when Q = Z and the columns of (cid:2) (cid:3) generate Q + ,taking τ = (cid:10)(cid:2) (cid:3)(cid:11) to be the face along the x -axis yields a quotient monoid Q + / Z τ ∼ = Z / Z ⊕ N with torsion, preventing k [ Q + ] τ k [ Q + ] /τ from having a section to anycategory of modules over a subgroup of Q . Lemma 2.24.
The quotient-restriction functors M M/τ are exact.Proof.
Localizing along τ is exact because the localization k [ Q + + Z τ ] of k [ Q + ] is flatas a k [ Q + ]-module. The exactness of the functor that takes each k [ Q + / Z τ ]-module M τ to M/τ can be checked on each Q/ Z τ -degree individually. (cid:3) Support and primary decomposition.Definition 2.25.
Fix a face τ of a partially ordered group Q . The submodule of M globally supported on τ isΓ τ M = \ τ ′ τ (cid:0) ker( M → M τ ′ ) (cid:1) = ker (cid:0) M → Y τ ′ τ M τ ′ (cid:1) . Fix a Q -module M for a polyhedral partially ordered group Q . The local τ -support of M is the module Γ τ M τ of elements globally supported on τ in the localization M τ , orequivalently [Mil20b, Proposition 4.6] the localization along τ of the submodule of M globally supported on τ . Example 2.26.
The global supports of the interval module for the interval in R on the ... PSfrag replacements xy ··· .. τ = { } . PSfrag replacements xy , τ = x -axisPSfrag replacements xy ··· , .. τ = y -axis . PSfrag replacements xy left-hand side of this display (the same interval as in Example 1.5) are the intervalmodules for the intervals on the right-hand side, each labeled by the face τ on which thesupport is taken. Extending the first two of these global supports modules downward, SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 15 so they become quotients instead of submodules of the original interval module, yieldsthe canonical primary components in Example 1.5.
Definition 2.27.
A module M over a polyhedral partially ordered group is coprimary if for some face τ , the localization map M ֒ → M τ is injective and Γ τ M τ is an essentialsubmodule of M τ , meaning every nonzero submodule of M τ intersects Γ τ M τ nontrivially. Definition 2.28.
Fix a face τ of the positive cone Q + in a partially ordered group Q .A homogeneous element y ∈ M q in a Q -module M is1. τ -persistent if it has nonzero image in M q ′ for all q ′ ∈ q + τ ;2. τ -transient if, for each f ∈ Q + r τ , the image of y vanishes in M q ′ whenever q ′ = q + λf for λ ≫ τ -coprimary if it is τ -persistent and τ -transient. Proposition 2.29 ([Mil20b, Theorem 4.13]) . Fix a face τ of the positive cone Q + in areal or discrete polyhedral partially ordered group Q . A Q -module M is τ -coprimary ifand only if every homogeneous element divides a τ -coprimary element, where y ∈ M q divides y ′ ∈ M q ′ if q (cid:22) q ′ and y has image y ′ under the structure morphism M q → M q ′ . Definition 2.30.
Fix a Q -module M over a polyhedral partially ordered group Q .A primary decomposition of M is an injection M ֒ → L ri =1 M/M i into a direct sum ofcoprimary quotients M/M i , called components of the decomposition. Example 2.31.
When M = k [ I ] in Definition 2.30 is an interval module, a primarydecomposition k [ I ] ֒ → L ri =1 k [ I i ] may also be expressed as a primary decomposition of I itself: I = S I i , where each I i is a coprimary interval in I . That is, I i is an intervalin Q but a downset in the subposet I . See Example 1.5, where the interval on the leftside is the union of the two intervals on the right side.2.5. Matlis duality.Definition 2.32.
A poset Q is self-dual if it is given a poset isomorphism Q −→∼ Q op with its opposite poset. On elements denote this isomorphism by q
7→ − q . Example 2.33.
Inversion makes partially ordered abelian groups self-dual as posets.
Definition 2.34.
Fix a poset Q with opposite poset Q op . The Matlis dual of a Q -module M is the Q op -module M ∨ defined by ( M ∨ ) q = Hom k ( M q , k ). When Q −→∼ Q op is a self-duality, then ( M ∨ ) q = Hom k ( M − q , k ) , so the homomorphism ( M ∨ ) q → ( M ∨ ) q ′ is transpose to M − q ′ → M − q . Example 2.35.
The Matlis dual over a partially ordered group Q is equivalently M ∨ = Hom Q ( M, k [ Q + ] ∨ )where Hom Q ( M, N ) = L q ∈ Q Hom (cid:0)
M, N ( q ) (cid:1) is the direct sum of all degree-preservinghomomorphisms from M to Q -graded translates of N , i.e., N ( q ) a = N a + q . This isproved using the adjuntion between Hom and ⊗ ; see [MS05, Lemma 11.16], notingthat the nature of the grading group is immaterial. And as in [MS05, Lemma 11.16],Hom Q ( M, N ∨ ) = ( M ⊗ Q N ) ∨ . Example 2.36.
It is instructive to compute the Matlis dual of localization along aface τ over a partially ordered abelian group: the Matlis dual of M τ is( M τ ) ∨ = Hom (cid:0) k [ Q + ] τ ⊗ M, k (cid:1) = Hom (cid:0) k [ Q + ] τ , Hom( M, k ) (cid:1) = Hom (cid:0) k [ Q + ] τ , M ∨ (cid:1) , the module of homomorphisms from a localization of k [ Q + ] into M ∨ . The unfamiliarityof this functor is one of the reasons for developing most of the theory in this paper interms of socles and cogenerators instead of tops and generators. Lemma 2.37. ( M ∨ ) ∨ is canonically isomorphic to M if M is Q -finite (Def. 2.9.2). (cid:3) Remark 2.38.
Every Q -finite injective module over a discrete polyhedral group Q is, by [MS05, Theorem 11.30], isomorphic to a direct sum of downset modules k [ D ]for downsets of the form D = a + τ − Q + , said to be cogenerated by a along theface τ . Taking Matlis duals, every Q -finite flat module over a discrete polyhedralgroup Q is isomorphic to a direct sum of upset modules k [ U ] for upsets of the form U = b + Z τ + Q + . These upset modules are the graded translates of localizationsof k [ Q + ] along faces. Lemma 2.39.
Hom (cid:0) k [ Q + ] τ , ( − ) ∨ (cid:1) is exact for all faces τ of any partially orderedgroup Q . Consequently, Hom( k [ Q + ] τ , − ) is exact on the category of Q -finite modules.Proof. Localization is exact and so is Matlis duality, so the first sentence followsfrom Example 2.36. The consequence comes from Lemma 2.37: for Q -finite modules,Hom( k [ Q + ] τ , − ) is the composite ( − ) ∨ followed by Hom (cid:0) k [ Q + ] τ , ( − ) ∨ (cid:1) . (cid:3) Remark 2.40.
What really drives the lemma is the observation that while the oppositenotion to injective is projective (reverse all of the arrows in the definition), the adjointnotion to injective is flat. That is, a module is flat if and only if its Matlis dual isinjective. This is an instance of a rather general phenomenon that can be phrased interms of a monoidal abelian category C possessing a Matlis object E for a Matlis dualpair of subcategories A and B such that Hom( − , E ) restricts to exact contravariantfunctors A → B and
B → A that are inverse to one another. The idea is to set M ∨ = Hom( M, E ), the
Matlis dual of any object M of C , and define an object of C tobe B -flat if F ⊗ − is an exact functor on B . Then an object F of A is B -flat if and onlyif Hom( F, − ) is an exact functor on A . Examples of this situation include artinian and SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 17 noetherian modules over a complete local ring; modules of finite length over any localring (in both cases E = E ( R/ m ) is the injective hull of the residue field); and of course Q -finite modules over a partially ordered abelian group Q . The latter two examplesfeature a single Matlis self-dual subcategory. Example 2.41.
It is important to note that Hom( k [ Q + ] τ , − ) is not exact on thecategory of all—that is, not necessarily Q -finite—modules over a partially orderedgroup Q . Indeed, let F ։ k [ Q ] be any free cover of the localization of k [ Q + ] along themaximal face. (When Q + = N , this writes the module k [ Z ] of Laurent polynomials asa quotient of a graded free module F over the ordinary polynomial ring k [ N ] in onevariable.) Then k [ Q ] = k [ Q + ] τ for τ = Q itself, and applying Hom( k [ Q ] , − ) to thesurjection F ։ k [ Q ] yields the homomorphism 0 → k [ Q ], which is not surjective.3. Geometry of real staircases
The difference between ordinary noetherian commutative algebra and algebra overreal polyhedral groups begins with the local geometry of downsets near their boundaries(Section 3.1) and the functorial view of this geometry (Section 3.2).3.1.
Tangent cones of downsets.Definition 3.1.
The tangent cone T a D of a downset D in a real polyhedral group Q (Example 2.3) at a point a ∈ Q is the set of vectors v ∈ − Q + such that a + ε v ∈ D for all sufficiently small (hence all) ε > Remark 3.2.
Since the real number ε in Definition 3.1 is strictly positive, the vector v = lies in T a D if and only if a itself lies in D , and in that case T a D = − Q + . Example 3.3.
The tangent cone defined here is not the tangent cone of D as a stratifiedspace, because the cone here only considers vectors in − Q + . A specific simple exampleto see the difference is the closed half-plane D beneath the line y = − x in R , wherethe usual tangent cone at any point along the boundary line is the half-plane, whereas T a D = − R . Furthermore, T a D can be nonempty for a point a in the boundary of D even if a does not lie in D itself. For an example of that, take D ◦ to be the interiorof D ; then T a D ◦ = − R r { } for any a on the boundary line.The most important conclusion concerning tangent cones at points of downsets,Proposition 3.10, says that such cones are certain unions of relative interiors of faces.Some definitions and preliminary results are required. Definition 3.4.
Fix a real polyhedral group Q .1. For any face σ of the positive cone Q + , write σ ◦ for the relative interior of σ .2. For any set ∇ of faces of Q + , write Q ∇ = S σ ∈∇ σ ◦ , the cone of shape ∇ .3. A cocomplex in Q + is an upset in the face poset F Q of Q + , where σ (cid:22) τ if σ ⊆ τ . Example 3.5.
The cocomplex ∇ σ = { faces σ ′ of Q + | σ ′ ⊇ σ } is the open star of theface σ . It determines the cone Q ∇ σ of shape ∇ σ , which plays an important role. Remark 3.6.
The next proposition is the reason for specializing this section to realpolyhedral groups instead of arbitrary polyhedral partially ordered groups, where limitsmight not be meaningful. For example, although limits make formal sense in the integerlattice Z n with the usual discrete topology, it is impossible for a sequence of points in therelative interior of a face to converge to the origin of the face. This quantum separationhas genuine finiteness consequences for the algebra of Z n -modules that usually do nothold for R n -modules. Proposition 3.7. If { a k } k ∈ N is any sequence converging to a , then S ∞ k =0 ( a k − Q + ) ⊇ a − Q ◦ + . If the sequence is contained in a − σ ◦ , then the union equals a − Q ∇ σ .Proof. For each point b ∈ a − Q ◦ + , every linear function ℓ : R n → R that is nonnegativeon Q + eventually takes values on the sequence that are bigger than ℓ ( b ); thus b lies inthe union as claimed. When the sequence is contained in a − σ ◦ , the union is containedin a − σ ◦ − Q + by hypothesis, but the union contains a − σ ◦ by the first claim appliedwith σ in place of Q + . The union therefore equals a − σ ◦ − Q + because it is a downset.The next lemma completes the proof. (cid:3) Lemma 3.8. If σ is any face of the positive cone Q + then σ ◦ + Q + = Q ∇ σ .Proof. Fix f + b ∈ σ ◦ + Q + . If ℓ ( f + b ) = 0 for some linear function ℓ : R n → R thatis nonnegative on Q + , then ℓ ( f ) = 0, too. Therefore the support face of f + b (thesmallest face in which it lies) contains σ . On the other hand, suppose b lies interior tosome face of Q + that contains σ . Then pick any f ∈ σ ◦ . If ℓ ( b ) = 0 then also ℓ ( f ) = 0,because the support face of b contains σ . But if ℓ ( b ) >
0, then ℓ ( b ) > ℓ ( ε f ) forany sufficiently small positive ε . As Q + is an intersection of only finitely many closedhalf-spaces, a single ε works for all relevant ℓ , and then b = ε f +( b − ε f ) ∈ σ ◦ + Q + . (cid:3) For Q + = R n + the following is essentially [MMc15, Lemma 5.1]. Corollary 3.9. If D ⊆ Q is a downset in a real polyhedral group, then a lies in theclosure D if and only if D contains the interior a − Q ◦ + of the negative cone with origin a . Proposition 3.10. If a ∈ D for a downset D , then T a D = − Q ∇ is the negative coneof shape ∇ for some cocomplex ∇ in Q + . In this case ∇ = ∇ a D is the shape of D at a .Proof. The result is true when n = 1 because there are only three possibilities for a ∈ R : either a ∈ D , in which case T a D = − R + = Q ∇ for ∇ = F Q (Remark 3.2); or a D but a lies in the closure of D , in which case T a D = Q ∇ for ∇ = { Q ◦ + } ⊆ F Q ; or a is separated from D by a nonzero distance, in which case T a D = Q {} is empty.Write D σ for the intersection of D with the a -translate of the linear span of σ : D σ = D ∩ ( a + R σ ) . SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 19 If σ $ Q + , then T a D σ = σ ∇ for some upset ∇ ⊆ F σ by induction on the dimensionof σ . In actuality, only the case dim σ = n − F σ fordim σ = n − F Q : only the open maximal face Q ◦ + itself lies outside oftheir union, and that case is dealt with by Corollary 3.9. (cid:3) Upper boundary functors.Definition 3.11.
For a module M over a real polyhedral group Q , a face σ of Q + ,and a degree a ∈ Q , the upper boundary atop σ at a in M is the vector space( δ σ M ) a = M a − σ = lim −→ a ′ ∈ a − σ ◦ M a ′ . Lemma 3.12.
The functor M δ σ M = L a ∈ Q ( δ σ M ) a is exact.Proof. Direct limits are exact in categories of vector spaces (or modules over rings). (cid:3)
Lemma 3.13.
The structure homomorphisms of M as a Q -module induce naturalhomomorphisms M a − σ → M b − τ for a (cid:22) b in Q and faces σ ⊇ τ of Q + .Proof. The natural homomorphisms come from the universal property of colimits. Firsta natural homomorphism M a − σ → M b − σ is induced by the composite homomorphisms M c → M c + b − a → M b − σ for c ∈ a − σ ◦ because adding b − a takes a − σ ◦ to b − σ ◦ .For M b − σ → M b − τ the argument is similar, except that existence of natural homomor-phisms M c → M b − τ for c ∈ b − σ ◦ requires Proposition 3.7 and Lemma 3.8. (cid:3) Remark 3.14.
The face poset F Q of the positive cone Q + can be made into a com-mutative monoid in which faces σ and τ of Q + have sum σ + τ = σ ∩ τ. Indeed, these monoid axioms use only that (cid:0) F Q , ∩ (cid:1) is a bounded meet semilattice, themonoid unit element being the maximal semilattice element—in this case, Q + itself.When F Q is considered as a monoid in this way, the partial order on it has σ (cid:22) τ if σ ⊇ τ , which is the opposite of the default partial order on the faces of a polyhedralcone. For utmost clarity, and because both monoid partial orders are relevant (seeRemark 11.3), F op Q is written when this monoid partial order is intended. Definition 3.15.
Fix a module M over a real polyhedral group Q and a degree a ∈ Q .The upper boundary functor takes M to the Q × F op Q -module δM whose fiber over a ∈ Q is the F op Q -module ( δM ) a = M σ ∈F Q M a − σ = M σ ∈F Q ( δ σ M ) a . The fiber of δM over σ ∈ F op Q is the upper boundary δ σ M of M atop σ . Example 3.16.
The upper boundary of the triangular module M in Examples 1.2and 1.7 has vector spaces of dimension either 0 or 1 in every graded component indexedby R × F op R . For each a ∈ R , depict the corresponding F op R -module using a solid dotfor a vector space of dimension 1 and no solid dot for a vector space of dimension 0. ThenPSfrag replacements xy R F op R PSfrag replacements xy R F op R yx δM is drawn at left and F op R is drawn at right. The F op R -module at a is drawn with a at the upper-right corner, to convey the idea that one should stand at a and seewhat direct limits result as a is approached along the various faces. When M a = 0,the point a is drawn as an empty dot. Remark 3.17.
Upper boundaries contain the later notion of ephemeral modules[BP19]: an R n module M is ephemeral if its upper boundary δ σ M atop the inte-rior σ of R n + vanishes. This notion is key to the difference between poset moduletheory [Mil20a] and the formulation of persistent homology via constructible sheaves[KS18], as detailed in [Mil20c]. Remark 3.18.
The face of Q + that contains only the origin is an absorbing element:it acts like infinity, in the sense that σ + { } = { } in the monoid F op Q for all faces σ .Adding the absorbing element in the F op Q component therefore induces a natural Q × F op Q -module projection from the upper boundary δM to M . At a degree a ∈ Q ,this projection is M a − σ → M a − = M a . Interestingly, the frontier of a downset D —those points in the topological closure but outside of D —is the set of nonzero degreesof a functor, namely ker( δ σ M → M ) for σ = Q ◦ + . The proof is by Corollary 3.9.There is no natural map M → δ σ M when σ = { } has positive dimension, becausean element of degree a in M comes from elements of δ σ M in degrees less than a .However, that leaves a way for Lemma 3.13 to afford a notion of divisibility of upperboundary elements by elements of M . Definition 3.19.
An element y ∈ M b divides x ∈ ( δ σ M ) a if b ∈ a − Q ∇ σ = a − σ ◦ − Q + (Lemma 3.8) and y x under the natural map M b → M a − σ (Lemma 3.13). Theelement y is said to σ -divide x if, more restrictively, b ∈ a − σ ◦ . Upper boundary functors were introduced in [Mil17].
SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 21
Now come the fundamental calculations of upper boundary functors, in the nextlemma and proposition, that drive all of the results in the rest of the paper.
Lemma 3.20. If σ ∈ F Q and D is a downset in Q then δ σ k [ D ] = k [ δ σ D ] , where δ σ D = [ x ∈ Q D ∩ ( x + R σ ) = D ∪ [ x ∈ ∂D D ∩ ( x + R σ ) It suffices to take the middle union over x in any subspace complement to R σ .Proof. For the second displayed equality, observe that the middle union contains theright-hand union because the middle one contains D . For the other containment, if x + R σ contains no boundary point of D , then D ∩ ( x + R σ ) = D ∩ ( x + R σ ) is alreadyclosed, so the contribution of D ∩ ( x + R σ ) to the middle union is contained in D .For the other equality, δ σ k [ D ] is nonzero in degree a if and only if a − σ ◦ ⊆ D .That condition is equivalent to a − σ ◦ ⊆ D ∩ ( a + R σ ) because a − σ ◦ ⊆ a + R σ .Translating D ∩ ( a + R σ ) back by a yields a downset in the real polyhedral group R σ ,with ( R σ ) + = σ , thereby reducing to Corollary 3.9. (cid:3) Proposition 3.21. If σ ∈ F Q and D is a downset in a real polyhedral group then δ σ k [ D ] = k [ δ σ D ] is the indicator quotient for a downset δ σ D that satisfies1. D ⊆ δ σ D ⊆ D and2. δ σ D = { a ∈ D | σ ∈ ∇ a D } .Proof. Item 1 follows from item 2. What remains to show is that δ σ D is a downsetin D characterized by item 2 and that it is semialgebraic if D is.First, σ ∈ ∇ a D means that a − σ ◦ ⊆ D , which immediately implies that a ∈ δ σ D byLemma 3.20. Conversely, suppose a ∈ δ σ D . Lemma 3.20 and Corollary 3.9, the latterapplied to the downset − a + D ∩ ( a + R σ ) in R σ , imply that a − σ ◦ ⊆ D , and hence σ ∈ ∇ a D by definition, proving item 2. Given that a − σ ◦ ⊆ D , Proposition 3.7 yields D ∩ ( a − Q + ) ⊇ a − Q ∇ σ . Consequently, if b ∈ a − Q + then D ∩ ( b − σ ) ⊇ b − σ ◦ ,whence b ∈ δ σ D . Thus δ σ D is a downset. (cid:3) Socles and cogenerators
The main contribution of this section is Definition 4.29, which introduces the no-tions of cogenerator functor and socle along a face with a given nadir. Its concomitantfoundations include ways to decompose the cogenerator functors into continuous anddiscrete parts (Proposition 4.38), interactions with localization (Proposition 4.40), andleft-exactness (Proposition 4.44), along with the crucial calculation of socles in the sim-plest case, namely the indicator module of a single face (Example 4.43). The theory isbuilt step by step, starting with ordinary (closed) socles over arbitrary posets in Sec-tion 4.1 and proceeding through Section 4.4 to cogenerator functors and socles with in-creasing levels of complexity and (necessarily) decreasing freedom regarding the poset.
Closed socles and closed cogenerator functors.
In commutative algebra, the socle of a module over a local ring is the set of elementsannihilated by the maximal ideal. These elements form a vector space over the residuefield k that can alternately be characterized by taking homomorphisms from k into themodule. Either characterization works for modules over partially ordered groups, butonly the latter generalizes readily to modules over arbitrary posets. Note that soclesover face posets of polyhedra occur naturally in the theory over real polyhedral groups. Definition 4.1.
Fix an arbitrary poset P . The skyscraper P -module k p at p ∈ P has k in degree p and 0 in all other degrees. The closed cogenerator functor Hom P ( k , − )takes each P -module M to its closed socle : the P -submodulesoc M = Hom P ( k , M ) = M p ∈ P Hom P ( k p , M ) . When it is important to specify the poset, the notation P -soc is used instead of soc.A closed cogenerator of degree p ∈ P is a nonzero element in (soc M ) p . Remark 4.2.
The bar over “soc” is meant to evoke the notion of closure or “closed”.The bar under Hom is the usual one in multigraded commutative algebra for the directsum of homogeneous homomorphisms of all degrees (see [GW78, Section I.2] or [MS05,Definition 11.14], for example).
Example 4.3.
The closed socle of M consists of the elements that are annihilated bymoving up in any direction, or that have maximal degree. In particular, the intervalmodule k [ I ] for any interval I ⊆ P (Example 2.6.3) has closed soclesoc k [ I ] = k [max I ] , the interval module supported on the set of elements of I that are maximal in I . Lemma 4.4.
The closed cogenerator functor over any poset is left-exact.Proof. A P -module is the same thing as a module over the path algebra of (the Hasse di-agram of) P with relations to impose commutativity, namely equality of the morphisminduced by p < p ′′ and the composite morphism induced by p < p ′ < p ′′ . A P -moduleis also the same as a sheaf on P in which the topology comprises the upsets of P . (See[Yuz81] as well as [Cur14, § P -modules is abelian, so Hom from a fixed source is automatically left-exact. (cid:3) SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 23
Socles and cogenerator functors.Definition 4.5.
The cogenerator functor takes a module over a real polyhedral groupto its socle : soc M = soc δM, which is the closed socle, computed over the poset Q × F op Q (see Remark 3.14) of theupper boundary module of M (Definition 3.15). Remark 4.6.
Notationally, the lack of a bar over “soc ” serves as a visual cue that thefunctor is over a real polyhedral group, as the upper boundary δ is not defined in moregenerality. This visual cue persists throughout the more general notions of socle: barmeans arbitrary poset, and no bar means real polyhedral group. The subscript “ ”refers to the minimal face of the positive cone of the real polyhedral group; to beprecise, it is the τ = special case of Definition 4.29. Lemma 4.7.
The cogenerator functor M soc M is left-exact.Proof. Use exactness of upper boundaries atop σ (Lemma 3.12), exactness of the directsums forming δM from δ σ M , and left-exactness of closed socles (Lemma 4.4). (cid:3) Sometimes is it useful to apply the closed socle functor to δM over Q × F op Q in twosteps, first over one poset and then over the other. These yield the same result. Lemma 4.8.
The functors Q -soc and F op Q -soc commute. In particular, F op Q -soc ( Q -soc δM ) ∼ = soc M ∼ = Q -soc ( F op Q -soc δM ) . Proof.
By taking direct sums over a and σ , this follows from the natural isomorphismsHom F op Q (cid:0) k σ , Hom Q ( k a , − ) (cid:1) ∼ = Hom Q ×F op Q ( k a ,σ , − ) ∼ = Hom Q (cid:0) k a , Hom F op Q ( k σ , − ) (cid:1) . (cid:3) The fundamental examples—indicator quotients for downsets—require a no(ta)tion.
Definition 4.9.
In the situation of Definition 3.4, write ∂ ∇ for the antichain of facesof Q + that are minimal under inclusion in ∇ . Remark 4.10.
The reason for writing ∂ ∇ instead of max or min is that it would be am-biguous either way, since both F Q and F op Q are natural here. Taking the “op” perpsec-tive (Remark 3.14), the F op Q -module k [ ∂ ∇ ] with basis ∂ ∇ resulting from Definition 4.9is really just a F op Q -graded vector space: the antichain condition ensures that every non-identity element of F op Q acts by 0, unless ∂ ∇ = { } , in which case all of F op Q acts by 1.The case of most interest here is ∇ = ∇ a D , the shape of D at a (Proposition 3.10). Example 4.11.
For a downset D in a real polyhedral group, F op Q -soc δ k [ D ] has k [ ∂ ∇ a D ]in each degree a , because δ k [ D ] itself has k [ ∇ a D ] in each degree a by Definition 3.15 andProposition 3.21. Note that δ k [ D ] and F op Q -soc δ k [ D ] are direct sums over faces σ , sincethey are F op Q -modules. What Q -soc then does, for each σ , is find the degrees a ∈ Q maximal among those where σ ∈ ∂ ∇ a D , by Proposition 3.21.2 and Example 4.3.Taking socles in the other order, first Q -soc k [ δ σ D ] asks whether σ ∈ ∇ a D but σ
6∈ ∇ b D for any b ≻ a in Q . That can happen even if σ contains a smaller face where it stillhappens. What F op Q -soc then does is return the smallest faces at a where it happens. Corollary 4.12.
The socle of the indicator quotient k [ D ] for any downset D in a realpolyhedral group Q is nonzero only in degrees lying in the topological boundary ∂D .Proof. By Proposition 3.21.1, δ k [ D ] is a direct sum of indicator quotients. Example 4.3and Proposition 3.10 show that the socle of an indicator quotient over a real polyhedralgroup lies along the boundary of the downset in question. (cid:3) Remark 4.13.
Corollary 4.12 is false for intervals with no interior.4.3.
Closed socles along faces of positive dimension.
The definition of closed socle and closed cogenerator are expressed in terms of Homfunctors analogous to those in Definition 4.1. They are more general in that they occuralong faces of Q , but more restrictive in that Q needs to be a partially ordered groupinstead of an arbitrary poset for the notion of face to make sense. Definition 4.14.
Fix a face τ of a partially ordered group Q . The skyscraper Q -module at a ∈ Q along τ is k [ a + τ ], the subquotient k [ a + Q + ] / k [ a + m τ ] of k [ Q ],where m τ = Q + r τ . SetHom Q (cid:0) k [ τ ] , − (cid:1) = M a ∈ Q Hom Q (cid:0) k [ a + τ ] , − (cid:1) . Definition 4.15.
Fix a partially ordered group Q , a face τ , and a Q -module M .1. The global closed cogenerator functor along τ takes M to its global closed soclealong τ : the k [ Q + / Z τ ]-modulesoc τ M = Hom Q (cid:0) k [ τ ] , M (cid:1) /τ.
2. If Q/ Z τ is partially ordered, then the local closed cogenerator functor along τ takes M to its local closed socle along τ : the Q/ Z τ -modulesoc ( M/τ ) = Hom Q/ Z τ ( k , M/τ ) . Elements of soc (
M/τ ) are identified with elements of
M/τ via ϕ ϕ (1).3. Regard the Q -module Hom Q (cid:0) k [ τ ] , M (cid:1) naturally as contained in M via ϕ ϕ (1).A homogeneous element in this Q -submodule that maps to a nonzero elementof soc τ M is a global closed cogenerator of M along τ . If I ⊆ Q is an inteval,then a closed cogenerator of I is the degree in Q of a closed cogenerator of k [ I ]. SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 25
4. Regard soc (
M/τ ) naturally as contained in
M/τ via ϕ ϕ (1). A nonzerohomogeneous element in soc ( M/τ ) is a local closed cogenerator of M along τ .Assume the default modifier “global” when neither “local” nor “global” is wrtiten. Remark 4.16.
Notationally, a subscript on “soc” serves as a visual cue that the functoris over a partially ordered group, as faces of posets are not defined in more generality.This visual cue persists throughout the more general notions of socle.
Remark 4.17.
The closed cogenerator functor over a partially ordered group is theglobal closed cogenerator functor along the trivial face: soc = soc { } and it equals thelocal cogenerator functor along { } . Remark 4.18.
In looser language, a closed cogenerator of M along τ is an element • annihilated by moving up in any direction outside of τ but that • remains nonzero when pushed up arbitrarily along τ .Equivalently, a closed cogenerator along τ is an element whose annihilator under theaction of Q + on M equals the prime ideal m τ = Q + r τ of the positive cone Q + .Elements like this are sometimes known as “witnesses” in commutative algebra. Example 4.19.
The closed socle along a face τ of the indicator quotient k [ I ] for anyinterval I in a partially ordered group Q with partially ordered quotient Q/ Z τ issoc τ k [ I ] = k [max τ I ] , where max τ I is the image in Q/ Z τ of the set of closed cogenerators of I along τ :max τ I = (cid:8) a ∈ I | ( a + Q + ) ∩ I = a + τ (cid:9) / Z τ. The set of closed cogenerators of a downset D along τ can also be characterized as theelements of D that become maximal in the localization D τ of D (Example 2.18).Every global closed cogenerator yields a local one. Proposition 4.20.
Fix a partially ordered group Q . There is a natural injection soc τ M ֒ → soc ( M/τ ) for any Q -module M if τ is a face with partially ordered quotient Q/ Z τ .Proof. Localizing any homomorphism k [ a + τ ] → M along τ yields a homomorphism k [ a + Z τ ] → M τ , so Hom Q (cid:0) k [ τ ] , M (cid:1) τ is naturally a submodule of Hom Q (cid:0) k [ Z τ ] , M τ (cid:1) .The claim now follows from Lemma 2.24 and the next result. (cid:3) Lemma 4.21. If Q and Q/ Z τ are partially ordered, there is a canonical isomorphism Hom Q (cid:0) k [ Z τ ] , M τ (cid:1) /τ ∼ = Hom Q/ Z τ ( k , M/τ ) . Proof.
Follows from the definitions, using that k [ Z τ ] /τ = k in ( Q/ Z τ )-degree . (cid:3) The following crucial remark highlights the difference between real-graded algebraand integer-graded algebra. It is the source of much of the subtlety in the theorydeveloped in this paper, particularly Sections 4–9.
Remark 4.22.
In contrast with taking support on a face [Mil20b, Proposition 4.6] andalso with socles in commutative algebra over noetherian local or graded rings localiza-tion need not commute with taking closed socles along faces of positive dimension inreal polyhedral groups. In other words, the injection in Proposition 4.20 need not besurjective: there can be local closed cogenerators that do not lift to global ones. Theproblem comes down to the homogeneous prime ideals of the monoid algebra k [ Q + ]not being finitely generated, so the quotient k [ τ ] fails to be finitely presented; it meansthat Hom k [ Q + ] (cid:0) k [ τ ] , − (cid:1) need not commute with A ⊗ k [ Q + ] − , even when A is a flat k [ Q + ]-algebra such as a localization of k [ Q + ]. The context of R n -modules complicates therelation between support on τ and closed cogenerators along τ because the “thickness”of the support can approach 0 without ever quantum jumping all the way there and,importantly, remaining there along an entire translate of τ , as it would be forced tofor a discrete group like Z n . See Example 2.26, for instance, where the support on the y -axis does not contribute any closed socle along the y -axis to the ambient module.This issue is independent of the density phenomenon explored in Section 7; indeed, theinterval in Example 2.26 is closed, so its socle equals its closed socle and is closed. Proposition 4.23.
The global closed cogenerator functor soc τ along any face τ of apartially ordered group is left-exact, as is the local version if Q/ Z τ is partially ordered.Proof. For the global case, Hom Q ( k [ τ ] , − ) is exact because it occurs in the categoryof graded modules over the monoid algebra k [ Q + ], and quotient-restriction is exact byLemma 2.24. For the local case, use exactness of M M/τ again (Lemma 2.24) andleft-exactness of closed socles (Lemma 4.4), the latter applied over Q/ Z τ . (cid:3) Remark 4.24.
Closed socles, without reference to faces, work over arbitrary posetsand are actually used that way in this work (over F op Q , for instance, in Section 4.2). Thatexplains why this separate section on closed socles along faces of positive dimension isrequired, instead of simply doing Section 4.1 in this specificity in the first place.4.4. Socles along faces of positive dimension.Lemma 4.25. If τ is a face of a real polyhedral group Q then the face poset of the quo-tient real polyhedral group Q/ R τ is isomorphic to the open star ∇ τ from Example 3.5by the map ∇ τ → ( Q/ R τ ) + sending σ ∈ ∇ τ to its image σ/τ in Q/ R τ .Proof. See Remark 2.20. (cid:3)
Definition 4.26.
In the situation of Lemma 4.25, endow ∇ τ with the monoid andposet structures from Remark 3.14, so σ (cid:22) σ ′ in ∇ τ if σ ⊇ σ ′ . The upper boundaryfunctor along τ takes M to the Q ×∇ τ -module δ τ M = L σ ∈∇ τ δ στ M = δM/ L σ τ δ σ M . SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 27
The notation is such that δ στ = 0 ⇔ σ ⊇ τ . Definition 4.27.
Fix a partially ordered group Q , a face τ , and an arbitrary commu-tative monoid P . The skyscraper ( Q × P )-module at ( a , σ ) ∈ Q × P along τ is k σ [ a + τ ] = k [ a + τ ] ⊗ k k σ , the right-hand side being a module over the ring k [ Q + ] ⊗ k k [ P ] = k [ Q + × P ] withtensor factors as in Definitions 4.1 and 4.14. SetHom Q × P (cid:0) k [ τ ] , − (cid:1) = M ( a ,σ ) ∈ Q × P Hom Q × P (cid:0) k σ [ a + τ ] , − (cid:1) . Remark 4.28.
When P is trivial, this notation agrees with Definition 4.14, because Q × { } ∼ = Q canonically, so Hom Q ×{ } (cid:0) k [ τ ] , − (cid:1) = Hom Q (cid:0) k [ τ ] , − (cid:1) . Definition 4.29.
Fix a real polyhedral group Q , a face τ , and a Q -module M .1. The global cogenerator functor along τ takes M to its global socle along τ :soc τ M = Hom Q ×∇ τ (cid:0) k [ τ ] , δ τ M (cid:1) /τ. The ∇ τ -graded components of soc τ M are denoted by soc στ M for σ ∈ ∇ τ .2. The local cogenerator functor along τ takes M to its local socle along τ :soc ( M/τ ) = soc δ ( M/τ ) = Hom Q/ R τ ×∇ τ (cid:0) k , δ ( M/τ ) (cid:1) , where the upper boundary is over Q/ R τ and the closed socle is over Q/ R τ × ∇ τ .Elements of soc ( M/τ ) are identified with elements of δ ( M/τ ) via ϕ ϕ (1).3. Regard Hom Q ×∇ τ (cid:0) k [ τ ] , δ τ M (cid:1) as a ( Q × ∇ τ )-submodule of δ τ M via ϕ ϕ (1).A homogeneous element s in this submodule that maps to a nonzero element ofsoc τ M is a global cogenerator of M along τ , and if s ∈ δ στ M then it has nadir σ .If I ⊆ Q is an interval, then a cogenerator of I along τ with nadir σ is the degreein Q of a cogenerator of k [ I ] with nadir σ along τ .4. Regard soc ( M/τ ) as contained in δ ( M/τ ) via ϕ ϕ (1). A nonzero homoge-neous element in soc ( M/τ ) is a local cogenerator of M along τ .Assume the default modifier “global” when neither “local” nor “global” is wrtiten. Example 4.30.
The boundary δM in Example 3.16 explains the nadirs in Example 1.7.The other F op R -modules in Example 3.16, which have solid points in their upper-rightcorners, do not yield socle elements of δM because the R -component can be movedup without annihilation. That is not a universal statement, though: it only holds inthis example because all of the closed socles of M (Definition 4.15) vanish. Example 4.31.
See Example 1.8 for a socle along a face of positive dimension.
Remark 4.32.
The reason to quotient by τ in Definition 4.29.1 is to lump togetherall cogenerators with nadir σ along the same translate of R τ . This lumping makes itpossible for a socle basis to produce a downset hull that is (i) as minimal as possible and(ii) finite. The lumping also creates a difference between the notion of socle elementand that of cogenerator: a socle element is a class of cogenerators, these classes beingindexed by elements in the quotient-restriction. In contrast, a local cogenerator is acogenerator of the quotient-restriction itself, so a local cogenerator is already an elementin the socle of the quotient-restriction. This difference between socle element andcogenerator already arises for closed socles along faces (Definition 4.15) but disappearsin the context of socles not along faces (see Remark 4.17), be they over real polyhedralgroups (Definition 4.5) or closed over posets (Definition 4.1). Remark 4.33.
If localization commuted with cogenerator functors, then the restrictionfrom F op Q to ∇ τ in Definition 4.29.1 would happen automatically, because localizing M along τ would yield a module over Q + + R τ , whose face poset is naturally ∇ τ . But inthis real polyhedral setting, the restriction from F op Q to ∇ τ must be imposed manuallybecause the Hom must be taken before localizing (Remark 4.22), when the default faceposet is still F op Q . Remark 4.34. If a is a cogenerator of a downset D along τ , then the topology of D at a is induced by downsets of the form a ′ − σ ◦ for faces σ ∈ ∇ τ and elements a ′ ∈ a + τ ◦ .This subtle issue regarding shapes of cogenerators along τ is a vital reason for using ∇ τ instead of F op Q . It is tempting to expect that if a face σ is minimal in the shape ∇ a D ,then any expression of D as an intersection of downsets must induce the topology of D at a by explicitly taking a − σ ◦ into account in one of the intersectands. One way toaccomplish that would be for an intersectand to be a union of downsets of the form b − Q ∇ a D (see Definition 3.4) in which one of the elements b is a . But if σ ∈ ∇ a ′ D for all a ′ ∈ a + τ , or even merely for a single element a ′ ∈ a + τ ◦ , then a − σ ◦ = a ′ − ( a ′ − a − σ ◦ ) ∈ a ′ − ( τ ◦ + σ ◦ ) ⊆ a ′ − ( τ ∨ σ ) ◦ . As the purpose of cogenerators is to construct downset decompositions as minimallyas possible, it is counterproductive to think of σ as being a valid F op Q -socle degreeunless σ ∈ ∇ τ , because otherwise it fails to give rise to an essential cogenerator. SeeTheorem 7.19 for the most general possible view of considerations in this Remark. Remark 4.35.
In terms of persistent homology, cogenerators are deaths of classes. Inthat context, the need for upper boundary functors and socle theory beyond closedsocles is particularly crucial, because the modules most pertinent to applied topologyare precisely those whose closed socles vanish [KS18, BP19, Mil20c]. That is, the upperboundaries of these modules are as far from closed as possible.
Remark 4.36.
Although soc τ M is a module over Q/ R τ × ∇ τ by construction, theactions of Q/ R τ and ∇ τ on it are trivial, in the sense that attempting to move a SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 29 nonzero homogeneous element up in one of the posets either takes the element to 0 orleaves it unchanged. (The latter only happens if the degree is unchanged, which occursonly when acting by the identity ∈ Q/ R τ or when acting by σ ∈ ∇ τ on an elementof ∇ τ -degree σ ′ ⊆ σ .) That is what it means to be a direct sum of skyscraper modules.It implies that any direct sum decomposition of soc τ M as a vector space graded by Q/ R τ × ∇ τ is also a decomposition of soc τ M as a Q/ R τ -module or as a ∇ τ -module. Lemma 4.37. If τ is a face of a is a real polyhedral group Q and N = L σ ∈∇ τ N σ is amodule over Q × ∇ τ , then Hom ∇ τ ( k σ , N ) /τ ∼ = Hom ∇ τ ( k σ , N/τ ) , and hence ( ∇ τ -soc N ) /τ ∼ = ∇ τ -soc ( N/τ ) . Proof.
Hom ∇ τ ( k σ , N ) is the intersection of the kernels of the Q -module homomor-phisms N σ → N σ ′ for faces σ ⊃ σ ′ , so the isomorphism of Hom modules follows fromLemma 2.24. The socle isomorphism follows by taking the direct sum over σ ∈ ∇ τ . (cid:3) Proposition 4.38.
The functors soc τ and ∇ τ -soc commute. In particular, ∇ τ -soc (soc τ δ τ M ) ∼ = soc τ M ∼ = soc τ ( ∇ τ -soc δ τ M ) . Proof.
By taking direct sums over a and σ , this is mostly the natural isomorphismsHom ∇ τ (cid:0) k σ , Hom Q (cid:0) k [ a + τ ] , − (cid:1)(cid:1) ∼ = Hom Q ×∇ τ (cid:0) k σ [ a + τ ] , − (cid:1) ∼ = Hom Q (cid:0) k [ a + τ ] , Hom ∇ τ ( k σ , − ) (cid:1) that result from the adjunction between Hom and ⊗ . Taking the quotient-restrictionalong τ (Definition 2.22) almost yields the desired result; the only issue is that theleft-hand side requires Lemma 4.37. (cid:3) Example 4.39. If a is a cogenerator of a downset D ⊆ Q along τ with nadir σ ,then reasoning as in Example 4.11 and using Definition 4.9, computing ∇ τ -soc first inProposition 4.38 shows that σ ∈ ∂ ( ∇ a D ∩ ∇ τ ). What soc τ then does is verify that theimage e a of a in Q/ R τ is maximal with this property, by Example 4.19. Proposition 4.40.
There is a natural injection soc τ M ֒ → soc ( M/τ ) for any module M over a real polyhedral group Q and any face τ of Q .Proof. By Proposition 4.20 soc τ N ֒ → soc ( N/τ ) for N = ∇ τ -soc δ τ M viewed as a Q/ R τ -module. Proposition 4.38 yields soc τ N = soc τ M . It remains to show that( Q/ R τ )-soc ( N/τ ) = soc ( M/τ ). To that end, first note that( ∇ τ -soc δ τ M ) /τ ∼ = ∇ τ -soc (cid:0) ( δ τ M ) /τ (cid:1) ∼ = ∇ τ -soc δ ( M/τ ) , the first isomorphism by Lemma 4.37 and the second by Lemma 4.41, which showsthat the modules acted on by ∇ τ -soc are isomorphic. Now apply the last isomorphismin Lemma 4.8, with Q replaced by Q/ R τ so that automatically F op Q must be replacedby ∇ τ via Lemma 4.25. (cid:3) Lemma 4.41. If σ ⊇ τ then ( δ σ M ) /τ ∼ = δ σ/τ ( M/τ ) .Proof. Explicit calculations from the definitions show that in degree a /τ both sides equallim −→ a ′ ∈ a − σ ◦ v ∈ τ M a ′ + v , although they take the colimits in different orders: v first or a ′ first. The hypothesisthat σ ⊇ τ enters to show that any direct limit over { a ′ ∈ Q | a ′ /τ ∈ a /τ − ( σ/τ ) ◦ } can equivalently be expressed as a direct limit over a ′ ∈ a − σ ◦ . (cid:3) Corollary 4.42.
An indicator quotient for a downset in a real polyhedral group has atmost one linearly independent socle element along each face with given nadir and degree.In fact, the degrees of independent socle elements along τ with fixed nadir are incompa-rable in Q/ R τ , and nadirs of socle elements with fixed degree are incomparable in ∇ τ .Proof. A socle element of an indicator quotient E along a face τ of Q is a local socle ele-ment of E along τ by Proposition 4.40. Local socle elements along τ are socle elements(along the minimal face { } ) of the quotient-restriction along τ by Definition 4.29.2.But E/τ is an indicator quotient of k [ Q/ R τ ], so its socle degrees with fixed nadir σ are incomparable, as are its nadirs with fixed socle degrees, by Example 4.11. (cid:3) Example 4.43.
Propositions 4.38 and 4.40 ease some socle computations. To see how,consider the indicator Q -module k [ ρ ] for a face ρ of Q . Proposition 4.40 immediatelyimplies that soc τ k [ ρ ] = 0 unless ρ ⊇ τ , because localizing along τ yields k [ ρ ] τ = 0unless ρ ⊇ τ .Next compute δ σ k [ ρ ]. When either a ρ or σ ρ , the direct limit in Definition 3.11is over a set a − σ ◦ of degrees in which k [ ρ ] = 0 in a neighborhood of a . Hence theonly faces that can appear in δ τ k [ ρ ] lie in the interval between τ and ρ , so assume τ ⊆ σ ⊆ ρ . If (cid:0) δ σ k [ ρ ] (cid:1) a = 0 then it equals k because k [ ρ ] is an indicator module for asubset of Q . Moreover, if ( δ σ k [ ρ ]) a = k then the same is true in any degree b ∈ a + ρ because ( b − a )+( a − σ ◦ ) ∩ ρ ⊆ ( b − σ ◦ ) ∩ ρ . Thus δ σ k [ ρ ] is torsion-free as a k [ ρ ]-module.The soc τ on the left side of Proposition 4.38, which by Definition 4.15.1 is a quotient-restriction of a module Hom Q (cid:0) k [ τ ] , δ τ k [ ρ ] (cid:1) , can only be nonzero if τ = ρ , as any nonzeroimage of k [ τ ] is a torsion k [ ρ ]-module. Hence the socle of k [ ρ ] along τ equals the closedsocle along τ = ρ , which is computed directly from Definition 4.29.1 and Definition 2.22to be Hom Q (cid:0) k [ τ ] , k [ τ ] (cid:1) /τ = k [ τ ] /τ . In summary,soc τ k [ ρ ] = ( k for ∈ Q/ R τ if τ = ρ Proposition 4.44.
The global cogenerator functor soc τ along any face τ of a realpolyhedral group is left-exact, as is the local cogenerator functor along τ .Proof. Proposition 4.23 and Lemma 3.12. (cid:3)
SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 31 Tame, semialgebraic, and PL socles
The tame, semialgebraic, and PL conditions are preserved under taking socles. Thatis the goal of this section, Theorem 5.13, which states such a result for the most generalform of socle over any real polyhedral group. But because the various forms of socles inSection 4 occur in contexts more general than real polyhedral groups, it is necessary torecord the statements separately for each form of socle. The order in which they are cov-ered here is the same as in Section 4: closed socles soc over an arbitrary poset Q (Propo-sition 5.7); socles soc over real polyhedral groups (Corollary 5.10); closed socles soc τ along faces of polyhedral groups (Proposition 5.12); and finally socles soc τ along facesof real polyhedral groups (Theorem 5.13). The proofs are based on the observation thateverything reduces to the effects of cogenerator functors on downset modules. Verifyingthe hypotheses for the criterion in Theorem 5.2 for cogenerator functors in the tamecase is relatively straightforward. The semialgebraic and PL cases require more power(Lemma 5.4 and onward). First, here is a handy concept [Mil20a, Definition 3.14]. Definition 5.1.
Let each of S and S ′ be a nonempty interval in Q (Example 2.6.3).A homomorphism ϕ : k [ S ] → k [ S ′ ] of interval modules is connected if there is a scalar λ ∈ k such that ϕ acts as multiplication by λ on the copy of k in degree q for all q ∈ S ∩ S ′ . Theorem 5.2.
Fix posets Q and Q ′ . Suppose a left-exact functor S from the categoryof Q -modules to the category of Q ′ -modules takes each • downset module k [ D ] to a subquotient S ( k [ D ]) = k [ S D ] of k [ Q ′ ] , and • connected morphism k [ D ] → k [ D ′ ] of downset modules to a connected morphism k [ S D ] → k [ S D ′ ] of interval modules.Then1. the restriction of S to the category of tame Q -modules (see Proposition 2.15)yields a functor to the category of tame Q ′ -modules; and2. if Q and Q ′ are partially ordered real vector spaces, and S D is semialgebraicin Q ′ for all semialgebraic downsets D ⊆ Q , then S restricts to a functor from thecategory of semialgebraic Q -modules to the category of semialgebraic Q ′ -modules.The previous claim remains true with “PL” in place of “semialgebraic”.Proof. Assume M is a tame Q -module. Then M = ker( E → E ) is the kernel of a tamemorphism of finite direct sums of downset modules by the syzygy theorem for posetmodules [Mil20a, Theorem 6.12.5]. Left-exactness implies that S M = ker( S E → S E ).The first goal is to show that S M is a tame module. For that it suffices by Proposi-tion 2.15 to show that S E → S E is a tame morphism. But that also follows fromProposition 2.15 because each component of S E → S E has the form k [ S D ] → k [ S D ′ ]and is hence a tame morphism by hypothesis. The argument works mutatis mutandisin the semialgebraic and PL cases.Now suppose that a morphism M → M ′ is given. By the syzygy theorem again[Mil20a, Theorem 6.12], there is a copresentation M ′ = ker( E ′ → E ′ ) such that the map M → M ′ is induced by a morphism of their copresentations. The compositemorphism S M → S E → S E ′ has image in S M ′ . The morphism S M → S M ′ is tame,semialgebraic, or PL by any common refinement of two encodings (Definition 2.11)of S E ′ subordinate to the morphisms (Definition 2.13) from S M and S M ′ . (cid:3) Example 5.3.
The upper boundary functor S = δ σ atop σ in Definition 3.15 satisfiesthe hypotheses of Theorem 5.2.1 with Q ′ = Q by Proposition 3.21. Therefore δ σ M isa tame Q -module if M is tame, and δ σ M → δ σ M ′ is a tame morphism if M → M ′ is.Hence the same is true with δ in place of δ σ , since δ = L σ ∈F Q δ σ as endofunctors on thecategory of Q -modules. And the same is true of δ τ for any face τ ∈ F Q (Definition 4.26),since it is a subdirect sum of δ . The corresponding semialgebraic conclusions are trueas well, this time using Theorem 5.2.2, but checking that uses Proposition 5.5, whichrequires a bit more power. Lemma 5.4. If X ⊆ R n and X → Y is a morphism of semialgebraic varieties, thenthe family X Y obtained by taking the closure in R n of every fiber of X is semialgebraic.Proof. This is a consequence of Hardt’s theorem [Har80, Theorem 4] (see also [Shi97,Remark II.3.13]), which says that over a subset of Y whose complement in Y hasdimension less than dim Y , the family X → Y is trivial. (cid:3) Proposition 5.5. If D is a semialgebraic or PL downset in a real polyhedral group Q and σ ∈ F Q is a face then δ σ D is similarly semialgebraic or PL.Proof. Semialgebraic case: Lemma 5.4 with Y = Q/ R σ and X = D by Lemma 3.20.PL case: in Lemma 3.20 the union can be broken over finitely many relatively openpolyhedral cells comprising D . So assume D is a single relatively open polyhedral cell.The union in Lemma 3.20 is plainly a subset of D . Indeed, if π : D → Q/ R σ , then theunion is π − (cid:0) π ( D ) (cid:1) . That is, the union is the complement in D of the (closed) facesof D whose projections mod R σ are contained in the boundary of π ( D ). (cid:3) Proposition 5.7 covers the case of closed socles over an arbitrary poset, with the nextlemma needed for the semialgebraic and PL cases.
Lemma 5.6. If D is a semialgebraic or PL downset in a real polyhedral group then max D is similarly semialgebraic or PL, as well.Proof. The semialgebraic proof relies on standard operations on subsets that preservethe semialgebraic property; see [Shi97, Chapter II], for instance. As it happens, theproof works verbatim for the PL case because the relevant (in)equalities are linear.Inside of R n × R n , consider the subset X whose fiber over each point a ∈ D is a + m , where m = Q r { } is the maximal monoid ideal of Q + . Note that m issemialgebraic because it is defined by linear inequalities and a single linear inequation.The subset X ⊆ R n × R n is semialgebraic because it is the image of the algebraicmorphism D × m → D × R n sending ( a , q ) ( a , a + q ). The intersection of X with SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 33 the semialgebraic subset D × D remains semialgebraic, as does the projection of thisintersection to D . The image of the projection is D r max D because ( a + m ) ∩ D = ∅ precisely when a ∈ max D . Therefore max D = D r ( D r max D ) is semialgebraic. (cid:3) Proposition 5.7.
If a module M over any poset is tame then so is its closed socle soc M . If M is semialgebraic or PL over a real polyhedral group then so is soc M . If M → M ′ is a tame, semialgebraic, or PL morphism, then so is soc M → soc M ′ .Proof. Apply Theorem 5.2: left-exactness is Lemma 4.4, the criteria on downset mod-ules and connected morphisms between them both follow from Example 4.3, and thesemialgebraic or PL criterion is Lemma 5.6. (cid:3)
The next three results cover the case of socles over real polyhedral groups.
Lemma 5.8.
The homomorphisms δ σ M → δ σ ′ M for faces σ ⊇ σ ′ afforded by Proposi-tion 3.13 are tame, semialgebraic, or PL if M is.Proof. Use [Mil20a, Theorem 6.12.5] to express M = ker( E → E ) as the kernel ofa downset copresentation that is tame, semialgebraic, or PL as the case may be. Fora single downset D , observe that δ σ ′ D ⊆ δ σ D whenever σ ⊇ σ ′ by Proposition 3.21.2.Therefore, by Proposition 3.21, the natural map δ σ k [ D ] → δ σ ′ k [ D ] is a quotient ofdownset modules, which is a connected homomorphism (Definition 5.1) and hencetame, semialgebraic, or PL, as the case may be. The homomorphism δ σ M → δ σ ′ M isinduced by the morphism δ σ E • → δ σ ′ E • of copresentations. The argument in the finaltwo sentences of the proof of Theorem 5.2 therefore works here. (cid:3) The next result is stated in the generality of ∇ τ -soc δ τ (see Section 4.4) for theeventual purpose of Theorem 5.13, even though for the time being all that is neededis the case τ = { } , where ∇ τ -soc δ τ = F op Q -soc δ (see Section 4.1). Proposition 5.9.
Fix a real polyhedral group Q and a face τ ∈ F Q . The endofunctoron the category of Q -modules that takes M to ∇ τ -soc δ τ restricts to endofunctors onthe categories of tame Q -modules, semialgebraic Q -modules, and PL Q -modules.Proof. The same proof works, mutatis mutandis, for the semialgebraic and PL cases.The ∇ τ -graded component of ∇ τ -soc δ τ M in ∇ τ -degree σ is the intersection of thekernels of the Q -module homomorphisms δ σ M → δ σ ′ M for σ ⊇ σ ′ . These are tamemorphisms, if M is tame, by Lemma 5.8. The intersection of their kernels is tame byProposition 2.15 because any intersection of kernels of morphisms from a single objectto finitely many objects in any abelian category is the kernel of the morphism to thedirect sum. So ∇ τ -soc δ τ M → δ τ M is tame.Any given tame morphism M → N induces a tame morphism δM → δN by Exam-ple 5.3. Hence the composite ∇ τ -soc δ τ M → δ τ M → δ τ N is a tame morphism thathappens to have its image in ∇ τ -soc δ τ N . On the other hand, ∇ τ -soc δ τ N → δ τ N is also a tame morphism. The morphism ∇ τ -soc δ τ M → ∇ τ -soc δ τ N is tame by any common refinement of two poset encodings of δ τ N subordinate to the morphisms from ∇ τ -soc δ τ M and ∇ τ -soc δ τ N . (cid:3) Corollary 5.10.
If a module M over a real polyhedral group is tame, semialgebraic, orPL then so is its socle soc M . If M → M ′ is a tame, semialgebraic, or PL morphism,then the natural map soc M → soc M ′ is, as well.Proof. By Lemma 4.8, soc M is the composite of the functors Q -soc and F op Q -soc δ ,which preserve the tame, semialgebraic, and PL categories by Propositions 5.7 and 5.9,the latter of which has ∇ τ -soc δ τ = F op Q -soc δ when τ = { } . (cid:3) The next two results cover closed socles along faces of arbitrary polyhedral groups.
Lemma 5.11. If D is a semialgebraic or PL downset in a real polyhedral group Q then max τ D is similarly semialgebraic or PL in Q/ R τ for any face τ of Q + .Proof. The projection of a semialgebraic set is semialgebraic, so by Example 4.19 itsuffices to prove that the set of degrees of closed cogenerators of k [ D ] along τ is semi-algebraic. The argument comes in two halves, both following the framework of theproof of Lemma 5.6. For the first half, simply replace m by m τ = Q + r τ to findthat (cid:8) a ∈ D | ( a + Q + ) ∩ D ⊆ a + τ (cid:9) is semialgebraic. The second half uses τ in-stead of m , and this time it intersects the subset X with D × ( Q r D ) to find that (cid:8) a ∈ D | ( a + Q + ) ∩ D ⊇ a + τ (cid:9) is semialgebraic. The desired set of degrees isthe intersection of these two semialgebraic sets. Replacing “semialgebraic” with “PL”works because, again, the relevant (in)equalities are linear. (cid:3) Proposition 5.12.
If a module M over a partially ordered group is tame then so is itsclosed socle soc τ M along any face τ . If M is semialgebraic or PL over a real polyhedralgroup then so is soc τ M . If M → M ′ is a tame, semialgebraic, or PL morphism, thenthe natural map soc τ M → soc τ M ′ is, as well.Proof. Apply Theorem 5.2: left-exactness is Proposition 4.23; the criteria on downsetmodules and connected morphisms between them follow from Example 4.19, notingthat the set max τ D is an upset in the downset D/ Z τ because it is contained in the setof maximal elements of D/ Z τ ; and the semialgebraic or PL criterion is Lemma 5.11. (cid:3) Finally, here is the version covering total socles over real polyhedral groups.
Theorem 5.13.
Over a real polyhedral group Q , the cogenerator functor soc τ alongany face τ restricts to endofunctors on the categories of tame, semialgebraic, or PLmodules over Q . For any face σ ⊇ τ , this statement remains true for the cogeneratorfunctor soc στ along τ with nadir σ .Proof. The cogenerator functor soc τ M is the composite of soc τ and ∇ τ -soc δ τ by Propo-sition 4.38. These functors preserve the tame, semialgebraic, and PL categories byPropositions 5.12 and 5.9. The soc στ claim follows by taking ∇ τ -graded pieces. (cid:3) SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 35 Essential property of socles
In this section, Q is a real polyhedral group unless otherwise stated.The culmination of the foundations developed in Section 4 says that socles and co-generators detect injectivity of homomorphisms between tame modules over real poly-hedral groups (Theorem 6.7), as they do for noetherian rings in ordinary commutativealgebra. The theory is complicated by there being no actual submodule containing agiven non-closed socle element; that is why socles are functors that yield submodulesof localizations of auxiliary modules rather than submodules of localizations of thegiven module itself. Nonetheless, it comes down to the fact that, when D ⊆ Q is adownset, every element can be pushed up to a cogenerator. Theorem 6.5 contains aprecise statement that suffices for the purpose of Theorem 6.7, although the definitiveversion of Theorem 6.5 occurs in Section 7, namely Theorem 7.19.The proof of Theorem 6.5 requires a definition—essentially the notion dual to that ofshape (Proposition 3.10). Informally, it is the set of faces σ such that a neighborhoodof a in a + σ ◦ is contained in the downset D . The formal definition reduces by negationto the discussion surrounding tangent cones of downsets (Section 3.1), noting that thenegative of an upset is a downset. Definition 6.1.
The upshape of a downset D in a real polyhedral group Q at a is∆ D a = F Q r ∇ − a − U , where U = Q r D is the upset complementary to D . Lemma 6.2.
The upshape ∆ D a is a polyhedral complex (a downset) in F Q . As a functionof a , for fixed D the upshape ∆ D a is decreasing, meaning a (cid:22) b ⇒ ∆ D a ⊇ ∆ D b .Proof. These claims are immediate from the discussion in Section 3.1. (cid:3)
Remark 6.3.
The upshape ∆ D a is a rather tight analogue of the Stanley–Reisner com-plex of a simplicial complex, or more generally the lower Koszul simplicial complex[MS05, Definition 5.9] of a monomial ideal in a degree from Z n . (The complex K b ( I )would need to be indexed by b − supp( b ) to make the analogy even tighter.) Similarly,the shape of a downset at an element of Q is analogous to the upper Koszul simplicialcomplex of a monomial ideal [MS05, Definition 1.33].The general statement about pushing up to cogenerators relies on the special caseof closed cogenerators for closed downsets. Lemma 6.4. If D ⊆ Q is a downset and the part of D above b ∈ D is closed, so ( b + Q + ) ∩ D = ( b + Q + ) ∩ D , then b (cid:22) a for some closed cogenerator a of D . Proof.
It is possible that b + Q + ⊆ D , in which case D = Q and b is by definition aclosed cogenerator along τ = Q + . Barring that case, the intersection ( b + Q + ) ∩ ∂D of the principal upset at b with the boundary of D is nonempty. Among the pointsin this intersection, choose a with minimal upshape. Observe that { } ∈ ∆ D a because a ∈ D , so ∆ D a is nonempty.Let τ ∈ ∆ D a be a facet. The goal is to conclude that ∆ D a = F τ has no facet otherthan τ , for then ∆ D a ′ = F τ for all a ′ (cid:23) a in D by upshape minimality and Lemma 6.2,and hence a is a cogenerator of D along τ by Definition 4.15 (see also Remark 4.18).Suppose that ρ ∈ F Q is a ray that lies outside of τ . If ρ ∈ ∆ D a then upshapeminimality implies ρ ∈ ∆ D a ′ for any a ′ ∈ ( a + τ ◦ ) ∩ D , and such an a ′ exists by definitionof upshape. Consequently, some face containing both ρ and τ lies in ∆ D a : if v is anysufficiently small vector along ρ , then a ′ + v = a + ( a ′ − a ) + v ∈ D , and the smallestface containing ( a ′ − a ) + v contains both the interior of τ (because it contains a ′ − a )and ρ (because it contains v ). But this is impossible, so in fact ∆ D a = F τ . (cid:3) Theorem 6.5. If D is a downset in a real polyhedral group Q and b ∈ D , then thereare faces τ ⊆ σ of Q + and a cogenerator a of D along τ with nadir σ such that b (cid:22) a .Proof. It is possible that b + Q + ⊆ D , in which case D = Q and b is by definition aclosed cogenerator along τ = Q + , which is the same as a cogenerator along Q + withnadir Q + . Barring that case, the intersection ( b + Q + ) ∩ ∂D of the principal upset at b with the boundary of D is nonempty. Among the points in this intersection, there isone with minimal shape, and it suffices to treat the case where this point is b itself.Minimality of ∇ b D implies that the shape does not change upon going up from b while staying in the closure D . Consequently, given any face σ ∈ ∇ a D , the shape of D atevery point in b + Q + that lies in D also contains σ . Equivalently by Proposition 3.21.2,( b + Q + ) ∩ δ σ D = ( b + Q + ) ∩ D . Lemma 6.4 produces a closed cogenerator a of δ σ D ,along some face τ , satisfying b (cid:22) a . Since ∇ a D is a nonempty cocomplex, its intersectionwith ∇ τ is nonempty, so assume σ ∈ ∇ a D ∩ ∇ τ . The closed cogenerator a of δ σ D neednot be a cogenerator of D , but if σ is minimal under inclusion in ∇ a D ∩ ∇ τ , then a isindeed a cogenerator of D along τ with nadir σ by Proposition 4.38—specifically thefirst displayed isomorphism—applied to Example 4.3. (cid:3) Remark 6.6.
The arguments in the preceding two proofs are essential to the wholetheory of socles, which hinges upon them. The structure of the arguments dictate theforms of all of the notions of socle, particularly those involving cogenerators along faces.Theorem 6.7 is intended for tame modules, but because it has no cause to deal withgenerators, in actuality it only requires half of a fringe presentation (or a little less;see Definition 2.16). The statement uses divisibility (Definition 3.19), which worksverbatim for δ τ M , by Definition 4.26, because it refers only to upper boundaries atopa single face σ . SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 37
Theorem 6.7 (Essentiality of socles) . Fix a homomorphism ϕ : M → N of modulesover a real polyhedral group Q .1. If ϕ is injective then soc τ ϕ : soc τ M → soc τ N is injective for all faces τ of Q + .2. If soc τ ϕ : soc τ M → soc τ N is injective for all faces τ of Q + and M is downset-finite, then ϕ is injective.If M is downset-finite then each homogeneous element of M divides a cogenerator of M .Proof. Item 1 is a special case of Proposition 4.44. Item 2 follows from the divisibilityclaim, for if y divides a cogenerator s along τ then ϕ ( y ) = 0 whenever soc τ ϕ ( e s ) = 0,where e s is the image of s in soc τ M .For the divisibility claim, fix a downset hull M ֒ → L kj =1 E j and a nonzero y ∈ M b .For some j the projection y j ∈ E j of y divides a cogenerator of E j along some face τ with some nadir σ by Theorem 6.5. Choose one such cogenerator s j , and suppose ithas degree a ∈ Q . There can be other indices i such that (soc στ E i ) e a = 0, where e a isthe image of a in Q/ R τ . For any such index i , as long as y i = 0 it divides a uniquecogenerator in s i ∈ δ στ E i by Corollary 4.42. Therefore the image of y in E = L kj =1 E j divides the sum of these cogenerators s j . But that sum is itself another cogeneratorof E along τ with nadir σ in degree a , and the fact that y divides it places the sum inthe image of the injection (Lemma 3.12) δ στ M ֒ → δ στ E . (cid:3) Remark 6.8.
In terms of persistent homology, Theorem 6.7 says that a homomorphismof real multipersistence modules is injective if and only if it takes the “right endpoints”of the source injectively to a subset of the “right endpoints” of the target.
Corollary 6.9.
Fix a downset-finite module M over a real polyhedral group.1. M = 0 if and only if soc τ M = 0 for all faces τ .2. soc τ M ′ ∩ soc τ M ′′ = soc τ ( M ′ ∩ M ′′ ) in soc τ M for submodules M ′ and M ′′ of M .Proof. That M = 0 ⇒ M = 0 is trivial. On the other hand, if soc τ M = 0 for all τ then M is a submodule of 0 by Theorem 6.7.2.The second equality follows from left-exactness (Proposition 4.44):soc τ ( M ′ ∩ M ′′ ) = soc τ ker( M ′ → M/M ′′ )= ker (cid:0) soc τ M ′ → soc τ ( M/M ′′ ) (cid:1) = ker(soc τ M ′ → soc τ M/ soc τ M ′′ )= soc τ M ′ ∩ soc τ M ′′ , where the penultimate equality is because soc τ M ′′ is the kernel of the homomorphismsoc τ M → soc τ ( M/M ′′ ), so that soc τ M/ soc τ M ′′ ֒ → soc τ ( M/M ′′ ). (cid:3) There is a much stronger statement connecting socles to essential submodules (The-orem 8.5), but it requires language to speak of density in socles as well as tools toproduce submodules from socle elements, which are the main themes of Section 7. Minimality of socle functors
Socles capture the entirety of a downset by maximal elements in closures along faces;that is in some sense the main content of socle essentiality (Theorem 6.7), or more pre-cisely Theorem 6.5. But since closures are involved, it is reasonable to ask if anythingsmaller still captures the entirety of every downset. Algebraically, for arbitrary mod-ules, this asks for subfunctors of cogenerator functors. The particular subfunctors hereconcern the graded degrees of socle elements, for which notation is needed.
Definition 7.1.
The degree set of any module N over a poset P isdeg N = { a ∈ P | N a = 0 } . Write deg P = deg if more than one poset could be intended.7.1. Neighborhoods of group elements.
The topological condition characterizing when enough cogenerators are present is asort of density in the set of all cogenerators. Lemma 3.20 has a related closure notion.Recall the statement and context of Lemma 3.8, which says that Q ∇ σ = σ ◦ + Q + . Definition 7.2.
Fix faces σ ⊇ τ of a real polyhedral group Q . A σ -vicinity of a point e a ∈ Q/ R τ is a subset of Q/ R τ of the form ( u + Q ∇ σ ) / R τ such that a − u ∈ σ ◦ and a is a representative for the the coset e a = a + R τ ∈ Q/ R τ . Lemma 7.3.
The σ -vicinities of points in Q/ R τ form a base for a topology on Q/ R τ .More strongly, the intersection of any finite set of σ -vicinities (for perhaps differentpoints in Q/ R τ ) contains a σ -vicinity of each point in their intersection.Proof. The σ -vicinities of the points of Q/ R τ cover Q/ R τ because e a lies in any of its σ -vicinities. So it suffices to prove the stronger claim, which by induction reduces to: theintersection of any σ -vicinity ( u + Q ∇ σ ) / R τ of e a and any σ -vicinity ( v + Q ∇ σ ) / R τ of e b contains a σ -vicinity of each point e c in their intersection. Pick a coset representative c ∈ Q for c + R τ = e c . Translating c by a vector far inside of τ ◦ , if necessary, assumethat c ∈ u + Q ∇ σ . Perhaps translating along τ ◦ further, assume that c ∈ v + Q ∇ σ also.Then u + Q ∇ σ contains the intersection with c − σ ◦ of an open neighborhood of c , asdoes v + Q ∇ σ . Any vector w ∈ c − σ ◦ in the intersection of these neighborhoods yieldsa σ -vicinity ( w + Q ∇ σ ) / R τ of e c contained in both of the given σ -vicinities. (cid:3) Definition 7.4.
The topology in Lemma 7.3 is called the ∇ σ -topology on Q/ R τ . Remark 7.5.
In this paper, all topological notions in real vector spaces—limit, closure,neighborhood, and so on—refer to the usual topology unless explicitly otherwise stated.For example, Remark 7.9 refers to σ -closure, ∇ σ -closure, and ∇ σ -open neighborhoods. SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 39
Remark 7.6.
The ∇ σ -topologies for various σ are more general than the γ -topologiesfrom [KS18] because the cone ∇ σ is not necessarily closed. Its non-closedness reflectsdirections that ought to be thought of as inverted, and the image of ∇ σ in the collapsemodulo the inverted directions is closed, but the ∇ σ -topology is needed on the vectorspace before this collapse. Remark 7.7.
The strength of Lemma 7.3 beyond providing a base for a topology restson a σ -vicinity of e a not being the same as a basic ∇ σ -open set containing e a . Indeed,a σ -vicinity u + Q ∇ σ is required to contain a representative for e a that lies in the face u + σ ◦ , not merely somewhere arbitrary in u + Q ∇ σ . Definition 7.8.
Fix faces σ ⊇ τ of a real polyhedral group Q .1. A σ -limit point of a subset X ⊆ Q/ R τ is a point e a ∈ Q/ R τ that is a limit (inthe usual topology) of points in X each of which lies in a σ -vicinity of e a .2. The σ -closure of X ⊆ Q/ R τ is the set of points e a ∈ Q/ R τ such that X has atleast one point in every σ -vicinity of e a . Remark 7.9.
The σ -closure of X equals its ∇ σ -closure, by which is meant the closureof X in the ∇ σ -topology. The reason: every basic ∇ σ -open neighborhood of a pointcontains a σ -vicinity of that point by Lemma 7.3. Remark 7.10.
The sets X to which Definition 7.8 is applied are typically decomposedas finite unions of antichains (but see Proposition 7.12 for an instance where this is notthe case). Such sets “cut across” subsets of the form ( u + Q ∇ σ ) / R τ , rather than beingswallowed by them, so σ -vicinities have a fighting chance of reflecting some concept ofcloseness in antichains. If σ = Q + and τ = { } , for example, and X is an antichainin Q , then a σ -vicinity of a point a ∈ X is the same thing as a usual open neighborhoodof a in X , so σ -closure is the usual topological closure. If, at the other extreme, σ = τ ,then every antichain in Q/ R τ is τ -closed. Example 7.11.
Let Q = R and τ = { } . Take for X ⊂ R the convex hull of , e , e but with the first standard basis vector e removed. If σ is the x -axis of R , then in X = the blue points ofconstitute a σ -vicinity of e in X . In addition, the bold blue segment in the half-openhypotenuse H is a σ -vicinity of e in H . The point e itself is a σ -limit point of H .The concept of σ -vicinity provides a means to connect socles (Definition 4.29) withsupport (Definition 2.25) and primary decomposition (Definition 2.30). Proposition 7.12.
In a real polyhedral group Q , every cogenerator of a downset D along τ with nadir σ has a σ -vicinity O in D ⊆ Q (so σ ⊇ { } are the faces inDefinition 7.8) such that k [ O ] ⊆ k [ D ] is τ -coprimary and globally supported on τ .Proof. Let a be such a cogenerator of D . Suppose { a k } k ∈ N ⊆ a − σ ◦ ⊆ D is anysequence converging to a . If a k is supported on a face τ ′ , then τ ′ ⊇ τ because a (cid:23) a k and a remains a cogenerator of the localization of D along τ by Proposition 4.40. Thesame argument shows that the σ -vicinity O = a k + Q ∇ σ yields a submodule k [ O ] ⊆ k [ D ]such that k [ O ] ֒ → k [ O ] τ . The goal is therefore to show that some a k is supported on τ ,for then O is supported on τ , as support can only decrease upon going up in Q .If each a k is supported on a face properly containing τ , then, restricting to a sub-sequence if necessary, assume that it is the same face τ ′ for all k . (This uses thefiniteness of the number of faces.) But then a + τ ′ = lim k ( a k + τ ′ ) is contained in δ σ D by Definition 3.11, contradicting the fact that a is supported on τ in δ σ D . (cid:3) Example 7.13.
All three of the downsets D D D in R have a cogenerator at the open corner a along the face τ = { } , but theirbehaviors near a differ in character. Write σ x and σ y for the faces of R that are itshorizontal and vertical axes, respectively.1. Here a has two nadirs: it is a cogenerator along τ = { } for both σ x and σ y byProposition 3.21 and Example 4.3. The blue set in Example 7.11 constitutes a τ -coprimary σ x -vicinity of a globally supported on τ , as in Proposition 7.12, ifthe open point there is also a .2. Here a has only the nadir σ x , because the downset has no points in a + R σ y to take the closure of in Lemma 3.20. Again, the blue set in Example 7.11constitutes the desired σ x -vicinity of a .3. Here a has only the nadir σ y . It is possible to compute this directly, but itis more apropos to note that Proposition 7.12 rules out σ x as a nadir. Indeed,every σ x -vicinity of a in D is an infinite vertical strip. None of these σ x -vicinitiesare supported on τ = { } , since elements therein persist forever along σ y . Incontrast, every choice of v ∈ σ ◦ y yields a σ y -vicinity ( − v + R ∇ σ y ) ∩ D supportedon { } ; that is, the entire negative y -axis is supported on { } .Compare the following with Example 4.43; it is the decisive more or less explicitcalculation that justifies the general theory of socles and provides its foundation. SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 41
Corollary 7.14.
Fix a face τ of a real polyhedral group Q and a subquotient M of k [ Q ] that is τ -coprimary and globally supported on τ . Then soc τ ′ M = 0 unless τ ′ = τ .Proof. Proposition 4.40 implies that soc τ ′ M = 0 unless τ ′ ⊇ τ by definition of globalsupport: localizing along τ ′ yields M τ ′ = 0 unless τ ′ ⊆ τ . On the other hand, M beinga subquotient of k [ Q ] means that M ⊆ k [ D ] for some downset D . By left-exactness ofsocles (Proposition 4.44), every cogenerator of M is a cogenerator of k [ D ]. ApplyingProposition 7.12 to any such cogenerator along τ ′ implies that τ = τ ′ , because no τ -coprimary module has a submodule supported on a face strictly contained in τ . (cid:3) Remark 7.15.
Remember that being τ -coprimary does not require the whole moduleto be globally supported on τ ; only an essential submodule need be globally supportedon τ . This occurs for the global support on τ = { } in Example 2.26, which is strictlycontained in the corresponding τ -primary component from Example 1.5. Dually, beingglobally supported on τ allows for elements with support strictly contained in τ . Remark 7.16.
Definition 7.8.1 stipulates no condition the generators of the relevant σ -vicinities—the vectors u in Definition 7.2. The a priori difference between being a σ -limit point and lying in the σ -closure is hence that for σ -closure, the convergence isstipulated on the generators of the σ -vicinities rather than on the points of X . Thatsaid, the a priori weaker (that is, more inclusive) notion of σ -limit point is equivalent:the generators can be forced to converge. Proposition 7.17.
If a sequence { a ′ k } k ∈ N in a real polyhedral group Q has a ′ k → a and a ′ k ∈ a k + Q ∇ σ for some a k ∈ a − σ ◦ , where σ is a fixed face, then it is possible tochoose the elements a k so that a k → a . Consequently, if σ ⊇ τ then the σ -closure ofany set X ⊆ Q/ R τ equals the set of its σ -limit points.Proof. Writing a ′ k = a − v k + z k with v k ∈ R σ and z k ∈ σ ⊥ , the only relevantconsequence of the hypothesis a ′ k ∈ a k + Q ∇ σ is to force z k to land in Q + / R σ whenprojected to Q/ R σ . Consider the set Z ⊆ σ ⊥ of vectors in σ ⊥ whose images in Q/ R σ lie in Q + / R σ and have magnitude ≤
1. Let V ⊆ R σ be the ball of radius 1. Find s ∈ σ ◦ so that s + V + Z ⊆ Q + . To see that such an s exists, first find s ′ ∈ σ ◦ sothat s ′ + Z ⊆ Q + . To construct s ′ , rescale any element s ′′ ∈ σ ◦ ; this works becausethe projection of ( s ′′ + σ ⊥ ) ∩ Q + to Q/ R σ contains a neighborhood of in Q + / R σ .Then observe that the condition s ′ + Z ⊆ Q + remains true after adding any elementof σ to s ′ . In particular, construct s by adding the center of any ball in σ ◦ of radius 1,which exists because σ ◦ is nonempty, open in R σ , and closed under positive scaling.Having fixed s with s + V + Z ⊆ Q + , set a k = a − ε k s , where ε k = | a ′ k − a | . The reasonfor this choice of ε k is that 2 ε k → a ′ k → a ) and ε k bounds the magnitudesof v k and z k . This latter condition implies a ′ k = a − v k + z k ∈ a + ε k V + ε k Z =( a − ε k s ) + ε k s + ( ε k s + ε k V + ε k Z ) ⊆ a k + σ ◦ + Q + = a k + Q ∇ σ by Lemma 3.8. The claim involving τ follows, when τ = { } , from Proposition 3.7: it implies thateach element b ∈ a − σ ◦ precedes some a k , and hence the σ -vicinity generated by b contains a ′ k . The case of arbitrary τ reduces to τ = { } by working modulo R τ . (cid:3) Dense cogeneration of downsets.
The subfunctor version of density in socles for modules requires first a geometric versionfor downsets. For geometric intuition, it is useful to again recall Lemma 3.8, whichsays that Q ∇ σ = σ ◦ + Q + . Thus a − Q ∇ σ is the “coprincipal” downset with apex a andshape ∇ σ . Adding τ to get a + τ − Q ∇ σ takes the union of these downsets along a + τ . Itis worth noting that the downset thus constructed is preserved under translation by R τ . Lemma 7.18.
Let Q be a real polyhedral group with faces σ ⊇ τ . Write e a ∈ Q/ R τ forthe coset a + R τ containing a ∈ Q . Then a + τ − Q ∇ σ = a ′ + τ − Q ∇ σ for all a ′ ∈ e a .Hence there is only one downset a + τ − Q ∇ σ = e a − Q ∇ σ per coset e a = a + R τ .Proof. Using Lemma 3.8 to write a + τ − Q ∇ σ = a + τ − σ ◦ − Q + , this set is translation-invariant along R τ because − Q + contains − τ . (cid:3) The question is which of these downsets must appear in any decomposition.
Theorem 7.19.
Let A στ ⊆ Q be a set of cogenerators of a downset D in a real polyhedralgroup Q along a face τ with nadir σ for each σ ∈ ∇ τ , and let A τ = S σ ⊇ τ A στ . Then D = [ faces σ, τ with σ ⊇ τ [ a ∈ A στ a + τ − Q ∇ σ as long as the σ -closure of the image of A τ in Q/ R τ contains the projection modulo R τ of every cogenerator of D along τ with nadir σ .Proof. Theorem 6.5 is equivalent to the desired result in the case that every A στ is theset of all cogenerators of D along τ with nadir σ , by Example 4.39 and Remark 4.34.Hence it suffices to show that [ σ ′ ⊇ τ [ a ′ ∈ A σ ′ τ a ′ + τ − Q ∇ σ ′ ⊇ a + τ − Q ∇ σ for any fixed cogenerator a of D along τ with nadir σ . In fact, by definition of σ -limitpoint, it is enough to show that ∞ [ k =1 a ′ k + τ − Q ∇ σ k ⊇ a + τ − Q ∇ σ , where { a ′ k } k ∈ N is a sequence of elements of A τ such that • a k lands in a σ -vicinity of the image e a of a when projected to Q/ R τ , and • these images e a ′ k converge to e a in Q/ R τ SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 43 and σ k is a nadir of the cogenerator a ′ k along τ .Note that there is something to prove even when σ = τ (see the end of Remark 7.10)because A ττ only needs to have at least one closed cogenerator in Q for each closedsocle degree in Q/ R τ , whereas the set of all closed cogenerators along τ mapping to agiven socle degree might not be a single translate of τ . On the other hand, τ − Q ∇ τ = τ − τ ◦ − Q + by Lemma 3.8, and this is just R τ − Q + . Therefore a + τ − Q ∇ τ containsthe translate of the negative cone − Q + at every point mapping to e a , cogenerator orotherwise, completing the case σ = τ .For general σ ⊇ τ , Lemma 7.18 reduces the question to the quotient Q/ R τ , whereit becomes ∞ [ k =1 e a ′ k − Q ∇ σ k /τ ⊇ e a − Q ∇ σ /τ. But as Q ∇ σ k /τ = σ ◦ k /τ + ( Q/ R τ ) + by Lemma 3.8, it does no harm (and helps thenotation) to assume that τ = { } . The desired statement is now ∞ [ k =1 a ′ k − Q ∇ σ k ⊇ a − Q ∇ σ , the hypotheses being those of Proposition 7.17. The proof is completed by applyingProposition 3.7 to the sequence { a k } k ∈ N produced by Proposition 7.17, noting that a k − Q + ⊆ a ′ k − Q ∇ σ k as soon as a k ∈ a ′ k − Q ∇ σ k , because a ′ k − Q ∇ σ k is a downset. (cid:3) Example 7.20.
Consider the downsets in Example 7.13. The question is whether a is forced to appear in the union from Theorem 7.19 or not.1. The point a is a σ x -limit point of D by Example 7.11, which shares its geometrywith D on the relevant set, namely the x -axis and above. Thus Theorem 7.19wants to force a to appear. And indeed it appears, but only because of the y - axis:every σ y -vicinity of a in D contains exactly one cogenerator, namely a itself.2. In contrast, a is not needed for D . Abstractly, this is because D is missingprecisely the negative y -axis that caused a to be forced in D . But geometricallyit is evident that D equals the union of the closed negative quadrants hangingfrom the open diagonal ray.3. Here a is the sole cogenerator of D along τ = { } , so it is forced to appear. Example 7.21.
The curve atop each of the following two downsets in R is a hyperbola.PSfrag replacements xy PSfrag replacements xy D D
1. Plucking out a single point from the hyperbola has an odd effect. At the frontierpoint Definition 4.29 detects two open cogenerators, with different nadirs as inExample 1.7, but they are redundant: D equals the union of the closed negativeorthants cogenerated by the points along the rest of the hyperbola.2. The ability to omit cogenerators is even more striking upon deleting an intervalfrom the hyperbola, instead of a single point, to get the downset D . Along thedeleted curve, Definition 4.29 detects cogenerators of the same shape as thosefor D . Hence D is the union of coprincipal downsets of these shapes alongthe deleted curve together with closed coprincipal downsets along the rest of thehyperbola. However, any finite number of the coprincipal downsets along thedeleted curve can be omitted, as can be checked directly. In fact, any subset ofthem that is dense in the deleted curve can be omitted by Theorem 7.19. Notethat a closed negative orthant is required at the lower endpoint of the deletedcurve, because the endpoint has not been deleted, whereas the cogenerator atthe upper endpoint of the deleted curve can always be omitted because of opennegative orthants hanging from points along the hyperbola just below it.Here is a restatement of Theorem 6.5, phrased as a special case of Theorem 7.19, interms of coprincipal downsets via Lemma 7.18. Corollary 7.22.
Every downset D in a real polyhedral group Q is the union of the coprincipal downsets e a − Q ∇ σ indexed by the degrees e a ∈ Q/ R τ of socle elementsof k [ D ] along all faces τ with all nadirs σ : D = [ faces σ, τ with σ ⊇ τ [ e a ∈ deg Q/ R τ soc στ k [ D ] e a − Q ∇ σ . Remark 7.23.
Theorem 7.19 is an analogue for real polyhedral groups of the factthat monomial ideals in affine semigroup rings admit unique irredundant irreducibledecompositions [MS05, Corollary 11.5]. To see the analogy, note that expressing adownset as a union is the same as expressing its complementary upset as an intersection.In Theorem 7.19 the union is neither unique nor irredundant, but only in the sensethat a topological space can have many dense subsets, each of which can usually bemade smaller by omitting some points. The union in Corollary 7.22 is still canonical,though redundant in a predictable manner, namely that of Theorem 7.19. The analogyin this remark is not the tightest possible; see Section 9.2 for the true analogy.
Remark 7.24.
In the case of Q = R n with componentwise partial order, Ingebretsonand Sather-Wagstaff characterized the downsets in Q + that admit decompositions as inTheorem 7.19 with finitely many terms [ISW13, Theorem 4.12]: they are the ones whosecomplementary upsets have finitely many generators in the sense of [Mil20d]. The newaspects here, for arbitrary downsets, are the related notions of minimality and density SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 45 of cogenerating sets. Theorem 7.19 implies the characterization of irreducible downsets[ISW13, Theorem 3.9] because irreducible decompositions are assumed finite there.The final result in this subsection is applied in the proof of Theorem 7.27.
Corollary 7.25.
Fix a cogenerator a of a downset D along a face τ with nadir σ ina real polyhedral group. If b ∈ D and b (cid:22) a , then the image e a of a in Q/ R τ has a σ -vicinity O in deg Q/ R τ soc τ k [ D ] such that e b (cid:22) e a ′ for all e a ′ ∈ O .Proof. Assume b ∈ D and b (cid:22) a . Theorem 7.19 implies that b ∈ a + τ − Q ∇ σ = a + R τ − σ ◦ − Q + . Therefore a + R τ = b + R τ + s + q for some s ∈ σ ◦ and q ∈ Q + .The σ -vicinity in question is ( e b + e q + Q ∇ σ ) ∩ deg Q/ R τ soc τ k [ D ]. (cid:3) Dense subfunctors of socles.
In general, a subfunctor Φ :
A → B of a covariant functor Ψ :
A → B is a naturaltransformation Φ → Ψ such that Φ( A ) ⊆ Ψ( A ) for all objects A ∈ A [EM45, Chap-ter III]; denote this by Φ ⊆ Ψ. (This notation assumes that the objects of B are sets,which they are here; in general, Φ( A ) → Ψ( A ) should be monic.) Definition 7.26.
A subfunctor S τ = L σ ∈∇ τ S στ ⊆ soc τ from modules over Q to modulesover Q/ R τ × ∇ τ is dense if the σ -closure of deg Q/ R τ S τ k [ D ] contains deg soc στ k [ D ] forall faces σ ⊇ τ and downsets D ⊆ Q . An S -cogenerator of a Q -module M is acogenerator of M along some face τ whose image in soc τ M lies in S τ M . Theorem 7.27.
Fix subfunctors S τ ⊆ soc τ for all faces τ of a real polyhedral group.Theorem 6.7 holds with S in place of soc if and only if S τ is dense in soc τ for all τ .Proof. Every subfunctor of any left-exact functor takes injections to injections; there-fore Theorem 6.7.1 holds for any subfunctor of soc τ by Proposition 4.44. The contentis that Theorem 6.7.2 is equivalent to density of S τ in soc τ for all τ .First suppose that S τ is dense in soc τ for all τ . It suffices to show that each ho-mogeneous element y ∈ M divides some S -cogenerator s , for then ϕ ( y ) = 0 whenever S τ ϕ ( e s ) = 0, where e s is the image of s in S τ M ⊆ soc τ M . There is no harm in assumingthat M is a submodule of its downset hull: M ⊆ E = L kj =1 E j . Theorem 6.7 producesa cogenerator x of E that is divisible by y , and x is automatically a cogenerator of M —say x ∈ δ στ M ⊆ δ στ E —because y divides x . Write x = P kj =1 x j ∈ δ στ E = L kj =1 δ στ E j .For any index j such that x j = 0, Corollary 7.25 and the density hypothesis yields a σ -vicinity of e a containing a socle element e s j mapped to by an S -cogenerator s j that isdivisible by y j . An S -cogenerator s of M divisible by y is constructed from s j just asan ordinary cogenerator is constructed from s j in the proof of Theorem 6.7.Now suppose that S τ is not dense in soc τ for some face τ , so some downset D ⊆ Q has a cogenerator a ∈ Q whose image e a ∈ deg soc στ k [ D ] ⊆ Q/ R τ has a σ -vicinitydeg Q/ R τ soc τ k [ D ] ∩ ( u + Q ∇ σ ) / R τ devoid of images of S -cogenerators along τ . Ap-pealing to Lemma 7.3, the intersection of u + Q ∇ σ with a σ -vicinity O of a in D from Proposition 7.12 contains another σ -vicinity O ′ of a that still satisfies the con-clusion of Proposition 7.12 because every submodule of any τ -coprimary module glob-ally supported on τ is also τ -coprimary and globally supported on τ . The injection k [ O ′ ] ֒ → k [ D ] yields an injection S τ k [ O ′ ] ֒ → S τ k [ D ], but by construction S τ k [ D ]vanishes in all degrees from deg Q/ R τ soc τ k [ O ′ ], so S τ k [ O ′ ] = 0. On the other hand,soc τ ′ k [ O ′ ] = 0 for τ ′ = τ by Corollary 7.14, so the subfunctor S τ ′ vanishes on k [ O ′ ] forall faces τ ′ . Consequently, applying S τ ′ to the homomorphism ϕ : k [ O ′ ] → ֒ → τ ′ even though ϕ is not injective. (cid:3) Essential submodules via density in socles
Given a closed cogenerator of M , there is an obvious submodule of M containing thesocle element, namely the submodule generated by the cogenerator itself. In contrast,open cogenerators are not elements of M itself. How, then, do cogenerators detectinjectivity in Theorem 6.7? Each cogenerator must still yield a submodule to witnessthe injectivity, because injectivity means there is no actual submodule of M that goesto 0. The cogenerator merely indicates the presence of such a submodule, rather thanbeing an element of it. This section reconstructs an honest submodule around eachcogenerator. It requires much of the theory in earlier sections. As a consequence, asubmodule M ′ ⊆ M is an essential submodule precisely when the socle of M ′ is densein that of M (Theorem 8.5).More precisely, the σ -vicinities in Proposition 7.12 transfer cogenerators back intohonest submodules; they are, in that sense, the reverse of Definition 3.11. In fact thistransference of cogenerators into submodules works not merely for indicator quotientsbut for arbitrary modules with finite downset hulls, as in Theorem 8.5. The key is thegeneralization of σ -vicinities to arbitrary downset-finite modules. Definition 8.1.
Fix a module M over a real polyhedral group Q and a face τ .1. A σ -divisor (Definition 3.19) y ∈ M of a cogenerator of M along τ with nadir σ (Definition 4.29) is nearby if y is globally supported on τ (Definition 2.25).2. A σ -vicinity in M of a cogenerator s ∈ δ στ M is a submodule of M generated bya nearby σ -divisor of s .3. A neighborhood in soc τ M of a homogeneous socle element e s ∈ soc στ M is soc τ N for a σ -vicinity N in M of a cogenerator in δ στ M that maps to e s .4. An inclusion S τ ⊆ soc τ M of ( Q/ R τ ×∇ τ )-modules is dense if for all σ ⊇ τ , everyneighborhood of every homogeneous element of soc στ M intersects S τ nontrivially. Lemma 8.2.
Every neighborhood in M of every homogeneous element in soc στ M is a τ -coprimary submodule of M globally supported on τ .Proof. Let y be a nearby σ -divisor of a cogenerator s ∈ δ στ M . Let x be a homogeneousmultiple of y . That x is supported on τ is automatic from the hypothesis that y issupported on τ . To say that h y i is τ -coprimary means, given that it is supported SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 47 on τ , that h y i is a submodule of its localization along τ . But s remains a cogeneratorafter localizing along τ by Proposition 4.40, so x must remain nonzero because it stilldivides s after localizing. (cid:3) Proposition 8.3.
Fix a downset-finite module M over a real polyhedral group withfaces σ ⊇ τ . Every cogenerator in δ στ M has a σ -vicinity in M .Proof. Let s ∈ δ στ M be the cogenerator, and let its degree be deg Q ( s ) = a ∈ Q .Choose a downset hull M ֒ → E = L kj =1 E j , so E j = k [ D j ] for a downset D j . Express s = s + · · · + s k ∈ δ στ E = L kj =1 δ στ E j . Proposition 7.12 produces a σ -vicinity O j of a in Q , for each index j , such that k [ O j ∩ D j ] is a σ -vicinity in E j of the image e s j ∈ soc στ E j . Lemma 7.3 then yields a single σ -vicinity O = a − v + Q + of a in Q thatlies in the intersection T kj =1 O j . The cogenerator s ∈ δ στ is a direct limit over a − σ ◦ ;since O contains a neighborhood (in the usual topology) of a in σ ◦ , some element y ∈ M with degree in O is a σ -divisor of s . This element y is nearby s by construction. (cid:3) The following generalization of Corollary 7.14 to modules with finite downset hullsis again the decisive computation.
Corollary 8.4.
Fix a downset-finite τ -coprimary Q -module M globally supported ona face τ of a real polyhedral group Q . Then soc τ ′ M = 0 unless τ ′ = τ .Proof. Proposition 4.40 implies that soc τ ′ M = 0 unless τ ′ ⊇ τ by definition of globalsupport: localizing along τ ′ yields M τ ′ = 0 unless τ ′ ⊆ τ . On the other hand, applyingProposition 8.3 to any cogenerator of M along a face τ ′ implies that τ = τ ′ , because no τ -coprimary module has a submodule supported on a face strictly contained in τ . (cid:3) Theorem 8.5.
In a downset-finite module M over a real polyhedral group, M ′ is anessential submodule if and only if soc τ M ′ ⊆ soc τ M is dense for all faces τ .Proof. First assume that M ′ is not an essential submodule, so N ∩ M ′ = 0 for somenonzero submodule N ⊆ M . Let s ∈ δ στ N be a cogenerator. Any σ -vicinity of s in N ,afforded by Proposition 8.3, has a socle along τ that is a neighborhood of e s in soc τ M whose intersection with soc τ M ′ is 0. Therefore soc τ M ′ ⊆ soc τ M is not dense.Now assume that soc τ M ′ ⊆ soc τ M is not dense for some τ . That means soc στ M for some nadir σ has an element e s with a neighborhood soc τ N that intersects soc τ M ′ in 0. But soc τ N ∩ soc τ M ′ = soc τ ( N ∩ M ′ ) by Corollary 6.9.2. The vanishing of thissocle along τ means that soc τ ′ ( N ∩ M ′ ) = 0 for all faces τ ′ by Corollary 8.4, and thus N ∩ M ′ = 0 by Corollary 6.9.1. Therefore M ′ is not an essential submodule of M . (cid:3) Example 8.6.
The convex hull of , e , e in R but with the first standard basisvector e removed defines a subquotient M of k [ R ]. It has submodule M ′ that is theindicator module for the same triangle but with the entire x -axis removed. All of thecogenerators of both modules occur along the face τ = { } because both modules are globally supported on { } . However the ambient module—but not the submodule—hasa cogenerator y ∈ δ στ M with nadir σ = x -axis of degree e : ⊆ with σ -vicinities of e .A typical σ -vicinity of y in M is shaded in light blue (in fact, M itself is also a σ -vicinity of y ), with the corresponding vicinity in soc στ M in bold blue. Every suchvicinity contains socle elements in soc στ M ′ , so M ′ ⊆ M is an essential submoduleby Theorem 8.5. Trying to mimic this example in a finitely generated context isinstructive: pixelated rastering of the horizontal lines either isolates the socle elementat the right-hand endpoint of the bottom edge or prevents it from existing in the firstplace by aligning with the right-hand end of the line above it.9. Primary decomposition over real polyhedral groups
This section takes the join of [Mil20b], which develops primary decomposition as faras possible over arbitrary polyhedral partially ordered groups, and Section 4, whichdevelops socles over real polyhedral groups. That is, it investigates how socles interactwith primary decomposition in real polyhedral groups.9.1.
Associated faces and coprimary modules.
What makes the theory for real polyhedral groups stronger than for arbitrary polyhe-dral partially ordered groups is the following notion familiar from commutative algebra,except that (as noted in Section 8) socle elements do not lie in the original module.
Definition 9.1.
A face τ of a real polyhedral group Q is associated to a downset-finite Q -module M if soc τ M = 0. If M = k [ D ] for a downset D then τ is associated to D .The set of associated faces of M or D is denoted by Ass M or Ass D . Theorem 9.2.
A downset-finite module M over a real polyhedral group is τ -coprimary(Definition 2.27) if and only if soc τ ′ M = 0 whenever τ ′ = τ or equivalently Ass( M ) = { τ } .Proof. If M is not τ -coprimary then either M → M τ has nonzero kernel N , or M → M τ is injective while M τ has a submodule N τ supported on a face strictly containing τ . Inthe latter case, moving up by an element of τ shows that N = N τ ∩ M is nonzero. Ineither case, any cogenerator of N lies along a face τ ′ = τ , so 0 = soc τ ′ N ⊆ soc τ ′ M .On the other hand, if M is τ -coprimary then Γ τ M is an essential submodule of M because every nonzero submodule of M ⊆ M τ has nonzero intersection with Γ τ M τ , andhence with M ∩ Γ τ M τ = Γ τ M , inside of the ambient module M τ by Definition 2.27.Theorem 8.5 says that soc τ ′ Γ τ M ⊆ soc τ ′ M is dense for all τ ′ . But soc τ ′ Γ τ M = 0 for τ ′ = τ by Corollary 8.4, so density implies soc τ ′ M = 0 for τ ′ = τ . (cid:3) SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 49
Lemma 9.3.
A downset D in a real polyhedral group is τ -coprimary if and only if D = [ faces σ with σ ⊇ τ [ a ∈ A στ a + τ − Q ∇ σ for sets A στ ⊆ Q such that the image in Q/ R τ × ∇ τ of S σ ⊇ τ A στ ×{ σ } is an antichain,and in that case A στ projects to a subset of deg Q/ R τ soc στ k [ D ] ⊆ Q/ R τ for each σ .Proof. If D is τ -coprimary, then it is such a union by Theorem 9.2 and Theorem 7.19,keeping in mind the antichain consequences of Example 4.11.On the other hand, if D is such a union, then first of all it is stable under trans-lation by R τ by Lemma 7.18. Working in Q/ R τ , therefore, assume that τ = { } .Example 4.11 implies that every element of A στ is a cogenerator of D with nadir σ .Proposition 7.12 produces a σ -vicinity O σ a of a in D that is globally supported on { } (and hence { } -coprimary). But every element b ∈ D that precedes a also precedessome element in O σ a ; that is, b (cid:22) a ⇒ ( b + Q + ) ∩O σ a = ∅ . The union of the σ -vicinities O σ a over all faces σ and elements a ∈ A στ therefore cogenerates D , so D is coprimaryby Proposition 2.29. (cid:3) Remark 9.4.
The antichain condition in Lemma 9.3 is necessary: Q itself is theunion of all translates of − Q , but Q is Q + -coprimary, whereas − Q is { } -coprimary.Moreover, the ∇ τ component of the antichain condition is important; that is, the nadirsalso come into play. For a specific example, take D ⊆ R to be the union of the opennegative quadrant cogenerated by and the closed negative quad-rant cogenerated by any point on the strictly negative x -axis. The Q -components of the two cogenerators are comparable in Q , but thenadirs are comparable the other way (it is crucial to remember that theordering on the nadirs is by F op Q , not F Q , so smaller faces are higher inthe poset). Of course, no claim can be made that deg Q/ R τ soc στ k [ D ] equals the imagein Q/ R τ of A στ ; only the density claim in Theorem 7.19 can be made. Lemma 9.5.
Over any real polyhedral group, every submodule of a coprimary moduleis coprimary.Proof.
Theorem 9.2, Theorem 6.7, and Definition 9.1. (cid:3)
Canonical decompositions of intervals.
The true real polyhedral generalizations of unique monomial minimal irreducible andprimary decomposition for monomial ideals in ordinary polynomial rings are Theo-rem 9.7 and Theorem 9.12, respectively. These concern intervals rather than downsetsbecause while the set of (exponents of) monomials outside of an ideal in k [ N n ] is adownset in N n , as a subset of Z n this set of exponents is merely an interval. Thesereal polyhedral decompositions make full use of the notions of socle, cogenerator, anddensity introduced in earlier sections: the topological graded algebra is essential. Theorem 6.5, on the existence of enough downset cogenerators, holds verbatim forintervals instead of downsets. The proof is a telling application of the dense-soclecharacterization of essential submodules in Theorem 8.5, combined with its precursor,Theorem 7.19, that dense subsets of cogenerators suffice to express a downset as aunion. With no additional effort, enough cogenerators of an interval can be located inany dense subset of the full set of cogenerators, generalizing Theorem 7.19 directly. Tostate this version it is useful to make a final definition concerning density.
Definition 9.6.
Fix a real polyhedral group Q . Let S be a family of sets S στ , indexedby pairs of faces σ and τ of Q + with σ ⊇ τ , such that S στ ⊆ Q/ R τ . This family S is dense in another such family ˆ S if the σ -closure (Definition 7.8.2) of S τ = S σ ⊇ τ S στ in Q/ R τ contains ˆ S στ for all σ ⊇ τ . Theorem 9.7.
Fix an interval I in a real polyhedral group Q and b ∈ I . There arefaces τ ⊆ σ of Q + and a cogenerator a of I along τ with nadir σ such that b (cid:22) a . Infact, the cogenerator a can be selected from any given family A = { A στ } σ ⊇ τ , where A στ ⊆ Q is a set of cogenerators of I along τ with nadir σ , as long as the family { A στ / R τ } σ ⊇ τ ofsets of cosets in Q/ R τ is dense in deg soc k [ I ] = { deg Q/ R τ soc στ k [ I ] } σ ⊇ τ . In this case, I = [ faces σ, τ with σ ⊇ τ [ a ∈ A στ ( a + τ − Q ∇ σ ) ∩ I. Proof.
Let D be the downset cogenerated by I . The interval module k [ I ] is an essen-tial submodule of k [ D ] by construction. Theorem 8.5 implies that soc k [ I ] is densein soc k [ D ], and hence any family that is dense in soc k [ I ] is dense in soc k [ D ]. The-orem 7.19 therefore expresses D as the union of downsets cogenerated by the givencogenerators of I . This means that every element of D precedes one of the givencogenerators of I , so certainly every element of I precedes such a cogenerator of I . (cid:3) Corollary 9.8.
Every interval I in a real polyhedral group has a canonical irreducibledecomposition as a union of intervals in I cogenerated by the family of global cogenera-tors of I . This decomposition is minimal in the sense that the only subfamilies yieldinga union that still equals I are dense in the canonical family.Proof. Let A στ in Theorem 7.19 comprise all cogenerators of I along τ with nadir σ . (cid:3) Remark 9.9.
The case of Theorem 9.7 corresponding to finitely generated monomialideals in real-exponent polynomial rings is treated in [ASW15].
Example 9.10.
Looking at the proof of Theorem 9.7, it is tempting to posit that if D is the downset cogenerated by an interval I in a real polyhedral group Q , then theinclusion I ⊆ D induces a socle isomorphism soc k [ I ] −→∼ soc k [ D ]. But alas, it fails:the module M ′ in Example 8.6 cogenerates a downset that shares with the ambientmodule M its cogenerator y , which is not a cogenerator of M ′ itself. Consequently,socle density as in Theorem 8.5 is the best one can hope for. (See Remark 9.14.) SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 51
Definition 9.11.
A primary decomposition (Example 2.31) I = S kj =1 I j of an intervalin a real polyhedral group is minimal if1. each face associated to I is associated to precisely one of the downsets I j , and2. the natural map soc τ k [ I ] → soc τ L kj =1 k [ I j ] is an isomorphism for all faces τ . Theorem 9.12.
Every interval I ⊆ Q in a real polyhedral group Q has a canonical pri-mary decomposition (Example 2.31) whose corresponding primary decomposition of theinterval module k [ I ] is minimal. Explicitly, this decomposition expresses I as a union I = [ τ ∈ Ass I [ σ ⊇ τ a ∈ A στ ( a + τ − Q ∇ σ ) ∩ I of coprimary intervals, where A στ ⊆ Q is the set of cogenerators of I along τ with nadir σ .Proof. That I equals the union is a special case of Theorem 9.7. The inner union forfixed τ is τ -coprimary by Lemma 9.5 because it is coprimary when the intersectionwith I is omitted, by Lemma 9.3. To see that the socle maps are isomorphisms, let I τ be the inner union for fixed τ ∈ Ass I . Applying soc τ to the primary decomposition k [ I ] ֒ → L τ k [ I τ ] yields soc τ k [ I ] ֒ → soc τ k [ I τ ] by Theorem 6.7. The question is whetherevery cogenerator of I τ is indeed a cogenerator of I . Each cogenerator of I τ hasa nadir σ for some face σ ⊇ τ , and Lemma 8.2 produces a corresponding σ -vicinity(Definition 8.1.2) in k [ I τ ]. But I τ is contained in I , explicitly by the way it is defined asa union, given Theorem 9.7, so this vicinity is contained in I . Therefore the cogeneratorof I τ in question must also be a cogenerator of I . (cid:3) Example 9.13.
The global support at the right-hand end of Example 2.26 yields aredundant primary component, and hence is not part of the minimal primary decom-position in Example 1.5 that comes from Theorem 9.12, because the interval I = y x has no cogenerators along the y -axis, in the functorial sense of Definition 4.29. Theillustrations in Example 2.26 show that k [ I ] has elements supported on the face of R that is the (positive) y -axis, but the socle of k [ I ] along the y -axis is 0, as the top halfof the boundary curve of k [ I ] is supported on the origin. Remark 9.14.
The sole reason why Example 9.10 happens is the penultimate sentenceof the proof of Theorem 9.12: the downset cogenerated by I τ need not be containedin I , so a vicinity of a cogenerator of this downset can have empty intersection with I . Example 9.15.
The canonical τ -primary component in Theorem 9.12 can differ fromthe τ -primary component P τ ( D ) in [Mil20b, Definition 3.2.6 and Corollary 3.11],namely the downset Γ τ ( D τ ) − Q + cogenerated by the local τ -support of D . How-ever, it takes dimension at least 3 to force a difference. For a specific case, let τ be the z -axis in R , and let D be the { } -coprimary (Lemma 9.3) downset in R cogeneratedby the nonnegative points on the surface z = 1 / ( x + y ). Then every point on thepositive z -axis is supported on τ in D . That would suffice, for the present purpose, but for the fact that τ fails to be associated to D . The remedy is to force τ to be asso-ciated by taking the union of D with any downset D = a + τ − R with a = ( x, y, z )satisfying xy <
0, the goal being for D D to be τ -coprimary but not contain the z -axis itself. The canonical τ -primary component of D = D ∪ D is just D itself, butby construction Γ τ D also contains the positive z -axis. (Note: D = D ∪ D is not thecanonical primary decomposition of D because D swallows an open set of cogenera-tors of D , so these cogenerators must be omitted from the { } -primary component toinduce an isomorphism on socles.) The reason why three dimensions are needed is that τ must have positive dimension, because elements supported on τ must be cloaked bythose supported on a smaller face; but τ must have codimension more than 1, becausethere must be enough room modulo R τ to have incomparable elements.9.3. Downset and interval hulls of modules.Definition 9.16.
A downset hull
M ֒ → E = L kj =1 E j (Definition 2.16) of a moduleover a real polyhedral group is1. coprimary if E j = k [ D j ] is coprimary for all j , so D j is a coprimary downset, and2. dense if the induced map soc τ M ֒ → soc τ E is dense (Definition 8.1.4) for all τ . Remark 9.17.
The density condition in Definition 9.16 is equivalent to M being anessential submodule of the downset hull E , by Theorem 8.5. Theorem 9.18.
Every downset-finite module M over a real polyhedral group admits adense coprimary downset hull.Proof. Suppose that M → L kj =1 E j is any finite downset hull. Replacing each E j bya primary decomposition of E j , using Theorem 9.12, assume that this downset hullis coprimary. Let E τ be the direct sum of the τ -coprimary summands of E . Thensoc τ E = soc τ E τ by Theorem 9.2. Replacing M with its image in E τ , it thereforesuffices to treat the case where M is τ -coprimary and E = E τ .The proof is by induction on the number k of summands of E . If k = 1 then M = k [ I ] ⊆ k [ D ] = k [ D τ ] is an interval submodule of a τ -coprimary downset moduleby hypothesis. If D ′ is the downset cogenerated by I , then k [ I ] ⊆ k [ D ′ ] is an essentialsubmodule by construction and coprimary by Lemma 9.5. Thus M = k [ I ] ⊆ k [ D ′ ] = E is a dense coprimary downset hull by Theorem 8.5.When k >
1, let M ′ = ker( M → E k ). Then M ′ ֒ → L k − j =1 E j , so it has a densecoprimary downset hull M ′ ֒ → E ′ by induction. The k = 1 case proves that thequotient M ′′ = M/M ′ has a dense coprimary downset hull M ′′ ֒ → E ′′ . The exactsequence 0 → M ′ → M → M ′′ → → soc τ M ′ → soc τ M → soc τ M ′′ which, if exact, automatically splits by Remark 4.36. Hence it suffices to prove thatsoc τ M → soc τ M ′′ is surjective. For that, note that the image of soc τ M in soc τ E SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 53 surjects onto its projection to soc τ E k , but the image of soc τ M → soc τ E k is the imageof the injection soc τ M ′′ ֒ → soc τ E k by construction. (cid:3) Remark 9.19.
Theorem 9.18 is the analogue of existence of minimal injective hulls forfinitely generated modules over noetherian rings [BH98, Section 3.2] (see also [Mil20a,Proposition 5.7 or Theorem 5.19] for finitely determined Z n -modules, which need notbe finitely generated). The difference here is that a direct sum—as opposed to directproduct—can only be attained by gathering cogenerators into finitely many bunches. Example 9.20.
The indicator module for the disjoint union of the strictly negativeaxes in the plane injects in an appropriate way into one downset module (the puncturednegative quadrant) or a direct sum of two (negative quadrants missing one boundaryaxis each). Thus the “required number” of downsets for a downset hull of a givenmodule is not necessarily obvious and might not be a functorial invariant. This maysound bad, but it should not be unexpected: the quotient by an artinian monomialideal in an ordinary polynomial ring can have socle of arbitrary finite dimension, so thenumber of coprincipal downsets required is well defined, but if downsets that are notnecessarily coprincipal are desired, then any number between 1 and the socle dimensionwould suffice. This phenomenon is related to Remark 4.36: breaking the socle of adownset into two reasonable pieces expresses the original downset as a union of thetwo downsets cogenerated by the pieces.
Definition 9.21. An interval hull of a module M over an arbitrary poset is an injection M ֒ → H = L j ∈ J H j with each H j being an interval module (Example 2.6.3). The hullis finite if J is finite. Over a real polyhedral group a finite interval hull is1. coprimary if H j = k [ I j ] is coprimary for all j , so I j is a coprimary interval, and2. minimal if the induced map soc τ M ֒ → soc τ H is an isomorphism for all faces τ . Remark 9.22.
Minimality of interval hulls is stronger than density of downset hulls:the induced socle inclusion is required to be an isomorphism rather than merely dense.
Theorem 9.23.
Every downset-finite module M over a real polyhedral group admits aminimal coprimary interval hull.Proof. The transition from downsets to intervals alters one key aspect of the proof ofTheorem 9.18, namely the base of the induction: when k = 1 the module is already aninterval module, so the identity map is a minimal interval hull, as the socle inclusion isthe identity isomorphism. The rest of the proof goes through mutatis mutandis, chang-ing “downset” to “interval”, “dense” to “minimal”, and all instances of “ E ” to “ H ”. (cid:3) Remark 9.24.
The proof of Theorem 9.23 shows more than its statement: any co-primary interval hull
M ֒ → H = H ⊕ · · · ⊕ H k of a coprimary module M induces afiltration 0 = M ⊂ M ⊂ · · · ⊂ M k = M such that soc τ M = L kj =1 soc τ ( M j /M j − ),and furthermore M ֒ → H can be “minimalized”, in the sense that a minimal hull H ′ can be constructed inside of H so that soc τ M ∼ = soc τ H ′ decomposes as direct sum of factors soc τ ( M j /M j − ) ∼ = soc τ H ′ j . Reordering the summands H j yields another fil-tration of M with the same property. That soc τ M breaks up as a direct sum in somany ways should not be shocking, in view of Remark 4.36. The main content is thatall of the socle elements of M/M k − are inherited from M , essentially because M k − is the kernel of a homomorphism to a direct sum of downset modules, so M k − has nocogenerators that are not inherited from M .9.4. Minimal primary decomposition of modules.Definition 9.25.
A primary decomposition
M ֒ → L ri =1 M/M i (Definition 2.30) of amodule over a real polyhedral group is minimal if soc τ M → soc τ L ri =1 M/M i is anisomorphism for all faces τ . Definition 9.26.
Given a coprimary interval hull
M ֒ → H of an arbitrary downset-finite module M over a real polyhedral group, write H τ for the direct sum of allsummands of H that are τ -coprimary. The kernel M τ of the composite homomorphism M → H → H τ is the τ -primary component of M . Theorem 9.27.
Every downset-finite module M over a real polyhedral group admits aminimal primary decomposition. In fact, if M ֒ → H is a coprimary interval hull then M ֒ → L τ M/M τ is a primary decomposition that is minimal if M ֒ → H is minimal.Proof. Fix a coprimary interval hull
M ֒ → H . The quotient M/M τ is τ -coprimary byLemma 9.5 since it is a submodule of the coprimary module H τ , and M → L τ M/M τ is injective because the injection M ֒ → L τ H τ = H factors through L τ M/M τ ⊆ H .Theorem 9.2 implies that soc τ ′ ( M/M τ ) = 0 unless τ = τ ′ , regardless of whether M ֒ → H is minimal. And if the hull is minimal, then soc τ M → soc τ H τ is an isomor-phism (by hypothesis) that factors through the injection soc τ ( M/M τ ) ֒ → soc τ H τ (byconstruction), forcing soc τ M ∼ = soc τ ( M/M τ ) to be an isomorphism for all τ . (cid:3) Remark 9.28.
Theorem 9.27 enables full access to interpretations of primary de-composition in persistent homology, now with a notion of minimality for multiple realparameters instead of versions without minimality for partially ordered groups [Mil20b]or with minimality in the discrete case [HOST19]. Primary decomposition has impor-tant statistical implications for applications of multipersistence [MT20].10.
Socles and essentiality over discrete polyhedral groups
The theory developed for real polyhedral groups in Sections 4–11 applies as well todiscrete polyhedral groups (Example 2.4). The theory is easier in the discrete case, inthe sense that only closed cogenerator functors are needed, and none of the densityconsiderations in Sections 7–8 are relevant. The deduction of the discrete case iselementary, but it is worthwhile to record the results, both because they are useful andfor comparison with the real polyhedral case.
SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 55
For the analogue of Theorem 6.7, the notion of divisibility in Definition 3.19 makessense, when σ = { } , verbatim in the discrete polyhedral setting: an element y ∈ M b divides x ∈ M a if b ∈ a − Q + and y x under the natural map M b → M a . Theorem 10.1 (Discrete essentiality of socles) . Fix a homomorphism ϕ : M → N ofmodules over a discrete polyhedral group Q .1. If ϕ is injective then soc τ ϕ : soc τ M → soc τ N is injective for all faces τ of Q + .2. If soc τ ϕ : soc τ M → soc τ N is injective for all faces τ of Q + and M is downset-finite, then ϕ is injective.Each homogeneous element of M divides some closed cogenerator of M .Proof. Item 1 is a special case of Proposition 4.23. Item 2 follows from the divisi-bility claim, for if y divides a closed cogenerator s along τ then ϕ ( y ) = 0 wheneversoc τ ϕ ( e s ) = 0, where e s is the image of s in soc τ M . The divisibility claim follows fromthe case where M is generated by y ∈ M b . But h y i is a noetherian k [ Q + ]-module andhence has an associated prime. This prime equals the annihilator of some homogeneouselement of h y i , and the quotient of k [ Q + ] modulo this prime is k [ τ ] for some face τ [MS05, Section 7.2]. That means, by definition, that the homogeneous element is aclosed cogenerator along τ divisible by y . (cid:3) The discrete analogue of Theorem 7.27 is simpler in both statement and proof.
Theorem 10.2.
Fix subfunctors S τ ⊆ soc τ for all faces τ of a discrete polyhedralgroup. Theorem 10.1 holds with S in place of soc if and only if S τ = soc τ for all τ .Proof. Every subfunctor of any left-exact functor takes injections to injections; there-fore Theorem 10.1.1 holds for any subfunctor of soc τ by Proposition 4.23. The contentis that Theorem 10.1.2 fails as soon as S τ M ( soc τ M for some module M and someface τ . To prove that failure, suppose e s ∈ soc τ M r S τ M for some closed cogenerator s of M along τ . Then h s i ⊆ M induces an injection S τ h s i ֒ → S τ M , but by constructionthe image of this homomorphism is 0, so S τ h s i = 0. But soc τ ′ h s i = 0 for all τ ′ = τ because h s i is abstractly isomorphic to k [ τ ], which has no associated primes otherthan the kernel of k [ Q + ] ։ k [ τ ]. Consequently, applying S τ ′ to the homomorphism ϕ : h s i → ֒ → τ ′ even though ϕ is not injective. (cid:3) The analogue of Theorem 8.5 is similarly simpler.
Theorem 10.3.
In any module M over a discrete polyhedral group, M ′ is an essentialsubmodule if and only if soc τ M ′ = soc τ M for all faces τ .Proof. First assume that M ′ is not an essential submodule, so N ∩ M ′ = 0 for somenonzero submodule N ⊆ M . Any closed cogenerator s of N along any face τ maps to anonzero element of soc τ M that lies outside of soc τ M ′ . Conversely, if soc τ M ′ = soc τ M ,then any closed cogenerator of M that maps to an element soc τ M r soc τ M ′ generatesa nonzero submodule of M whose intersection with M ′ is 0. (cid:3) The analogue of Theorem 9.12 uses slightly modified definitions but its proof is easier.
Definition 10.4.
A primary decomposition (Definition 2.31) I = S kj =1 I j of an intervalin a discrete polyhedral group is minimal if1. the intervals I j are coprimary for distinct associated faces of I , and2. the natural map soc τ k [ I ] → soc τ L kj =1 k [ I j ] is an isomorphism for all faces τ ,where τ is associated if some element generates an upset in I that is a translate of τ . Theorem 10.5.
Every interval I in a discrete polyhedral group has a canonical minimalprimary decomposition I = S τ I τ as a union of coprimary intervals I τ = [ a τ ∈ deg soc τ k [ I ] ( a τ − Q + ) ∩ I, where a τ is viewed as an element in Q/ Z τ to write a τ ∈ deg soc τ k [ I ] but a τ ⊆ Q isviewed as a coset of Z τ to write a τ − Q + .Proof. The interval I is contained in the union by the final line of Theorem 10.1, butthe union is contained in I because every closed cogenerator of I is an element of I . Itremains to show that I τ is coprimary and that the socle maps are isomorphisms.Each nonzero homogeneous element y ∈ k [ I τ ] divides an element s y whose degreelies in some coset a τ ∈ deg soc τ k [ I ] by construction. As I τ ⊆ I , each such element s y isa closed cogenerator of I τ along τ . Therefore k [ I τ ] is coprimary, inasmuch as no primeother than the one associated to k [ τ ] can be associated to I τ . The same argument showsthat these elements s y generate an essential submodule of I τ , and then Theorem 10.3yields the isomorphism on socles. (cid:3) Corollary 10.6.
Every interval I in a discrete polyhedral group Q has a unique irre-dundant irreducible decomposition as a union of its irreducible components , namelythe coprincipal intervals ( a τ − Q + ) ∩ I in Theorem 10.5.Proof. The irredundant condition is a consequence of the socle isomorphisms. (cid:3)
Definition 10.7.
A downset hull
M ֒ → E = L kj =1 E j (Definition 2.16) of a moduleover a discrete polyhedral group is1. coprimary if E j = k [ D j ] is coprimary for all j , so D j is a coprimary downset, and2. minimal if the induced map soc τ M → soc τ E is an isomorphism for all faces τ .The discrete analogue of Theorem 9.18 appears to be new. Theorem 10.8.
Every downset-finite module M over a discrete polyhedral group ad-mits a minimal coprimary downset hull. SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 57
Proof.
The argument follows that of Theorem 9.18, using Theorem 10.5 instead ofTheorem 9.12 and Theorem 10.3 instead of Theorem 8.5. In the course of the proof,note that the discrete analogue of Theorem 9.2 is the definition of associated prime,making the analogue of Lemma 9.5 trivial, and that the analogue of Remark 4.36 holds(more easily) in the discrete polyhedral setting. (cid:3)
Remark 10.9.
Remark 9.24 holds verbatim over discrete polyhedral groups.Finally, here is the discrete version of minimal primary decomposition.
Definition 10.10.
A primary decomposition
M ֒ → L ri =1 M/M i (Definition 2.30) ofa module over a discrete polyhedral group is minimal if soc τ M → soc τ L ri =1 M/M i isan isomorphism for all faces τ . Definition 10.11.
Given a coprimary downset hull
M ֒ → E of an arbitrary downset-finite module M over a discrete polyhedral group, write E τ for the direct sum of allsummands of E that are τ -coprimary. The kernel M τ of the composite homomorphism M → E → E τ is the τ -primary component of M . Theorem 10.12.
Every downset-finite module M over a discrete polyhedral groupadmits a minimal primary decomposition. If M ֒ → E is a coprimary downset hull then M ֒ → L τ M/M τ is a primary decomposition that is minimal if M ֒ → E is minimal.Proof. Follow the proof of Theorem 9.27, using downset hulls instead of interval hullsbecause, in contrast with the real polyhedral case (Theorems 9.18 and 9.23), downsethulls are minimal—not merely dense—in the discrete context (Theorem 10.8). (cid:3)
Generator functors and tops
The theory of generators is Matlis dual (Section 2.5) to the theory of cogenerators.Every result for socles, downsets, and cogenerators therefore has a dual. All of thesedual statements can be formulated so as to be straightforward, but sometimes they areless natural (see Remarks 11.19 and 11.20, for example), sometimes they are weaker(see Remark 11.5), and sometimes there are natural formulations that must be provedequivalent to the straightforward dual (see Definition 11.28 and Theorem 11.31, for ex-ample). This section presents those Matlis dual notions that are used in later sections.11.1.
Lower boundary functors.
The following are Matlis dual to Definition 3.11. Lemma 3.13, and Definition 3.15.
Definition 11.1.
For a module M over a real polyhedral group Q , a face ξ of Q + , anda degree b ∈ Q , the lower boundary beneath ξ at b in M is the vector space( ∂ ξ M ) b = M b + ξ = lim ←− b ′ ∈ b + ξ ◦ M b ′ . Lemma 11.2.
The structure homomorphisms of M as a Q -module induce naturalhomomorphisms M b + ξ → M c + η for b (cid:22) c in Q and faces ξ ⊆ η of Q + . (cid:3) Remark 11.3.
In contrast with Remark 3.14, the relevant monoid structure here onthe face poset F Q of the positive cone Q + is opposite to the monoid denoted F op Q . Inthis case the monoid axioms use that F Q is a bounded join semilattice, the monoidunit being { } . The induced partial order on F Q is the usual one, with ξ (cid:22) η if ξ ⊆ η . Definition 11.4.
Fix a module M over a real polyhedral group Q and a degree b ∈ Q .The lower boundary functor takes M to the Q ×F Q -module ∂M whose fiber over b ∈ Q is the F Q -module ( ∂M ) b = M ξ ∈F Q M b + ξ = M ξ ∈F Q ( ∂ ξ M ) b . The fiber of ∂M over ξ ∈ F Q is the lower boundary ∂ ξ M of M beneath ξ . Remark 11.5.
Direct and inverse limits play differently with vector space duality.Consequently, although the notion of lower boundary functor is categorically dualto the notion of upper boundary functor, the duality only coincides unfettered withvector space duality in one direction, and some results involving tops are necessarilyweaker than the corresponding results for socles; compare Theorem 6.7 with 12.3 andExample 12.7, for instance. To make precise statements throughout this section ongenerator functors, starting with Lemma 11.7, it is necessary to impose a finitenesscondition that is somewhat stronger than Q -finiteness (Definition 2.9.2). Definition 11.6.
A module M over a real polyhedral group Q is infinitesimally Q -fin-ite if its lower boundary module ∂M is Q -finite. Lemma 11.7. If ξ is a face of a real polyhedral group Q , then1. ∂ ξ ( M ∨ ) = ( δ ξ M ) ∨ for all Q -modules M , and2. ( ∂ ξ M ) ∨ = δ ξ ( M ∨ ) if M is infinitesimally Q -finiteProof. Degree by degree b ∈ Q , the first of these is because the vector space dualof a direct limit is the inverse limit of the vector space duals. Swapping “direct”and “inverse” only works with additional hypotheses, and one way to ensure these isto assume infinitesimal Q -finiteness of M . Indeed, then M = ∂ { } M is Q -finite, soreplacing M with M ∨ in the first item yields ∂ ξ M = (cid:0) δ ξ ( M ∨ ) (cid:1) ∨ by Lemma 2.37. Thus( ∂ ξ M ) ∨ = δ ξ ( M ∨ ), as ∂ ξ M —and hence (cid:0) δ ξ ( M ∨ ) (cid:1) ∨ and δ ξ ( M ∨ )—is also Q -finite. (cid:3) Example 11.8.
Any module M that is a quotient of a finite direct sum of upsetmodules (“upset-finite” in Definition 12.2) over a real polyhedral group is infinitesimally Q -finite. Indeed, the Matlis dual of such a quotient is a downset hull demonstratingthat M ∨ is downset-finite and hence Q -finite. Proposition 3.21 and exactness of upperboundary functors (Lemma 3.12) implies that δ ( M ∨ ) remains downset-finite and hence Q -finite. Applying Lemma 11.7.1 to M ∨ and using that ( M ∨ ) ∨ = M (Lemma 2.37) on SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 59 the left-hand side yields that ∂M is Q -finite. This example includes all tame modulesby the syzygy theorem [Mil20a, Theorem 6.12.4]. Proposition 11.9.
The category of infinitesimally Q -finite modules over a real poly-hedral group Q is a full abelian subcategory of the category of Q -modules. Moreover,the lower boundary functor is exact on this subcategory.Proof. Use Matlis duality, in the form of Lemma 11.7, along with Lemma 3.12. (cid:3)
Closed generator functors.
Here is the Matlis dual of Definition 4.1. Recall the skyscraper P -module k p there. Definition 11.10.
Fix an arbitrary poset P . The closed generator functor k ⊗ P − takes each P -module N to its closed top : the quotient P -moduletop N = k ⊗ P N = M p ∈ P k p ⊗ P N. When it is important to specify the poset, the notation P -top is used instead of top.A closed generator of degree p ∈ P is a nonzero element in (top N ) p . Example 11.11.
Elements of N p that persist from lower in the poset die in the tensorproduct k ⊗ P N . Consequently, R -modules like the maximal monomial ideal m ⊆ k [ R + ]have vanishing closed top, because every monomial with nonzero positive degree isdivisible by a monomial of smaller positive degree (its square root, for instance). Thisis merely the statement that m is not minimally generated. Remark 11.12. P -soc N ֒ → N is the universal P -module monomorphism that is 0when composed with all nonidentity maps induced by going up in P . The Matlis dual is N ։ P -top N , the universal P -module epimorphism that is 0 when composed with allnonidentity maps induced by going up in P . This is the essence of Proposition 11.15. Remark 11.13.
Matlis duality has an intrinsic asymmetry regarding the behavior oftops and socles. In the presence of sufficient finiteness, the asymmetry disappears, butin general it requires care to insert some finiteness appropriately. The following defini-tion, proposition, and proof are presented in (perhaps too much) detail to highlight howfiniteness enters. Local finiteness (Definition 11.14) can fail for modules over a partiallyordered group, but it is useful for the discrete (face lattice) half of the poset used tocompute tops and socles over real polyhedral groups; see Proposition 11.25. The otherfiniteness restriction, namely P -finiteness (Definition 2.9.2) has already appeared, withconsequences (see Example 2.41). Definition 11.14.
A module N over a poset P is locally finite if, for each poset element p ∈ P , there is a finite subset P ′ ( p, N ) ⊆ P such that, if N p → N p ′′ is nonzero for some p ′′ ∈ P , there is some p ′ ∈ P ′ ( p, N ) with p ′ ≺ p ′′ .Loosely, P ′ ( p, N ) sits between N p and any of its nonzero images higher in P . Proposition 11.15.
Fix a poset P with opposite poset P op . For a P -module N , Matlisduality interacts with tops and socles as follows:1. P -top ( N ) ∨ = P op -soc ( N ∨ ) for any P -module N , and2. P -top ( N ∨ ) = ( P op -soc N ) ∨ for any P op -finite or locally finite P op -module N .All of these hold with P and P op swapped.Proof. View the P -module N as a diagram of vector spaces indexed by P . Tensoringwith k in Definition 11.10 takes each vector space N p to the cokernel of the homomor-phism L p ′ ≺ p N p ′ → N p induced by the maps going up in P from p ′ to p . Let L p be theimage in N p of L p ′ ≺ p N p ′ . The vector space dual ( − ) ∗ of the surjection N p ։ N p /L p isthe kernel of the homomorphism Q p ′ ≻ p N ∗ p ′ ← N ∗ p , where p ′ ≻ p in the partial order on P op here. This kernel is Hom P op ( k , N ∨ ) p , proving the first equation by Definition 4.1.The second equation is similar: Hom P op ( k , − ) takes each vector space N p in thediagram N indexed by P op to the kernel of the homomorphism Q p ≺ p ′ N p ′ ← N p . Ifa finite sub-product—over a subset P ′ ( p, N ), say—suffices to compute the kernel, asthe P op -finite or locally finite conditions guarantee, then the vector space dual of thekernel is the cokernel of the homomorphism L p ′ ∈ P ′ ( p,N ) ( N p ′ ) ∗ → ( N p ) ∗ induced by themaps going up in P from p ′ to p in N ∨ . (cid:3) Closed generator functors along faces.
Generators along faces of partially ordered groups make sense just as cogeneratorsalong faces do; however, they are detected not by localization but by the Matlis dualoperation in Example 2.36, which is likely unfamiliar (and is surely less familiar thanlocalization). An element in the following can be thought of as an inverse limit ofelements of M taken along the negative of the face ρ . This is Matlis dual to theconstruction of the localization M ρ as a direct limit. Definition 11.16.
Fix a face ρ of a partially ordered group Q and a Q -module M . Set M ρ = Hom Q (cid:0) k [ Q + ] ρ , M (cid:1) . The following is Matlis dual to Definition 4.15.1; see Theorem 11.22. Duals for therest of Definition 4.15 are omitted for reasons detailed in Remarks 11.19 and 11.20.
Definition 11.17.
Fix a partially ordered group Q , a face ρ , and a Q -module M . The closed generator functor along ρ takes M to its closed top along ρ :top ρ M = (cid:0) k [ ρ ] ⊗ Q M (cid:1) ρ (cid:14) ρ. SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 61
Example 11.18.
When Q = R and ρ is the face of R along the x -axis, the module M ρ = Hom R (cid:0) k [ R ] ρ , M (cid:1) for the depicted module M isHom k , k = k . The effect is push the jagged right-hand boundary off to + ∞ . It is interesting tonote that the bottom edge of M does become homogenized in M ρ , because there isno way to place the closed purple upper half plane onto the blue module (either one,actually, though M with the jagged upper boundary is meant here). Indeed, the dottedleft-pointing horizontal ray has no location to place the solid left-pointing boundaryray of the half plane. In contrast, in any given homomorphism k [ R ] ρ → M , theright-pointing solid purple ray goes to 0 once it exits the blue region.The tensor product k [ ρ ] ⊗ R M is nonzero only on the solid blue segment along thebottom edge of M . Therefore (cid:0) k [ ρ ] ⊗ R M (cid:1) ρ = 0. Thus, while a nonzero homomorphism k [ Q + ] ρ → M detects any “infinitely backward-extending” portions of M , the closedtop only detects the “bottom-justified” such portions. Remark 11.19.
The notion of global closed cogenerator has a Matlis dual, but sincethe dual of an element—equivalently, a homomorphism from k [ Q + ]—is not an element,the notion of closed generator is not Matlis dual to a standard notion related to socles.A generator along a face ρ can be defined as an element of M ρ that is not a multipleof any generator of lesser degree, but making this precise requires care regarding what“degree of generator” and “lesser” mean; see [Mil20d]. Remark 11.20.
The notion of local socle has a Matlis dual, but it is not in any sense alocal top, because localization does not Matlis dualize to localization (Example 2.36).Instead, Matlis dualizing the local socle yields a functor k ⊗ Q/ Z ρ M ρ that surjectsonto top ρ M by the Matlis dual of Proposition 4.40. Local socles found uses in proofshere and there, such as Corollary 4.42, Proposition 7.12, Corollary 7.14, Lemma 8.2,and Corollary 8.4, via Proposition 4.40. But since Matlis duals of statements holdregardless of their proofs, given appropriate finiteness conditions (Definition 11.6),local socles and their Matlis duals have no further use in this paper.In the next lemma, a prerequisite to duality of closed socles and tops, a localizationof N along ρ on the left-hand side is hiding in the quotient-restriction (Definition 2.22). Lemma 11.21.
For any module N over a partially ordered group Q and any face ρ , ( N/ρ ) ∨ = ( N ∨ ) ρ /ρ. If N is Q -finite, then ( N ∨ ) /ρ = ( N ρ /ρ ) ∨ . Proof.
This is Example 2.36 plus the observation that quotient-restriction along ρ com-mutes with Matlis duality on modules that are already localized along ρ as can be seendirectly from Definitions 2.34 and 2.22. The detailed calculation goes like this:( N/ρ ) ∨ = ( N ρ /ρ ) ∨ = ( N ρ ) ∨ /ρ = ( N ∨ ) ρ /ρ. To derive the second displayed equation, use Lemma 2.37 to replace N by N ∨ in the firstequation, and then use Lemma 2.37 again to take the Matlis duals of both sides. (cid:3) Theorem 11.22.
For a module M over a partially ordered group Q ,1. (top ρ M ) ∨ = soc ρ ( M ∨ ) if k [ ρ ] ⊗ Q M is Q -finite, and2. top ρ ( M ∨ ) = (soc ρ M ) ∨ if M is Q -finite.Proof. The two are similar, but to indicate why the finitenesses must be assumed, bothare written out. The first and last lines of each half are by Definitions 11.17 and 4.15.1:(top ρ M ) ∨ = (cid:0) ( k [ ρ ] ⊗ Q M ) ρ /ρ (cid:1) ∨ = (cid:0) k [ ρ ] ⊗ Q M (cid:1) ∨ (cid:14) ρ by Lemma 11.21 and Q -finiteness of k [ ρ ] ⊗ Q M = Hom Q (cid:0) k [ ρ ] , M ∨ (cid:1) /ρ by Example 2.35= soc ρ ( M ∨ ) , and (soc ρ M ) ∨ = (cid:0) Hom Q (cid:0) k [ ρ ] , M (cid:1) /ρ (cid:1) ∨ = (cid:0) Hom Q (cid:0) k [ ρ ] , M (cid:1) ∨ (cid:1) ρ (cid:14) ρ by Lemma 11.21= (cid:0) k [ ρ ] ⊗ Q M ∨ (cid:1) ρ (cid:14) ρ by Example 2.35 and Q -finiteness of M = top ρ ( M ∨ ) . (cid:3) Remark 11.23.
A blanket hypothesis that M be Q -finite would suffice for both partsof Theorem 11.22, because tensoring the surjection k [ Q ] ։ k [ ρ ] with M yields asurjection M ։ k [ ρ ] ⊗ Q M , but the additional generality may be useful.11.4. Generator functors over real polyhedral groups.
Here is the Matlis dual to Definition 4.26, using Definition 11.4.
Definition 11.24.
For a face ρ of real polyhedral group, set ∆ ρ = ( ∇ ρ ) op , the openstar of ρ (Example 3.5) with the partial order opposite to Definition 4.26, so ξ (cid:22) η in ∆ ρ if ξ ⊆ η. The lower boundary functor along ρ takes M to the Q × ∆ ρ -module ∂ ρ M = L ξ ∈ ∆ ρ ∂ ξ M . Proposition 11.25.
Fix a face ρ of a real polyhedral group Q . The Matlis dual N ∨ over Q of any module N over Q × ∆ ρ is naturally a module over Q × ∇ ρ (withoutaltering degrees in ∆ ρ = ∇ ρ ). Moreover,1. (∆ ρ -top N ) ∨ = ∇ ρ -soc ( N ∨ ) for any module N over Q × ∆ ρ , and2. ∆ ρ -top ( N ∨ ) = ( ∇ ρ -soc N ) ∨ for any module N over Q × ∇ ρ , SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 63 where the Matlis duals are taken over Q . All of these hold with ∇ ρ and ∆ ρ swapped.Proof. Matlis duality over Q reverses the arrows in the ∆ ρ -module structure on N ,making N ∨ into a module over Q × ∇ ρ . An adjointness calculation then yields(∆ ρ -top N ) ∨ = ( k ⊗ ∆ ρ N ) ∨ = Hom ∇ ρ ( k , N ∨ )= ∇ ρ -soc ( N ∨ ) . The other adjointness is similar, but it uses finiteness of ∆ ρ via Proposition 11.15:∆ ρ -top ( N ∨ ) = k ⊗ ∆ ρ ( N ∨ )= Hom ∇ ρ ( k , N ) ∨ = ( ∇ ρ -soc N ) ∨ . (cid:3) Corollary 11.26.
For a face ρ over a real polyhedral group Q , the lower boundary isMatlis dual over Q to the upper boundary: as modules over Q × ∆ ρ ,1. ∂ ρ ( M ∨ ) = ( δ ρ M ) ∨ for all Q -modules M (see Definitions 11.24 and 4.26), and2. ( ∂ ρ M ) ∨ = δ ρ ( M ∨ ) if M is infinitesimally Q -finite.Proof. Lemma 11.7 plus the first part of Proposition 11.25. (cid:3)
Next is the Matlis dual of Definition 4.27, using the skyscraper modules k ξ [ b + ρ ]there, followed by the Matlis dual of Definition 4.29. Definition 11.27.
Fix a partially ordered group Q , a face ρ , and an arbitrary com-mutative monoid P . Define a functor k [ ρ ] ⊗ Q × P on modules N over Q × P by k [ ρ ] ⊗ Q × P N = M ( b ,ξ ) ∈ Q × P k ξ [ b + ρ ] ⊗ Q × P N. Definition 11.28.
Fix a real polyhedral group Q , a face ρ , and a Q -module M . The generator functor along ρ takes M to its top along ρ : the ( Q/ R ρ × ∆ ρ )-moduletop ρ M = (cid:0) k [ ρ ] ⊗ Q × ∆ ρ ∂ ρ M (cid:1) ρ (cid:14) ρ. The ∆ ρ -graded components of top ρ M are denoted by top ξρ M for ξ ∈ ∆ ρ . Proposition 11.29.
Fix a real polyhedral group Q and a module N such that k [ ρ ] ⊗ Q N is Q -finite. The functors top ρ and ∆ ρ -top commute on N . In particular, if M is a Q -module such that k [ ρ ] ⊗ Q ∂ ρ M is Q -finite (e.g., if M is Q -finite), then ∆ ρ -top (top ρ ∂ ρ M ) ∼ = top ρ M = top ρ (∆ ρ -top ∂ ρ M ) . Proof.
This is Matlis dual to Proposition 4.38, but to prove it without an infinitesimal Q -finiteness restriction requires a direct argument:∆ ρ -top (top ρ N ) = k ⊗ ∆ ρ (cid:0) k [ ρ ] ⊗ Q N (cid:1) ρ (cid:14) ρ by Definitions 11.10 and 11.17= k ⊗ ∆ ρ (cid:0) Hom (cid:0) k [ Q + ] ρ , k [ ρ ] ⊗ Q N (cid:1)(cid:14) ρ (cid:1) by Definition 11.16= (cid:0) k ⊗ ∆ ρ Hom (cid:0) k [ Q + ] ρ , k [ ρ ] ⊗ Q N (cid:1)(cid:1)(cid:14) ρ by Lemma 2.24= (cid:0) k [ ρ ] ⊗ Q N ⊗ ∆ ρ k (cid:1) ρ (cid:14) ρ by Lemma 2.39= top ρ (∆ ρ -top N ) . The penultimate line is equal to (cid:0) k [ ρ ] ⊗ Q × ∆ ρ N (cid:1) ρ (cid:14) ρ , and N = ∂ ρ M yields top ρ M . (cid:3) Remark 11.30.
The condition in Proposition 11.29 that k [ ρ ] ⊗ Q ∂ ρ M be Q -finite isweaker than M being Q -finite. Roughly, in each degree the former counts generatorssupported on ρ while the latter takes into account all elements, generator or otherwise.The difference is visible when Q = R and ρ = , in which case k [ ρ ] = k . The module M = L α ∈ R k [ α + R + ] with one generator in each real degree α yields a module k ⊗ R M that is R -finite, since it has dimension 1 in every graded degree, but M itself hasuncountable dimension in each graded degree. Theorem 11.31.
Over a real polyhedral group, the generator functor along a face ρ isMatlis dual to the cogenerator functor along ρ : if M is infinitesimally Q -finite, then1. top ρ ( M ∨ ) = (soc ρ M ) ∨ and2. (top ρ M ) ∨ = soc ρ ( M ∨ ) .Proof. (top ρ M ) ∨ = (cid:0) ∆ ρ -top (top ρ ∂ ρ M ) (cid:1) ∨ by Proposition 11.29= ∇ ρ -soc (cid:0) (top ρ ∂ ρ M ) ∨ (cid:1) by Proposition 11.25.1= ∇ ρ -soc (cid:0) soc ρ (cid:0) ( ∂ ρ M ) ∨ (cid:1)(cid:1) by Theorem 11.22.1= ∇ ρ -soc (cid:0) soc ρ δ ρ ( M ∨ ) (cid:1) by Corollary 11.26.2= soc ρ ( M ∨ ) by Proposition 4.38,and (soc ρ M ) ∨ = (cid:0) ∇ ρ -soc (soc ρ δ ρ M ) (cid:1) ∨ by Proposition 4.38= ∆ ρ -top (cid:0) (soc ρ δ ρ M ) ∨ (cid:1) by Proposition 11.25.2= ∆ ρ -top (cid:0) top ρ (cid:0) ( δ ρ M ) ∨ (cid:1)(cid:1) by Theorem 11.22.2= ∆ ρ -top (cid:0) top ρ ∂ ρ ( M ∨ ) (cid:1) by Corollary 11.26.1= top ρ ( M ∨ ) by Proposition 11.29. (cid:3) Essential properties of tops
Secondary decomposition and attached primes are Matlis dual to primary decomposi-tion and associated primes. To start, here is the dual to Definition 9.1 and Theorem 9.2.
SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 65
Definition 12.1.
Fix a face ρ and a module M over a real or discrete polyhedral group.1. The face ρ is attached to M if top ρ M = 0 (Definition 11.28).2. If M = k [ I ] for an interval I then ρ is attached to I .3. The set of attached faces of M or I is denoted by Att M or Att I .4. The module M is ρ -secondary if Att( M ) = { ρ } .Next comes the Matlis dual of Definitions 2.16 and 9.21. Definition 12.2. An interval cover of a module M over an arbitrary poset is a surjec-tion L j ∈ J F j ։ M with each F j an interval module. The cover is finite if J is finite. Itis an upset cover if the intervals are all upsets. The module M is upset-finite if it admitsa finite upset cover. If the poset is a real or discrete polyhedral group, the cover is1. secondary if F j = k [ I j ] is secondary for all j , so I j is a secondary upset, and2. minimal if the induced map top ρ F → top ρ M is an isomorphism for all faces ρ .The Matlis dual to Theorems 6.7 has an infinitesimal Q -finiteness hypotheses becauseduality between tops and socles in Theorem 11.31 requires it (see also Example 12.7),but the dual to Theorem 10.1 has only the upset-finiteness dual to downset-finiteness. Theorem 12.3 (Essentiality of real tops) . Fix a homomorphism ϕ : N → M of mod-ules over a real polyhedral group Q .1. If ϕ is surjective with M and N both being infinitesimally Q -finite modules, then top ρ ϕ : top ρ N → top ρ M is surjective for all faces ρ of Q + .2. If top ρ ϕ : top ρ N → top ρ M is surjective for all faces ρ of Q + and M is upset-finite, then ϕ is surjective. (cid:3) Remark 12.4.
One of the versions of Nakayama’s lemma says that a homomorphism M → N of finitely generated modules over a local ring R is surjective if and only ifit becomes surjective upon tensoring with the residue field k . In the language of topsand socles, M ⊗ R k = top M . Therefore Theorem 12.3 is the direct generalizationof Nakayama’s lemma to multigraded modules over real-exponent polynomial rings.Some finiteness is still required, but it is vastly weaker than finitely generated, ratherrequiring roughly that the generators can be gathered into finitely many coherentclumps. There is, in addition, a quintessentially real-exponent further weakening thatallows the top to be replaced by a dense image (Theorem 12.15). Theorem 12.5 (Essentiality of discrete tops) . Fix a homomorphism ϕ : N → M ofmodules over a discrete polyhedral group Q .1. If ϕ is surjective then top ρ ϕ : top ρ N → top ρ M is surjective for all faces ρ of Q + .2. If top ρ ϕ : top ρ N → top ρ M is surjective for all faces ρ of Q + and M is upset-finite, then ϕ is surjective. (cid:3) Remark 12.6.
In terms of persistent homology, Theorems 12.3 and 12.5 say that ahomomorphism of multipersistence modules is surjective if and only if it maps the “leftendpoints” of the source surjectively onto the “left endpoints” of the target.
Example 12.7.
Some hypothesis is needed in Theorem 12.3.1, in contrast to Theo-rem 6.7.1 or indeed Theorem 12.5.1. Let M = k [ U ] for the open half-plane U ⊂ R above the antidiagonal line y = − x . Then M is { } -secondary, with (top ξ { } M ) b = 0precisely when b lies on the antidiagonal and ξ is the x -axis or y -axis. The direct sum L b = (cid:0) k [ b + Q ∇ x ] ⊕ k [ b + Q ∇ y ] (cid:1) surjects onto M , but the map on tops fails to hit anyelement in R -degree . This kind of behavior might lead one to wonder: why is itsMatlis dual not a counterexample to Theorem 6.7.1? Because M ∨ does not possess awell defined map to a direct sum indexed by a = along the antidiagonal line, only toa direct product. Any sequence of points v k ∈ − U converging to yields a sequence ofelements z k ∈ M ∨ . The image of the sequence { z k } ∞ k =1 in any particular one (or finitedirect sum) of the downset modules of the form k [ a − Q ∇ x ] with a = is eventually 0,but in the direct product the sequence { z k } ∞ k =1 survives forever. The direct limit of theimage sequence witnesses the nonzero socle of the direct product at the missing point . Theorem 12.8.
Every upset-finite module M over a real or discrete polyhedral groupadmits a minimal secondary interval hull. When the polyhedral group is discrete, it ispossible to use upsets for all of the intervals.Proof. This is the Matlis dual of Theorems 9.23 and 10.8, using Example 11.8 to allowthe results of Section 11 to be applied at will, as the strongest hypothesis there isinfinitesimal Q -finiteness. (cid:3) Remark 12.9.
Matlis duality in persistent homology might appear to indicate thatgenerators (births) are in adamantine antisymmetry with cogenerators (deaths), butwhen it comes to interactions between the two, the symmetry is broken by the partialorder on Q : elements in Q -modules move from birth inexorably toward death. Defi-nition 12.10 treats elements functorially, as homomorphisms from the monoid algebraof the positive cone. Doing so makes it clear that the dual of an element is not anelement. It is instead a homomorphism to the injective hull of the residue field, as inDefinition 12.11. This complication in dealing with generators rather than cogeneratorscements the choice to develop the theory in terms of cogenerators in Sections 3–10.Density considerations are important for use in connection with resolutions (Sec-tion 13). They dualize directly, but phrasing them accurately is touchy because ofissues like those in Remarks 11.19 and 11.20. Clearer duality comes from a functorialrecasting of Definition 3.19; the following is precisely equivalent to that definition. Definition 12.10. An element k [ b + Q + ] β → M is said to divide a boundary element k [ a + Q + ] α → δ σ M if b ∈ a − Q ∇ σ = a − σ ◦ − Q + (Lemma 3.8) and α equals the composite k [ a + Q + ] ֒ → k [ b + Q + ] β −→ M b → M a − σ of the inclusion of principal upset Q -modules induced by a + Q + ⊆ b + Q + with β and the natural map from Lemma 3.13. The element β is said to σ -divide α if, morerestrictively, b ∈ a − σ ◦ . SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 67
Definition 12.11.
Fix a module M over a real polyhedral group Q .1. A basin M β → k [ b − Q + ] is said to attract a boundary basin ∂ ξ M α → k [ a − Q + ]if b ∈ a + Q ∇ ξ = a + ξ ◦ + Q + (Lemma 3.8) and α equals the composite M a + ξ → M β −→ k [ b − Q + ] ։ k [ a − Q + ]of the natural map from Lemma 11.2 with β and the surjection of coprincipaldownset Q -modules induced by b − Q + ⊇ a − Q + .2. The basin β is said to ξ -attract α if, more restrictively, b ∈ a + ξ ◦ .3. The basin β is ξ -secondary if its image in k [ b − Q + ] is a ξ -secondary module. Example 12.12.
Because of quotients modulo faces in Definition 11.28, a basin e t : top ξρ M → k [ e a − Q + / R ρ ]takes values modulo ρ . This basin lifts canonically to a homomorphism( k [ ρ ] ⊗ Q × ∆ ρ ∂ ξ M ) ρ → k [ a − Q + + R ρ ]that is not itself a basin but is induced by (perhaps many) basins k [ ρ ] ⊗ Q × ∆ ρ ∂ ξ M → k [ a − Q + ]under applying the Matlis dual ( − ) ρ of localization (Definition 11.16). Note that ∂ ξ M ։ k [ ρ ] ⊗ Q × ∆ ρ ∂ ξ M by right-exactness of colimits. (For the current purpose, surjectivity of this last mapis irrelevant, but it might be handy to keep in mind for intuition.) Composing thesevarious lifts, the basin e t lifts to (perhaps many) boundary basins t : ∂ ξ → k [ a − Q + ]. Definition 12.13.
Fix a module M over a real polyhedral group Q .1. A neighborhood in top ρ M of a basin e t of top ξρ M is top ρ ( βM ) for a ξ -secondarybasin β that ξ -attracts a lifted boundary basin t of ∂ ρ M (Example 12.12).2. A surjection top ρ M ։ T ρ of ( Q/ R ρ × ∆ ρ )-modules is dense if for all ξ ⊇ ρ ,every neighborhood of every basin of top ξρ M has nonzero image in T ρ .3. A quotient functor top ρ ։ T ρ from modules over Q to modules over Q/ R ρ × ∆ ρ is dense if top ρ k [ U ] ։ T ρ k [ U ] is dense for all faces ρ and upsets U ⊆ Q . Remark 12.14.
Definition 12.13 skips the duals to notions of “nearby” and “vicin-ity” from Definition 8.1 because the work of defining “coprimary”—and hence “sec-ondary”, by taking duals—has already been done in Section 9. The contrast betweenDefinitions 12.13 and 8.1 is simple: principal primary submodules become coprincipalsecondary quotients. Definition 12.13.3 is Matlis dual to Definition 7.26.
Theorem 12.15.
Fix quotient functors top ρ ։ T ρ for all faces ρ of a real polyhedralgroup. Theorem 12.3 holds with T instead of top if and only if top ρ ։ T ρ is dense for all ρ . Proof.
Apply the exact Matlis duality functor to the statement of Theorem 7.27 in thepresence of the finiteness hypothesis in Theorem 12.3. (cid:3)
The straightforward dualization of primary decomposition in Sections 9.4 and 10 tosecondary decomposition is omitted.13.
Minimal presentations over discrete or real polyhedral groups
Algebra of modules over arbitrary posets [Mil20a] and primary decomposition overpartially ordered groups [Mil20b] lack a crucial aspect of noetherian commutative al-gebra, namely minimality. Much of the edifice of modern commutative algebra is builton numerical, homological, combinatorial, or geometric behavior whose quantificationrests firmly on notions of minimality: Betti numbers, Castelnuovo–Mumford regu-larity, primary and irreducible decomposition, homological dimension, computationalcomplexity bounds—all of these depend on minimal resolutions, or minimal decom-positions, or minimal degrees of some nature. When the partially ordered group is areal vector space, earlier sections rescue notions of minimality, perhaps with densityamendments, for generators and decompositions. This section explores to what extentminimality applies to presentations and resolutions.
Definition 13.1 ([Mil20a, Definitions 3.16, 6.1, 6.4]) . Fix a module M over a partiallyordered abelian group Q .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.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.3. A fringe presentation of M is a direct sum F of upset modules k [ U ], a directsum 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 5.1).4. An upset resolution of M is a complex F • of Q -modules, each a direct sum ofupset modules, whose differential F i → F i − decreases homological degrees andhas only one nonzero homology H ( F • ) ∼ = M .5. A downset resolution of M is a complex E • of Q -modules, each a direct sum ofdownset modules, whose differential E i → E i +1 increases cohomological degreesand has only one nonzero homology H ( E • ) ∼ = M . Definition 13.2.
Each indicator presentation or indicator resolution in Definition 13.11. is finite if it has only finitely many summands in total;2. dominates a constant subdivision (Definition 2.8) or poset encoding (Defini-tion 2.11) of M if the morphism does (Definition 2.13);3. is semialgebraic or PL if the morhpism has that type (Definition 2.13). SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 69
Definition 13.3.
Over a real polyhedral group, a module morphism ϕ : M → N is1. injectively minimal or injectively dense if the canonical inclusion im ϕ ֒ → N in-duces an isomorphism soc(im ϕ ) −→∼ soc N or dense inclusion soc(im ϕ ) ֒ → soc N ;2. surjectively minimal or surjectively dense if the canonical surjection M ։ im ϕ in-duces an isomorphism top M −→∼ top(im ϕ ) or dense surjection top M ։ top(im ϕ ).Over a discrete polyhedral group the definition of injectively minimal and surjectivelyminimal are unchanged. In either the real or discrete polyhedral setting, a complex ofmodules is injectively or surjectively minimal or dense if all of its differentials are. Remark 13.4.
Category-theoretically, injective minimality or density should naturallybe phrased in terms of the image morphism of ϕ , while surjective minimality anddensity should be phrased in terms of the coimage morphism of ϕ . Remark 13.5.
The notion of minimal morphism makes sense in ordinary commuta-tive algebra much more generally: minimal resolutions and essential submodules aretransparently special cases. Irredundant irreducible decompositions 0 = T W j in amodule M also correspond to also injectively minimal morphisms M ֒ → L M/W j . Incontrast, for historical reasons, a minimal primary decomposition 0 = T P j in a mod-ule M is usually defined to have a minimal number of intersectands, a condition thatneed not induce an injectively minimal morphism M ֒ → L M/P j . Consequently, min-imal primary decompositions by this definition suffer from annoying non-uniqueness.For example, the p -primary component in one minimal primary decomposition canstrictly contain the p -primary component in another. Defining a primary decomposi-tion to be minimal precisely when it induces an injectively minimal morphism wouldrectify this containment problem and other defects. Definition 13.6.
Fix a module M over a real or discrete polyedral group Q .1. A downset copresentation or resolution E • of M is minimal or dense if the exactaugmented complex 0 → M → E • is correspondingly injecitvely minimal or dense.2. An upset presentation or resolution F • of M is minimal or dense if the exactaugmented complex 0 ← M ← F • is correspondingly top-minimal or top-dense.3. A fringe presentation F → E of M is minimal or dense if it is the composite ofa correspondingly minimal or dense upset cover and downset hull of M . Theorem 13.7.
A module over a real polyhedral group Q is tame if and only if it admits1. a dense finite fringe presentation; or2. a dense finite upset presentation; or3. a dense finite downset copresentation.Over a discrete polyedral group these presentations can be chosen minimal instead ofdense. When the module is semialgebraic or PL these presentations can all be chosensemialgebraic or PL, respectively. Proof.
In both the real and discrete cases, any one of these presentations is, in par-ticular, finite, so the existence of any of them implies that the module is tame by thesyzygy theorem [Mil20a, Theorem 6.12]. It is the other direction that requires thetheory in this paper.In the real polyhedral case, any finite downset hull can be densitized by Theorem 9.18and Remark 9.24. The Matlis dual of this statement says that any finite upset covercan be densitized, as well. Composing these from a given finite fringe presentationsyields a dense finite fringe presentation. In addition, the cokernel of any downset hull(dense or otherwise) of a tame module is tame by Proposition 2.15, so the cokernelhas a dense finite downset hull by Theorem 9.18 again. That yields a dense finitedownset copresentation. The Matlis dual of a dense finite downset copresentation ofthe Matlis dual M ∨ is a dense upset presentation of M by Theorem 11.31 (whichapplies unfettered to tame modules by Example 11.8).The minimal discrete polyhedral case follows the parallel proof, using Theorem 10.8and Remark 10.9 instead of Theorem 9.18 and Remark 9.24.If M is semialgebraic, then the densitization procedure in Theorem 9.18 and Re-mark 9.24 is semialgebraic by induction on the number k of summands there, thebase case being the canonical primary decomposition of a semialgebraic interval inTheorem 9.12, which is semialgebraic by Theorem 5.13. (cid:3) Remark 13.8.
If minimal instead of dense presentations are desired in the real poly-hedral setting, then they can be achieved by combining Definitions 9.21 and 13.1 toform interval copresentations instead of downset copresentations, or the Matlis dual in-terval presentations instead of upset presentations. Splicing these yields interval fringepresentations instead of fringe presentations. Theorem 9.23 and the interval version ofRemark 9.24 forms the basis for a minimalizing version of the proof of Theorem 13.7.
Remark 13.9.
Comparing Theorem 13.7 to the syzygy theorem for tame modulesover arbitrary posets [Mil20a, Theorem 6.12], various items are missing.1. Theorem 13.7 makes no claim concerning whether the presentations can be den-sitized if a poset encoding π : Q → P (Definition 2.11) has been specifiedbeforehand. It is a priori possible that deleting redundant generators of upsetsand cogenerators of downsets could prevent an indicator summand from beingconstant on fibers of π .2. Theorem 13.7 makes no claim concerning finite poset encodings dominating anyone of the three presentations there, but as each of these presentations is finite,existence is already implied by the syzygy theorem for tame modules [Mil20a,Theorem 6.12], including semialgebraic and PL considerations.3. Theorem 13.7 makes no claim concerning finiteness of minimal or dense indicatorresolutions. Dense resolutions of tame modules over real polyhedral groups (orminimal ones in the discrete polyhedral setting) can be constructed from scratch SSENTIAL GRADED ALGEBRA OVER POLYNOMIAL RINGS WITH REAL EXPONENTS 71 by Theorem 9.18, Theorem 10.8, and their Matlis duals, but there is no a prioriguarantee that such resolutions must terminate after finitely many steps.
Conjecture 13.10.
Every tame, semialgebraic, or PL module M over a real polyhedralgroup Q has finite dense downset and upset resolutions of the corresponding type. Conjecture 13.11.
Every tame module M over a discrete polyhedral group Q hasfinite minimal downset and upset resolutions of the corresponding type. Remark 13.12.
Remark 13.9.3 raises an intriguing point about indicator resolutions:the bound on the length in the syzygy theorem over arbitrary posets [Mil20a, Theo-rem 6.12] comes from the order dimension of an encoding poset, which is more or lessunrelated to the dimension of the real or discrete polyhedral group. It seems plausiblethat the geometry of the polyhedral group asserts control to prevent the lengths fromgoing too high, just as it does to prevent the cohomological dimension of an affinesemigroup ring from going too high via Ishida complexes to compute local cohomology[MS05, Section 13.3.1]. This points to potential value of developing a derived functorside of the top-socle / birth-death / generator-cogenerator story for indicator resolu-tions to solve Conjecture 13.14, which would be an even tighter indicator analogue ofthe Hilbert Syzygy Theorem.
Definition 13.13.
Fix a module M over a poset Q .1. The downset-dimension of M is the smallest length of a downset resolution of M .2. The upset-dimension of M is the smallest length of an upset resolution of M .3. The indicator-dimension of M is maximum of its downset- and upset-dimensions.4. The indicator-dimension of Q is the maximum of the indicator-dimensions of itstame modules. Conjecture 13.14.
The indicator-dimension of any real or discrete polyhedral group Q equals the rank of Q as a free module (over the field R or group Z , respectively). Remark 13.15.
No uniform bound on the lengths of finite upset and downset res-olutions over an arbitrary posets Q is known when Q has quotients with unboundedorder dimension. It is already open to find a module over R whose indicator-dimensionis provably as high as 2. It would not be shocking if the rank of Q were an upper boundinstead of an equality for the indicator-dimension in Conjecture 13.14: the use of up-set modules instead of free modules could prevent the final syzygies that, in finitelygenerated situations, come from elements supported at the origin by local duality. References [ASW15] Zechariah Andersen and Sean Sather-Wagstaff,
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